Sets, Relations, and Order Structures [version 0.30]

Sets, Relations, and Order Structures

Daniel J. Greenhoe Mathematical Structure and Design series

1


Daniel J. Greenhoe

Copyright © 2016 Daniel J. Greenhoe All rights reserved

This text was typeset using XƎLATEX, which is part of the TEXfamily of typesetting engines, which is arguably the greatest development since the Gutenberg Press. Graphics were rendered using the pstricks and related packages, and LATEX graphics support. The main roman, italic, and bold font typefaces used are all from the Heuristica family of typefaces (based on the Utopia typeface, released by Adobe Systems Incorporated). The math font is XITS from the XITS font project. The font used in quotation boxes is adapted from Zapf Chancery Medium Italic, originally from URW++ Design and Development Incorporated. The font used for the text in the title is Adventor (similar to Avant-Garde) from the TEX-Gyre Project. The font used for the ISBN in the footer of individual pages is Liquid Crystal (Liquid Crystal) from FontLab Studio. The Latin handwriting font is Lavi from the Free Software Foundation. The traditional Chinese font used for the author's Chinese name is 王漢宗中仿宋繁.1 Chinese glyphs appearing elsewhere in the text are from the font 王漢宗中明體繁.2 The ship on the cover is loosely based on the Golden Hind, a sixteenth century English galleon famous for circumnavigating the globe.

1

pinyin: Wáng Hàn Zōng Zhōng Fǎng Sòng Fán; translation: Hàn Zōng Wáng's Medium-weight Sòng-style Traditional Characters; literal: 王漢宗∼font designer's name; 中∼medium; 仿∼to imitate; 宋∼Sòng (a dynasty); 繁∼traditional 2 pinyin: Wáng Hàn Zōng Zhōng Mińg Tǐ Fán; translation: Hàn Zōng Wáng's Medium-weight Mińg-style Traditional Characters; literal: 王漢宗∼font designer's name; 中∼medium; 明∼Mińg (a dynasty); 體∼style; 繁∼traditional

❝Here, on the level sand,

Between the sea and land, What shall I build or write Against the fall of night? ❞

❝Tell me of runes to grave

That hold the bursting wave, Or bastions to design For longer date than mine. ❞

Alfred Edward Housman, English poet (1859–1936)

3

❝The uninitiated imagine that one must await inspiration in order to

create. That is a mistake. I am far from saying that there is no such thing as inspiration; quite the opposite. It is found as a driving force in every kind of human activity, and is in no wise peculiar to artists. But that force is brought into action by an effort, and that effort is work. Just as appetite comes by eating so work brings inspiration, if inspiration is not discernible at the beginning. ❞ Igor Fyodorovich Stravinsky (1882–1971), Russian-born composer

4

❝As

I think about acts of integrity and grace, I realise that there is nothing in my knowledge to compare with Frege's dedication to truth. His entire life's work was on the verge of completion, much of his work had been ignored to the benefit of men infinitely less capable, his second volume was about to be published, and upon finding that his fundamental assumption was in error, he responded with intellectual pleasure clearly submerging any feelings of personal disappointment. It was almost superhuman and a telling indication of that of which men are capable if their dedication is to creative work and knowledge instead of cruder efforts to dominate and be known.❞ Bertrand Russell (1872–1970), British mathematician, in a 1962 November 23 letter to Dr. van Heijenoort.

5

page viii

3 4

5

quote: image: quote: image: quote: image:

version 0.30

📘 Housman (1936), page 64 ⟨“Smooth Between Sea and Land”⟩, 📘 Hardy (1940) ⟨section 7⟩ http://en.wikipedia.org/wiki/Image:Housman.jpg 📘 Ewen (1961), page 408, 📘 Ewen (1950) http://en.wikipedia.org/wiki/Image:Igor_Stravinsky.jpg 📘 Heijenoort (1967), page 127 http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Russell.html Sets, Relations, and Order Structures

Daniel J. Greenhoe

2016 December 27 Tuesday UTC

SYMBOLS

❝rugula

XVI. Quae vero praesentem mentis attentionem non requirunt, etiamsi ad conclusionem necessaria sint, illa melius est per brevissimas notas designare quam per integras figuras: ita enim memoria non poterit falli, nec tamen interim cogitatio distrahetur ad haec retinenda, dum aliis deducendis incumbit.❞

❝Rule XVI. As for things which do not

require the immediate attention of the mind, however necessary they may be for the conclusion, it is better to represent them by very concise symbols rather than by complete figures. It will thus be impossible for our memory to go wrong, and our mind will not be distracted by having to retain these while it is taken up with deducing other matters.❞

René Descartes (1596–1650), French philosopher and mathematician

6

❝In

signs one observes an advantage in discovery which is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed the labor of thought is wonderfully diminished.❞ Gottfried Leibniz (1646–1716), German mathematician,

7

Symbol list symbol numbers: ℤ 𝕎

description integers whole numbers

… , −3, −2, −1, 0, 1, 2, 3, … 0, 1, 2, 3, …

…continued on next page… 6

quote: 📘 Descartes (1684a) ⟨rugula XVI⟩, translation: 📘 Descartes (1684b) ⟨rule XVI⟩, image: Frans Hals (circa 1650), http://en.wikipedia.org/wiki/Descartes, public domain 7 quote: 📘 Cajori (1993) ⟨paragraph 540⟩, image: http://en.wikipedia.org/wiki/File:Gottfried_Wilhelm_von_Leibniz.jpg, public domain

page x

SYMBOL LIST

symbol description ℕ natural numbers 1, 2, 3, … ⊣ ℤ non-positive integers … , −3, −2, −1, 0 − ℤ negative integers … , −3, −2, −1 ℤo odd integers … , −3, −1, 1, 3, … ℤe even integers … , −4, −2, 0, 2, 4, … 𝑚 with 𝑚 ∈ ℤ and 𝑛 ∈ ℤ⧵0 ℚ rational numbers 𝑛 ℝ real numbers completion of ℚ ℝ⊢ non-negative real numbers [0, ∞) ⊣ ℝ non-positive real numbers (−∞, 0] ℝ+ positive real numbers (0, ∞) − ℝ negative real numbers (−∞, 0) ℝ∗ extended real numbers ℝ∗ ≜ ℝ ∪ {−∞, ∞} ℂ complex numbers 𝔽 arbitrary field (often either ℝ or ℂ) ∞ positive infinity −∞ negative infinity 𝜋 pi 3.14159265 … relations: Ⓡ relation ? relational and 𝑋×𝑌 Cartesian product of 𝑋 and 𝑌 ordered pair (△, ▽) absolute value of a complex number 𝑧 |𝑧| = equality relation ≜ equality by definition → maps to ∈ is an element of ∉ is not an element of (Ⓡ) domain of a relation Ⓡ (Ⓡ) image of a relation Ⓡ (Ⓡ) range of a relation Ⓡ (Ⓡ) null space of a relation Ⓡ set relations: ⊆ subset ⊊ proper subset ⊇ super set ⊋ proper superset ⊈ is not a subset of ⊄ is not a proper subset of operations on sets: 𝐴∪𝐵 set union 𝐴∩𝐵 set intersection 𝐴 △𝐵 set symmetric difference 𝐴⧵𝐵 set difference 𝖼 𝐴 set complement set order |⋅| 𝟙𝐴 (𝑥) set indicator function or characteristic function logic: 1 “true” condition …continued on next page… version 0.30


Daniel J. Greenhoe


page xi

SYMBOL LIST

symbol description 0 “false” condition ¬ logical NOT operation ∧ logical AND operation ∨ logical inclusive OR operation ⊕ logical exclusive OR operation ⟹ “implies”; “only if” ⟸ “implied by”; “if” ⟺ “if and only if”; “implies and is implied by” ∀ universal quantifier: “for each” ∃ existential quantifier: “there exists” order on sets: ∨ join or least upper bound ∧ meet or greatest lower bound ≤ reflexive ordering relation “less than or equal to” ≥ reflexive ordering relation “greater than or equal to” < irreflexive ordering relation “less than” > irreflexive ordering relation “greater than” measures on sets: order or counting measure of a set 𝑋 |𝑋 | distance spaces: 𝖽 metric or distance function linear spaces: vector norm ‖⋅‖ operator norm ⦀⋅⦀ ⟨△ | ▽⟩ inner-product 𝗌𝗉𝖺𝗇(𝙑 ) span of a linear space 𝙑 algebras: ℜ real part of an element in a ∗-algebra ℑ imaginary part of an element in a ∗-algebra set structures: 𝑻 a topology of sets 𝑹 a ring of sets 𝑨 an algebra of sets ∅ empty set 𝑋 𝟚 power set on a set 𝑋 sets of set structures: (𝑋) set of topologies on a set 𝑋 (𝑋) set of rings of sets on a set 𝑋 (𝑋) set of algebras of sets on a set 𝑋 classes of relations/functions/operators: 𝟚𝑋𝑌 set of relations from 𝑋 to 𝑌 𝑋 𝑌 set of functions from 𝑋 to 𝑌 (𝑋, ) 𝑌 set of surjective functions from 𝑋 to 𝑌 j j (𝑋, 𝑌 ) set of injective functions from 𝑋 to 𝑌 j (𝑋, 𝑌 ) set of bijective functions from 𝑋 to 𝑌 (𝙓 , 𝙔 ) set of bounded functions/operators from 𝙓 to 𝙔 (𝙓 , 𝙔 ) set of linear bounded functions/operators from 𝙓 to 𝙔 (𝙓 , 𝙔 ) set of continuous functions/operators from 𝙓 to 𝙔 specific transforms/operators: …continued on next page… 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page xii

SYMBOL LIST

symbol 𝐅̃ 𝐅̂ 𝐅̆ 𝐙 ̃ 𝖿 (𝜔) ̆ 𝗑(𝜔) ̂ 𝗑(𝑧)

version 0.30

description Fourier Transform operator Fourier Series operator Discrete Time Fourier Series operator Z-Transform operator Fourier Transform of a function 𝖿(𝑥) ∈ 𝙇𝟤ℝ Discrete Time Fourier Transform of a sequence ⦅𝑥𝑛 ∈ ℂ⦆𝑛∈ℤ Z-Transform of a sequence ⦅𝑥𝑛 ∈ ℂ⦆𝑛∈ℤ


Daniel J. Greenhoe


SYMBOL INDEX

⦹

⦹

⦹

𝛿𝑛̄ , 259 ∧, 9, 186, 187 𝙇𝘕2 , 115 ↔, 179 ⇔, 186, 187 1, 186, 187 →, 174 ⇒, 186, 187 ⇐, 186, 187 ⊖, 186, 187 −, 186, 187 |, 186, 187 ↓, 186, 187 ¬, 9 , 186, 187 , 186, 187 ∨, 9, 186, 187 ◨, 186, 187 ◧, 186, 187 ⊕, 9, 186, 187 0, 186, 187 𝖽, 153 Ⓜ, 79 Ⓜ∗ , 79 ≠, 10 ∉, 4 Ⓜ, / 79 Ⓜ/ ∗ , 79 𝐈, 248 (𝑋, ⊑), 45 (△, ▽), 6 𝓓, 243, 244 𝓘, 243, 244 𝓝 , 243, 244 𝓡, 243, 244 𝑟, 271 𝜌, 271 𝜎, 271 ≤, 46 ≥, 46 ⟂, 191 ?, 6

⦹

⦹

𝒮j (𝑋, 𝑌 ), 252 Ⓒ, 193 ≺, 47 ×, 6 ∧, 187 ⇔, 187 1, 187 ⇒, 187 ÷, 187 ⊖, 187 −, 187 |, 187 ↓, 187 , 187 , 187 ∨, 187 ◨, 187 ◧, 187 ⊕, 187 0, 187 𝛿, 263 Ⓓ, 89 Ⓓ∗ , 89 ∅, 5 𝖾𝗉𝗂(𝖿), 267 ≖, 7 −, 124 𝖿 𝑛 , 259 𝖿 , 8, 249 𝗁𝗒𝗉(𝖿), 267 ⟺,9 ⟸,9 ⟹,8 ∈, 4 [⋅ ∶ ⋅], 265, 266 [⋅ ∶ ⋅), 265, 266 (⋅ ∶ ⋅], 265, 266 (⋅ ∶ ⋅), 265, 266 ∗, 272 ≡, 72 ∨, 58 ⋁ 𝐴, 58

⦹

ℤ, 22 ℕ, 14 𝐷,̌ 285 ∗, 37 ⊕, 277 , 46 𝐶 ∗ , 275 ℂ, 31 ℑ, 273 ℚ, 23 ℝ, 24 ℜ, 273 𝕎, 22 |⋅| ∈ ℝℂ , 35 |⋅| ∈ ℝℝ , 29 ÷, 124 𝑨, 214 ∧, 187 ⇔, 124, 187 1, 187 ⇒, 124, 187 ÷, 187 ⊖, 124, 187 −, 187 |, 187 ↓, 187 , 187 , 187 ∨, 187 ◨, 187 ◧, 187 △, 124, 187 0, 187 ∘, 242 ℬj (𝑋, 𝑌 ), 252 𝑌 𝑋, 8 𝑌 𝑋 , 249 ℐj (𝑋, 𝑌 ), 252 𝟚𝑋𝑌 , 7, 238

⊑, 45 𝟚𝑋 , 5, 201 ↓, 124 Ⓡ, 7, 238 Ⓡ−1 , 241 � 248 ○, 𝑹, 215 ÷, 187, 206 ∩, 187, 206 ⊘, 206 ⇔, 187, 206 ⧵, 10, 203 ∩, 10, 203 𝟙, 258 | 𝑋 |, 202 𝖼, 203 △, 10, 203 ∪, 10, 203 ⊗, 206 𝑋, 187, 203 ⇒, 187, 206 ⊖, 187, 206 ⧵, 187, 206 |, 187, 206 ↓, 187, 206 𝖼𝑥 , 187, 206 𝖼𝑦 , 187, 206 ∪, 187, 206 𝒜(𝑋), 214, 215 𝒯 (𝑋), 211 ◨, 187, 206 ◧, 187, 206 |, 124 ⊂, 10 ⊆, 10 sup 𝐴, 58 △, 187, 206 ∅, 187, 203 𝑻 , 211 𝑖, 31

page xiv

version 0.30


Daniel J. Greenhoe


CONTENTS

Front matter Front cover . . . . . . . . Title page . . . . . . . . . Copyright and typesetting Quotes . . . . . . . . . . Symbol list . . . . . . . . Symbol index . . . . . . . Contents . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

i . i . v . vi . vii . ix . xiii . xv

I Foundational Structure 1 Foundational Construction 1.1 Sets . . . . . . . . . . . . . . . 1.2 Relations . . . . . . . . . . . . 1.3 Equivalence relations . . . . . 1.4 Functions . . . . . . . . . . . . 1.5 Logic . . . . . . . . . . . . . . . 1.6 Set operations . . . . . . . . . 1.7 Abstract Mathematical Spaces

1 . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

3 3 6 7 8 8 10 11

2 Number Systems 2.1 Natural numbers . . . . . . . . . . . 2.1.1 Definitions . . . . . . . . . . . 2.1.2 Addition . . . . . . . . . . . . 2.1.3 Multiplication . . . . . . . . . 2.1.4 Order . . . . . . . . . . . . . 2.2 Whole Numbers . . . . . . . . . . . 2.3 Integers . . . . . . . . . . . . . . . . 2.4 Rational Numbers . . . . . . . . . . 2.5 Real numbers . . . . . . . . . . . . . 2.5.1 Construction . . . . . . . . . 2.5.2 Order structure . . . . . . . . 2.5.3 Arithmetic . . . . . . . . . . . 2.5.4 Least upper bound properties 2.5.5 Completeness . . . . . . . . 2.5.6 Normed algebra structure . . 2.6 Complex numbers . . . . . . . . . . 2.6.1 Definitions . . . . . . . . . . . 2.6.2 Field structure . . . . . . . . 2.6.3 Normed algebra structure . . 2.6.4 Star-algebra structure . . . . 2.6.5 C-star algebraic structure . . 2.6.6 Hermitian structure . . . . . . 2.7 Literature . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

13 14 14 15 17 21 22 22 23 23 24 25 25 25 27 29 31 31 32 35 37 40 40 41

. . . . . . .

. . . . . . .

page xvi

II

CONTENTS

Order Structures

43

3 Order 3.1 Preordered sets . . . . . . . . . . 3.2 Order relations . . . . . . . . . . 3.3 Linearly ordered sets . . . . . . . 3.4 Representation . . . . . . . . . . 3.5 Examples . . . . . . . . . . . . . 3.6 Functions on ordered sets . . . . 3.7 Decomposition . . . . . . . . . . 3.7.1 Subposets . . . . . . . . 3.7.2 Operations on posets . . 3.7.3 Primitive subposets . . . 3.7.4 Decomposition examples 3.8 Bounds on ordered sets . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

45 45 46 47 47 48 51 54 54 55 56 56 57

4 Lattices 4.1 Semi-lattices . . . . . . 4.2 Lattices . . . . . . . . . 4.3 Examples . . . . . . . . 4.4 Characterizations . . . . 4.5 Bounded lattices . . . . 4.6 Functions on lattices . . 4.6.1 Isomorphisms . 4.6.2 Metrics . . . . . 4.6.3 Lattice products 4.7 Literature . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

59 59 61 66 68 70 72 72 75 76 76

5 Modular Lattices 5.1 Modular relation . . . . . 5.2 Semimodular lattices . . . 5.3 Modular lattices . . . . . . 5.3.1 Characterizations . 5.3.2 Special cases . . . 5.4 Examples . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

79 79 80 80 80 84 84

6 Distributive Lattices 6.1 Distributivity relation . . . 6.2 Distributive Lattices . . . . 6.2.1 Definition . . . . . 6.2.2 Characterizations . 6.2.3 Properties . . . . . 6.2.4 Examples . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

89 . 89 . 89 . 89 . 90 . 104 . 107

7 Complemented Lattices 7.1 Definitions . . . . . . 7.2 Examples . . . . . . 7.3 Properties . . . . . . 7.4 Literature . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

109 109 109 110 112

8 Boolean Lattices 8.1 Definition and properties 8.2 Order properties . . . . 8.3 Additional operations . . 8.4 Representation . . . . . 8.5 Characterizations . . . . 8.6 Literature . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

115 115 116 124 125 131 137

. . . .

9 Orthocomplemented Lattices 139 9.1 Orthocomplemented Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 9.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 9.1.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 version 0.30


Daniel J. Greenhoe


page xvii

CONTENTS 9.1.3 Characterization . . . . . . . . . . . . . . . 9.1.4 Restrictions resulting in Boolean algebras . 9.2 Orthomodular lattices . . . . . . . . . . . . . . . . 9.2.1 Properties . . . . . . . . . . . . . . . . . . . 9.2.2 Characterizations . . . . . . . . . . . . . . . 9.2.3 Restrictions resulting in Boolean algebras . 9.3 Modular orthocomplemented lattices . . . . . . . . 9.4 Relationships between orthocomplemented lattices

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

III Functions on Lattices

. . . . . . . .

151

10 Valuations on Lattices 11 Negation 11.1 Definitions . . . . . . . . 11.2 Properties of negations 11.3 Examples . . . . . . . . 11.4 Projections . . . . . . .

143 146 148 148 149 149 150 150

153 . . . .

157 157 159 163 171

12 Logic 12.1 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Classical two-valued logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

173 173 179 184

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

IV Relations

189

13 Relations on lattices with negation 191 13.1 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 13.2 Commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 13.3 Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 14 Set Structures 14.1 General set structures . . . . . . . . 14.2 Operations on the power set . . . . . 14.2.1 Standard operations . . . . . 14.2.2 Non-standard operations . . 14.2.3 Generated operations . . . . 14.2.4 Set multiplication . . . . . . . 14.3 Standard set structures . . . . . . . 14.3.1 Topologies . . . . . . . . . . 14.3.2 Algebras of sets . . . . . . . 14.3.3 Rings of sets . . . . . . . . . 14.3.4 Partitions . . . . . . . . . . . 14.4 Numbers of set structures . . . . . . 14.5 Operations on set structures . . . . . 14.6 Lattices of set structures . . . . . . . 14.6.1 Ordering relations . . . . . . 14.6.2 Lattices of topologies . . . . 14.6.3 Lattices of algebra of sets . . 14.6.4 Lattices of rings of sets . . . 14.6.5 Lattices of partitions of sets . 14.7 Relationships between set structures 14.8 Literature . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

201 201 201 201 204 206 210 211 211 214 215 216 218 220 224 224 225 228 228 230 231 233

15 Relations and Functions 15.1 Relations . . . . . . . . . . . . 15.1.1 Definition and examples 15.1.2 Calculus of Relations . 15.2 Functions . . . . . . . . . . . . 15.2.1 Definition and examples

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

237 237 237 240 249 249


. . . . .

. . . . .

. . . . .

Daniel J. Greenhoe


version 0.30

page xviii

CONTENTS

15.2.2 Properties of functions 15.2.3 Types of functions . . 15.2.4 Image relations . . . . 15.2.5 Indicator functions . . 15.2.6 Calculus of functions . 15.3 Tempered Distributions . . . 15.4 Literature . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

251 252 255 257 259 262 264

A Intervals and Convexity A.1 Intervals . . . . . . . A.2 Convex sets . . . . . A.3 Convex functions . . A.4 Literature . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

265 265 266 267 269

B Normed Algebras B.1 Algebras . . . . . B.2 Star-Algebras . . B.3 Normed Algebras B.4 C* Algebras . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

271 271 272 274 275

C Translation Spaces C.1 Translation . . . . . . . . . . C.1.1 Definitions . . . . . . . C.1.2 Examples . . . . . . . C.1.3 Additive properties . . C.1.4 Subtractive properties C.2 Operations . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

277 277 277 278 280 282 285

Back Matter References . . . Reference Index Subject Index . . End of document

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

287 288 321 325 340

version 0.30

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .


Daniel J. Greenhoe


Part I Foundational Structure

CHAPTER 1 FOUNDATIONAL CONSTRUCTION

Fundamental mathematical structures. This chapter presents six of the most fundamental structures in mathematics: page 4) ➀ sets (Definition 1.1 ➁ relations (Definition 1.8 page 7) ➂ equivalence relations (Definition 1.9 page 7) ➃ functions (Definition 1.11 page 8) ➄ logic operators (Definition 1.14 page 9) ➅ set operations (Definition 1.17 page 10) These structures are “fundamental” in the sense that much of the remaining concepts of mathematics is built on them. We construct these structures as outlined next: We start with the concept of a set: a collection of objects that is uniquely defined by the objects that constitute the collection and that can be specified by any intelligible statement (Definition 1.1 page 4). Using sets, we can construct an ordered pair (Definition 1.5 page 6). The collection of all ordered pairs on a pair of sets is a Cartesian product (Definition 1.7 page 6). Any subset of a Cartesian product is a relation (Definition 1.8 page 7). An equivalence relation is a relation that is reflexive, symmetric, and transitive (Definition 1.9 page 7). A relation is a function if no value from the primary set is mapped to two different values in the secondary set (Definition 1.11 page 8). Using the concept of the function, we define the the logical operators not ¬, logical and ∧, and logical or ∨,…which are also functions (Definition 1.14 page 9). Using logic operators, we can easily define the set operators complement 𝖼, union ∪, intersection ∩, set difference ⧵, and symmetric difference △ (Definition 1.17 page 10).

1.1 Sets We accept without definition the concepts of true and false and that any given statement is either true or false, but never both, and never anything inbetween (next axiom).

page 4

1.1. SETS

Axiom 1.1 (axiom of the excluded middle). 1 There exists a pair of properties 1 and 0, a such that any given statement has exactly one of these properties. x The symbol 1 is called true and 0 is called false.

❝I call a manifold (an aggregate, a set) of elements, which belong to any conceptual

sphere, well-defined, if on the basis of its definition and in consequence of the logical principle of excluded middle, it must be recognized that it is internally determined whether an arbitrary object of this conceptual sphere belongs to the manifold or not, and also, whether two objects in the set , in spite of formal differences in the manner in which they are given, are equal or not.❞ Georg Cantor (1845–1918), German set theory pioneer

2

There is more than one definition of a set. The most basic form of set theory is called naive set theory (Definition 1.1—next). However, in general, Naive set theory is flawed as shown by Russell's Paradox and illustrated by the Barber of Seville anecdote. The most commonly used replacement for the flawed Naive set theory is Zermelo-Fraenkel (ZF) set theory.3 Definition 1.1 (Naive set theory). 4 A set is any collection of objects that satisfies the following axioms: d 1. Any two sets are equal if and only if their elements are the same. e f 2. For every set 𝑋 and statement 𝑠(𝑥), there exists a set 𝐴 consisting exactly of elements 𝑥 from set 𝑋 such that 𝑠(𝑥) is true.

(axiom of extension) (axiom of specification)

The axiom of extension states that the collection completely defines the set, and there is no other characteristic (no extension) which further defines the set. So if two sets have exactly the same elements, then the two sets are equal. The axiom of specification states that, you can form any subset 𝐴 of a set 𝑋 as long as you can specify which elements of 𝑋 you want included in 𝐴 by any possible intelligible statement 𝗌(𝑥): 𝐴 = {𝑥 ∈ 𝑋|𝗌(𝑥) is true} An element 𝑥 that is included in a set 𝑋 is denoted 𝑥 ∈ 𝑋 (next definition). A set that has no elements is called the empty set and is denoted by the symbol ∅ (Definition 1.3 page 5). (Proposition 1.1 page 5) shows that, under the axioms of naive set theory, the emptyset exists and that there is only one set that is the emptyset. Definition 1.2. 5 d An element 𝑥 is a member of a set 𝑋 is denoted by 𝑥 ∈ 𝑋. e f An element 𝑥 is not a member of a set 𝑋 is denoted by 𝑥 ∉ 𝑋. 1

Cantor (1882) pages 114–115 quote: 📃 Cantor (1882) pages 114–115 translation: 📘 Tait (2000), page 271 image: http://en.wikipedia.org/wiki/Image:Georg_Cantor.jpg 3 📃 Zermelo (1908a) pages 263–267 (7 axioms), 📘 Zermelo (1908b) (English translation), 📃 Fraenkel (1922), 📘 Wolf (1998), page 139 4 📘 Halmos (1960) pages 1–6 5 📘 Halmos (1960) page 2, 📘 Hausdorff (1937) page 12, 📃 Kuratowski (1921) page 161 📃

2

version 0.30


Daniel J. Greenhoe


page 5

CHAPTER 1. FOUNDATIONAL CONSTRUCTION

Definition 1.3. 6 d The empty set ∅ is defined as e ∅ ≜ {𝑥 ∈ 𝑋|𝑥 ≠ 𝑥} f Proposition 1.1. 7 Let ∅ be the empty set. p 1. ∅ exists. r p 2. ∅ is unique. ✎PROOF: 1. ∅ exists by the axiom of specification (Definition 1.1 page 4) ∅ ≜ {𝑥 ∈ 𝑋|𝑥 ≠ 𝑥}. 2. ∅ is unique by the axiom of extension (Definition 1.1 page 4).

✏ Definition 1.4 (next) defines the power set 𝟚𝑋 of a set 𝑋. The power set is simply the set of all subsets of 𝑋. The notation 𝟚𝑋 is meaningful because the number of elements (subsets of 𝑋) in a finite set 𝑋 is 2| 𝑋 | (Proposition 1.2 page 5). The tuple ( 𝟚𝑋 , ∩, ∪ ; ⊆) forms a lattice (Definition 4.3 page 61). The lattice ( 𝟚{𝑥,𝑦,𝑧} , ∩, ∪ ; ⊆) is illustrated to the right.

{𝑥, 𝑦, 𝑧} {𝑥, 𝑦}

{𝑥, 𝑧} {𝑦, 𝑧}

{𝑥}

{𝑧} ∅

{𝑦}

Definition 1.4. d The power set 𝟚𝑋 on a set 𝑋 is defined as e 𝟚𝑋 ≜ {𝐴|𝐴 ⊆ 𝑋} (the set of all subsets of 𝑋) f Proposition 1.2. Let | 𝑋 | represent the number of elements in a finite set 𝑋. p r | 𝟚𝑋 | = 2| 𝑋 | p ✎PROOF: 1. Let 𝑛 = | 𝑋 | be the number of elements in a set. 2. Let 𝑋 ≜ {𝑥1 , 𝑥2 , 𝑥3 , … , 𝑥𝑛 }. 3. Each subset of 𝑋 (each element in 𝟚𝑋 ) has 2 possibilities for each of the 𝑛 elements of 𝑋: Either the subset has a given element of 𝑋 (represent by 1), or does not have it (represented by a 0). 4. Therefore, the number of possible subsets (the number of elements in 𝟚𝑋 ) is 𝑋 | 𝟚 | = ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ | {has 𝑥1 , not have 𝑥1 } | ⋅ | {has 𝑥2 , not have 𝑥2 } | ⋯ | {has 𝑥𝑛 , not have 𝑥𝑛 } | 𝑛 times

= 2⏟⏟⏟ ⋅ 2⋯2 =2

𝑛 times 𝑛

= 2| 𝑋 |

✏ 6 7

📘 Halmos (1960) page 8, 📘 Kelley (1955) page 3, 📘 Kuratowski (1961), page 26 📘 Halmos (1960) page 8, 📘 Hausdorff (1937) page 13


Daniel J. Greenhoe


version 0.30

page 6

1.2

1.2. RELATIONS

Relations

Ordered pair. In the set {𝑎, 𝑏}, the order in which the elements are listed does not matter. That is, {𝑎, 𝑏} is equivalent to {𝑏, 𝑎}. However, in some applications the order does (very much) matter. To help with this, we have the concept of the ordered pair. In an ordered pair (𝑎, 𝑏), the order in which the elements are listed is significant. That is, (𝑎, 𝑏) is equivalent to (𝑏, 𝑎) if and only if 𝑎 = 𝑏. The ordered pair can be defined in different ways. One of the most common definitions is due to mathematician Kazimierz Kuratowski in 1921 and is presented next:8 Definition 1.5. 9 d e The ordered pair (𝑎, 𝑏) is defined as f

(𝑎, 𝑏) ≜ {{𝑎}, {𝑎, 𝑏}}

This definition may seem a little strange—it even has the strange consequence that {𝑎, 𝑏} ∈ (𝑎, 𝑏). But the crucial test of validity is if (𝑎, 𝑏) = (𝑐, 𝑑)

⟺

𝑎 = 𝑐 and 𝑏 = 𝑑.

This statement is true under Definition 1.5 but is most easily proved once we have defined the set intersection operator ∩ and set symmetric difference operator △. These are not defined until Definition 1.17 (page 10). And subsequently the above ordered pair equality is proved in Corollary 1.1 (page 10).

Cartesian product. Definition 1.6. Let 𝑋 and 𝑌 be sets, 1 denote the logical property of “true”, and 0 the logical property of “false” (Axiom 1.1 page 3). d ? is the “relational and” defined as e ? ≜ {((0, 0) , 0) , ((0, 1) , 0) , ((1, 0) , 0) , ((1, 1) , 1)} f

The “relational and” of Definition 1.6 can be illustrated using the truth table to the right →. Later, the relational and will be replaced by the logical and; but the logical and is a function, and functions are not defined until Definition 1.11 (page 8).

𝑥 0 0 1 1

𝑦 𝑥?𝑦 0 0 0 1 0 0 1 1

Definition 1.7. 10 Let 𝑋 and 𝑌 be sets, and let ? be the “relational and” relation of Definition 1.6 (page 6). d The Cartesian product 𝑋 × 𝑌 is defined as e 𝑋 × 𝑌 ≜ {(𝑥, 𝑦) |(𝑥 ∈ 𝑋) ? (𝑦 ∈ 𝑌 )} f 8

Alternative ordered pair definition: As an alternative to the Kuratowski definition, the ordered pair can also be taken as an axiom. References: 📘 Bourbaki (1968), page 72, 📘 Munkres (2000), page 13 9 📘 Suppes (1972) page 32, 📘 Halmos (1960) page 23, 📘 Kuratowski (1961), page 39, 📃 Kuratowski (1921) ⟨Def. V, page 171⟩, 📃 Wiener (1914) 10 📘 Halmos (1960) page 24, G. Frege, 2007 August 25, http://groups.google.com/group/sci.logic/msg/ 3b3294f5ac3a76f0 version 0.30


Daniel J. Greenhoe


page 7


Relations. A set of ordered pairs represents a relationship between the set formed by the first elements of the ordered pairs and the set formed by the second elements of the ordered pairs. Any such set is called a relation (next definition). The set of all relations in 𝑋 × 𝑌 is denoted 𝟚𝑋𝑌 . This notation is meaningful because the number of relations in 𝟚𝑋𝑌 is 2| 𝑋 || 𝑌 | (Proposition 15.1 page 238). Definition 1.8. 11 Let 𝑋 and 𝑌 be sets. A relation Ⓡ ∶ 𝑋 → 𝑌 is any subset of 𝑋 × 𝑌 . That is, d Ⓡ⊆𝑋×𝑌 e The set of all relations that are subsets of 𝑋 × 𝑌 is denoted 𝟚𝑋𝑌 . That is, f 𝟚𝑋𝑌 ≜ {Ⓡ|Ⓡ ⊆ (𝑋 × 𝑌 )}

1.3 Equivalence relations

❝To

avoide the tedious repetition of these woordes: is equal to: I will sette as I doe often in woorke use, a paire of parralles, or Gemowe lines of one lengthe, thus: =====, bicause noe 2 thynges, can be moare equalle.❞ Robert Recorde (1510–1558), Welsh physician and mathematician 12

Definition 1.9. 13 A relation ≖∈ 𝟚𝑋𝑋 is an equivalence relation on a set 𝑋 if d 1. 𝑥 ≖ 𝑥 ∀𝑥∈𝑋 (reflexive) e 2. 𝑥 ≖ 𝑦 ⟹ 𝑦 ≖ 𝑥 ∀𝑥,𝑦∈𝑋 (symmetric) f 3. 𝑥 ≖ 𝑦 and 𝑦 ≖ 𝑧 ⟹ 𝑥 ≖ 𝑧 ∀𝑥,𝑦,𝑧∈𝑋 (transitive)

and and

Example 1.1. Examples of equivalence relations include 1. The equality relation = on the set of integers ℤ. e 2. The set equality relation = (𝐴 = 𝐵 ⟹ 𝐴 ⊊ 𝐵 and 𝐵 ⊊ 𝐴) on the set of all sets. x 3. The similarity relation ∼ (all angles equal) on the set of triangles. 4. The modulo relation ∼ of all whole numbers that divide 60 (60|⋅). Definition 1.10. d A set 𝐸 is an equivalence class with respect to the equivalence relation = if e 𝑥=𝑦 ∀𝑥, 𝑦 ∈ 𝐸 f Equivalence relations occur, of course, in the context of relations. In the context of sets, a structure that is “equivalent” to the equivalence relation is the partition (Definition 14.11 page 216). In particular, an equivalence relation on a set generates a partition, and a partition on a set defines an equivalence relation. 11 12

13

📘 Halmos (1960) page 26 quote: 📘 Potts (1860) page 109 📘 Recorde (1557) image: http://nsm1.nsm.iup.edu/gsstoudt/history/images/witte.jpg http://www-gap.dcs.st-and.ac.uk/~history/PictDisplay/Recorde.html http://members.aol.com/jeff94100/witte.jpg 📘 Aliprantis and Burkinshaw (1998), page 7, 📘 Peano (1889b), page 91


Daniel J. Greenhoe


version 0.30

page 8

1.4

1.4. FUNCTIONS

Functions

❝La

logique parfois engendre des monstres. Depuis un demi-sie`cle on a vu surgir une foule de fonctions bizarres qui semblent s'efforcer de ressembler aussi peu que possible aux honnêtes fonctions qui servent à quelque chose.❞

❝Logic sometimes breeds monsters.

For half a century there has been springing up a host of weird functions, which seem to strive to have as little resemlence as possible to honest functions that are of some use.❞

Jules Henri Poincaré (1854–1912), French physicist and mathematician

14

A relation is a kind of mapping between two sets. A relation Ⓡ ⊆ 𝑋 × 𝑌 permits a single point in 𝑋 to map to two or more points in 𝑌 . That is, it is possible that (𝑥, 𝑦1 ) and (𝑥, 𝑦2 ) with 𝑦1 ≠ 𝑦2 are both members of some relation Ⓡ ⊆ 𝑋 × 𝑌 . An example of this is the relation ≤ in the set ℤ × ℤ where 1 ≤ 2 and 1 ≤ 3; so in this example the element 1 ∈ 𝑋 has a relationship with both 2 ∈ 𝑌 and 3 ∈ 𝑌 . Alternatively, we can say that both (1, 2) and (1, 3) are elements of the relation ≤. However, there is a special case of relations where this is not allowed; functions (next definition) are a kind of relation where the only way that both (𝑥, 𝑦1 ) and (𝑥, 𝑦2 ) can be elements of the relation is if 𝑦1 = 𝑦2 . The set of all functions in 𝑋 × 𝑌 is denoted 𝑌 𝑋 . This notation is meaningful because the number of functions in 𝑌 𝑋 is | 𝑌 || 𝑋 | (Proposition 15.5 page 250). Definition 1.11. 15 Let 𝑋 and 𝑌 be sets. Let ? be the “relational and” (Definition 1.6 page 6). A relation 𝖿 ∈ 𝟚𝑋𝑌 is a function if d (for each 𝑥, there is only one 𝖿(𝑥)) (𝑥, 𝑦1 ) ∈ 𝖿 ? (𝑥, 𝑦2 ) ∈ 𝖿 ⟹ 𝑦1 = 𝑦2 e 𝑋𝑌 f The set of all functions in 𝟚 is denoted 𝑌 𝑋 ≜ {𝖿 ∈ 𝟚𝑋𝑌 |𝖿 is a function} .

1.5

Logic ❝My

most nearly immortal contributions are an abbreviation and a typographical symbol. I invented “iff”, for “if and only if”—but I could never believe that I was really its first inventor.…The symbol is definitely not my invention—it appeared in popular magazines (not mathematical ones) before I adopted it,…It is the symbol that sometimes looks like ◻, and is used to indicate an end, usually the end of a proof.❞ Paul R. Halmos (1916–2006), Hungarian-born Jewish-American mathematician

16

Definition 1.12 (logic order relation). Let 1 and 0 represent the logical properties of “true” and “false”, respectively (Axiom 1.1 page 3). d The implies, or only if, relation ⟹ is defined as e ⟹ ≜ {((0, 0) , 1) , ((0, 1) , 1) , ((1, 0) , 0) , ((1, 1) , 1)} f 14

15 16

quote: 📘 Poincaré (1908a) ⟨book 2, chap. 2, sec. 5, par. 3⟩ translation 📘 Poincaré (1908b) page 125 image: http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Poincare.html 📘 Suppes (1972) page 86, 📘 Kelley (1955) page 10, 📘 Bourbaki (1939) quote: 📘 Halmos (1985) page 403 image: http://en.wikipedia.org/wiki/Image:Paul_Halmos.jpeg

version 0.30


Daniel J. Greenhoe


page 9


Definition 1.13. Let 1 and 0 represent the logical properties of “true” and “false”, respectively. d 𝑥 ⟸ 𝑦 if 𝑦 ⟹ 𝑥 ∀𝑥,𝑦∈{0, 1} e ∀𝑥,𝑦∈{0, 1} f 𝑥 ⟺ 𝑦 if 𝑥 ⟹ 𝑦 and 𝑦 ⟹ 𝑥 Alternatively, the relations presented in (Definition 1.12 page 8) and (Definition 1.13 page 9) can be represented in the form of truth tables: “implies” or “only if” ⟹ ∶ {0, 1}2 → {0, 1} 𝑥 𝑦 𝑥 ⟹ 𝑦 0 0 1 0 1 1 1 0 0 1 1 1

“implied by” or “if” ⟸ ∶ {0, 1}2 → {0, 1} 𝑥 𝑦 𝑥 ⟸ 𝑦 0 0 1 0 1 0 1 0 1 1 1 1

“if and only if” ⟺ ∶ {0, 1}2 → {0, 1} 𝑥 𝑦 𝑥 ⟺ 𝑦 0 0 1 0 1 0 1 0 0 1 1 1

Definition 1.14 (Propositional logic). Let 1 and 0 represent the logical properties of “true” and “false”, respectively (Axiom 1.1 page 3). The following binary operators are defined according to the expressions below: ¬ ≜ {(0, 1) , (1, 0)} (logical not) d ∨ ≜ {((0, 0) , 0) , ((0, 1) , 1) , ((1, 0) , 1) , ((1, 1) , 1)} (logical or) e ∧ ≜ {((0, 0) , 0) , ((0, 1) , 0) , ((1, 0) , 0) , ((1, 1) , 1)} (logical and) f ⊕ ≜ {((0, 0) , 0) , ((0, 1) , 1) , ((1, 0) , 1) , ((1, 1) , 0)} (logical exclusive or) Alternatively, the relations presented in Definition 1.14 can be represented in the form of truth tables: logical not 𝑥 ¬𝑥 0 1 1 0

logical or 𝑥 𝑦 𝑥∨𝑦 0 0 0 0 1 1 1 0 1 1 1 1

logical and 𝑥 𝑦 𝑥∧𝑦 0 0 0 0 1 0 1 0 0 1 1 1

logical exclusive-or 𝑥 𝑦 𝑥⊕𝑦 0 0 0 0 1 1 1 0 1 1 1 0

Note that the “logical and” function ∧ as defined in Definition 1.14 (previous) and the “relational and ”relation ? as defined in Definition 1.7 (page 6) are equivalent relations. More information about logic structure is presented in CHAPTER 12 (page 173). Key concepts concerning the structure of logic is more conveniently demonstrated once the concepts of order ( Chapter 3 page 45), lattices ( Chapter 4 page 59), and Boolean algebras ( Chapter 8 page 115) have been introduced. Definition 1.15 (predicate logic). ∃𝑖 ∈ 𝐼 such that 𝑝(𝑥𝑖 ) is true d e ∀𝑖 ∈ 𝐼, 𝑝(𝑥𝑖 ) is true f

⟺

[𝑝(𝑥𝑖 ) is true] ⋁

(existensial quantifier)

𝑖∈𝐼

⟺

⋀

[𝑝(𝑥𝑖 ) is true]

(universal quantifier)

𝑖∈𝐼

Theorem 1.1. (Wolf, 1998)62 Let 𝑥 be a condition and 𝑃 (𝑥) be a logical statement dependent on 𝑥. t ¬ [∀𝑥, 𝑃 (𝑥)] is equivalent to ∃𝑥 such that ¬𝑃 (𝑥) h is equivalent to ∀𝑥, ¬𝑃 (𝑥) m ¬ [∃𝑥, such that 𝑃 (𝑥)] 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 10

1.6. SET OPERATIONS

Theorem 1.2. 17 t h Any statement constructed using first order predicate logic is complete. m

1.6

Set operations

Equipped with the concept of the relation, we are now in a position to construct the following fundamental relationships between sets: The set inclusion relation 𝐴 ⊆ 𝐵 means 𝑥 ∈ 𝐴 ⟹ 𝑥 ∈ 𝐵. The equality relation 𝐴 = 𝐵 means 𝐴 ⊆ 𝐵 and 𝐵 ⊆ 𝐴. The non-equality relation 𝐴 ≠ 𝐵 means 𝐴 = 𝐵 is false (0). The proper subset relation 𝐴 ⊊ 𝐵 means 𝐴 ⊆ 𝐵 and 𝐴 ≠ 𝐵. These informal set relation descriptions are reinforced with their formal definitions next. Definition 1.16. 18 Let 𝐴 and 𝐵 be sets. The relation “subset” ⊆ d The relation “set equality” = e f The relation “set non-equality” ≠ The relation “proper subset” ⊊

is defined as the set is defined as the set is defined as the set is defined as the set

{(𝐴, 𝐵) |𝑥 ∈ 𝐴 ⟹ 𝑥 ∈ 𝐵} {(𝐴, 𝐵) |(𝐴 ⊆ 𝐵) ∧ (𝐵 ⊆ 𝐴)} {(𝐴, 𝐵) |𝐴 = 𝐵 is false (0)} {(𝐴, 𝐵) |(𝐴 ⊆ 𝐵) ∧ (𝐴 ≠ 𝐵)}

Definition 1.17. 19 20 Let 𝟚𝑋 be the power set (Definition 1.4 page 5) on a set 𝑋. 𝐴𝖼 ≜ {𝑥 ∈ 𝑋|¬(𝑥 ∈ 𝐴)} ∀𝐴∈𝟚𝑋 (complement) {𝑥 𝐴 ∪ 𝐵 ≜ ∈ 𝑋|(𝑥 ∈ 𝐴) ∨ (𝑥 ∈ 𝐵)} ∀𝐴,𝐵∈𝟚𝑋 (union) d 𝑋 e 𝐴 ∩ 𝐵 ≜ {𝑥 ∈ 𝑋|(𝑥 ∈ 𝐴) ∧ (𝑥 ∈ 𝐵)} ∀𝐴,𝐵∈𝟚 (intersection) f 𝐴⧵𝐵 ≜ {𝑥 ∈ 𝑋|(𝑥 ∈ 𝐴) ∧ ¬(𝑥 ∈ 𝐵)} ∀𝐴,𝐵∈𝟚𝑋 (difference) 𝑋 𝐴△𝐵 ≜ {𝑥 ∈ 𝑋|(𝑥 ∈ 𝐴) ⊕ (𝑥 ∈ 𝐵)} ∀𝐴,𝐵∈𝟚 (symmetric difference) Definition 1.5 (page 6) defined the ordered pair using Kuratowski's somewhat cryptic expression (𝑎, 𝑏) ≜ {{𝑎}, {𝑎, 𝑏}}. Theorem 1.3 (next) helps show that this expression is reasonable in that 𝑎 and 𝑏 can be extracted from the set { {𝑎}, {𝑎, 𝑏} } by simple set operations on its elements. Also, a key test to the usefulness of the definition of ordered pairs is whether or not (𝑎, 𝑏) = (𝑐, 𝑑) if and only if 𝑎 = 𝑐 and 𝑏 = 𝑑. In fact, this statement is true and demonstrated by Corollary 1.1 (page 10). Theorem 1.3. t {𝑎} = ⋂ (𝑎, 𝑏) = ⋂ {{𝑎}, {𝑎, 𝑏}} = {𝑎} ∩ {𝑎, 𝑏} h m {𝑏} = △ (𝑎, 𝑏) = △{{𝑎}, {𝑎, 𝑏}} = {𝑎}△{𝑎, 𝑏} Corollary 1.1. 21 c o (𝑎, 𝑏) = (𝑐, 𝑑) r

⟺

𝑎 = 𝑐 and 𝑏 = 𝑑

✎PROOF: 17

Gödel (1930a), 📃 Gödel (1930b), 📃 Henkin (1949), 📘 Takeuti (1975) page 43 ⟨Theorem 8.2⟩ 📘 Kelley (1955) page 2 19 📘 Vereščagin and Shen (2002) pages 1–2, 📘 Aliprantis and Burkinshaw (1998), pages 2–4 20 Origin of ∪ and ∩: 📘 Peano (1888a), 📘 Peano (1888b) 21 📘 Apostol (1975) page 33, 📘 Hausdorff (1937) page 15 📃

18

version 0.30


Daniel J. Greenhoe


page 11


1. Proof that (𝑎, 𝑏) = (𝑐, 𝑑) ⟹ 𝑎 = 𝑐 and 𝑏 = 𝑑: {𝑎} =

(𝑎, 𝑏) ⋂ = (𝑐, 𝑑) ⋂ = {𝑐}

by Theorem 1.3 page 10

{𝑏} = △ (𝑎, 𝑏)


by left hypothesis by Theorem 1.3 page 10

= △(𝑐, 𝑑)

by left hypothesis

= {𝑑}


2. Proof that (𝑎, 𝑏) = (𝑐, 𝑑) ⟸ 𝑎 = 𝑐 and 𝑏 = 𝑑: (𝑎, 𝑏) = (𝑐, 𝑑)

by right hypothesis

✏

1.7 Abstract Mathematical Spaces abstract spaces linear spaces

topological spaces metric spaces

metric linear spaces complete metric spaces normed linear spaces inner-product spaces

Banach spaces Hilbert spaces 𝟬

Figure 1.1: Lattice of mathematical spaces The abstract space was introduced by Maurice Fréchet in his 1906 Ph.D. thesis.22 An abstract space in mathematics does not really have a rigorous definition; but in general it is a set together with some other unifying structure.

❝…A

collection of these abstract elements will be called an abstract set. If to this set there is added some rule of association of these elements, or some relation between them, the set will be called an abstract space.❞ Marice René Fréchet (1878–1973), French mathematician who in his 1906 Ph.D. dissertation introduced the concept of the metric space 23 22

📃

Fréchet (1906), 📘 Fréchet (1928)


Daniel J. Greenhoe


version 0.30

page 12

1.7. ABSTRACT MATHEMATICAL SPACES

Examples of common abstract spaces include the following: 1. linear space: a set of “vectors” over a field with a vector-vector addition operator + such vectors can be added together and a scalar-vector multiplication operator × such that field elements can be multiplied with vectors. 📘 Peano (1888a) 2. metric space: A set of elements together with a metric 𝖽 such giving the “distance” between any two elements. 📃 Fréchet (1906) 3. measure space: A set of sets and a measure 𝜇 that measures the “size” of a set. 4. normed vector space: A vector space together with a norm ‖⋅‖ that measures the “length” of a vector. 📘 Banach (1922) 5. inner-product space: A vector space together with an inner-product ⟨△ | ▽⟩. 6. Banach space: a normed vector space that is “complete” with respect to the norm ‖⋅‖. 📘 Banach (1922) 7. Hilbert space: an inner-product space that is “complete” with respect to the norm induced by the inner-product ⟨△ | ▽⟩. 📘 von Neumann (1929)

23

quote: image:

version 0.30

Fréchet (1950) page 147 http://en.wikipedia.org/wiki/Frechet 📃


Daniel J. Greenhoe


CHAPTER 2 NUMBER SYSTEMS

❝Wherever there is number, there is beauty.❞ Proclus Lycaeus (412 – 485 AD), Greek philosopher 1

The most common number systems are the following: The set of natural numbers ℕ ≜ {1, 2, 3, …} The set of whole numbers 𝕎 ≜ {0, 1, 2, 3, …} The set of integers ℤ ≜ {… , −3, −2, −1, 0, 1, 2, 3, …} The set of rational numbers ℚ ≜ { 𝑚𝑛 |𝑛 ∈ ℤ, 𝑚 ∈ ℕ} The set of real numbers ℝ ≜ completion of ℚ The set of complex numbers ℂ ≜ {(𝑥, 𝑦) |𝑥, 𝑦 ∈ ℝ}

Definition 2.1

page 14

Definition 2.5

page 22

Definition 2.7

page 22

Definition 2.10

page 23

Definition 2.13

page 24

Definition 2.17

page 31

Generally there are two ways to construct these numbers systems:2 1. ordinal approach: Start by defining the natural numbers axiomatically using the Peano axioms. From there construct the other number systems. The ordinal approach presents the natural numbers as having an inherent order such that each natural number has a unique successor that is also a natural number. 2. cardinal approach: Start by defining the real numbers axiomatically using their field properties as axioms. From there define the natural numbers, whole numbers, integers, and rational numbers as subsets of the reals, and define the complex numbers as ordered pairs of the reals. The cardinal approach presents the real numbers as having inherent size such that any number except zero can be sliced into smaller and smaller pieces. Either approach is perfectly acceptable. However, under the assumption that, say, four thousand years ago most every culture on earth had some kind of counting system, but relatively few, if any, were familiar with the real number system, one might argue that the ordinal approach is a more “natural” choice for number system construction. And indeed it is the ordinal approach that is presented in the following material. 1 2

quote: 📘 Kline (1990) page 131 📘 Bunt et al. (1988) pages 3–4

page 14

2.1

2.1. NATURAL NUMBERS

Natural numbers ❝I

regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding; the simplest act is the passing from an already-formed individual to the consecutive new one to be formed.❞ Richard Dedekind (1831–1915), German mathematician 3

2.1.1

Definitions ❝The

Congress was a turning point in my intellectual life, because I there met Peano.…In discussions at the Congress I observed that he was always more precise than anyone else, and that he invariably got the better of any argument upon which he embarked, As the days went by, I decided that this must be owing to his mathematical logic.…It became clear to me that his notation afforded an instrument of logical analysis such as I had been seeking for years, and that by studying him I was acquiring a new and powerful technique for the work that I had long wanted to do.❞ 4 Bertrand Russell (1872–1970), British mathematician,

Axiom 2.1 (Peano's axioms). 5 The set ℕ of natural numbers satisfies the following axioms: 1. 1 ∈ ℕ (ℕ is not empty) 2. 𝗌(𝑛) ∈ ℕ ∀𝑛∈ℕ (every element 𝑛 in ℕ has a successor 𝗌(𝑛) in ℕ) 3. 𝗌(𝑛) = 𝗌(𝑚) ⟺ 𝑛 = 𝑚 ∀𝑛,𝑚∈ℕ (𝗌 is a bijection) 4. 𝗌(𝑛) ≠ 1 ∀𝑛∈ℕ (1 is not a successor of any element in ℕ) a x 5. ∀𝑀 ⊆ ℕ (axiom of induction) (5a) 1 ∈ 𝑀 and ⟹ 𝑀 =ℕ } (5b) 𝑚 ∈ 𝑀 ⟹ 𝗌(𝑚) ∈ 𝑀 The set ℕ is called the set of natural numbers. The value 𝗌(𝑛) is called the successor of 𝑛. The element 1 is called “one”. 3

quote: 📘 Dedekind (1872a) translation: 📘 Dedekind (1872b) page 2, 📘 Dedekind (1872c) page 2?, 📘 Dedekind (1872d) page 768 image: http://turnbull.mcs.st-and.ac.uk/history/PictDisplay/Dedekind.html 4 quote: 📘 Russell (1951), pages 216–217 image: http://en.wikipedia.org/wiki/File:Russell1907-2.jpg, public domain 5 📘 Landau (1966) page 2, 📘 Halmos (1960) page 46, 📘 Thurston (1956) page 51, 📘 Peano (1889a), 📘 Peano (1889b) page 94, 📘 Dedekind (1888a), 📘 Dedekind (1888b) page 67, 📘 Cori and Lascar (2001) pages 8–15 ⟨recursion theory⟩ version 0.30


Daniel J. Greenhoe


page 15

CHAPTER 2. NUMBER SYSTEMS

2.1.2 Addition

❝And

even now I do not see how arithmetic can be scientifically established; how numbers can be apprehended as logical objects, and brought under review; unless we are permitted – at least conditionally – to pass from a concept to its extension.❞ Gottlob Frege (1848–1925), German mathematician, logician, and philosopher 6

Definition 2.1 (Addition over ℕ). 7 Let 𝗌(𝑛) be the successor of 𝑛 (Definition 2.1 page 14). Let + ∶ ℕ × ℕ → ℕ be an operator that satisfies the following conditions: d 1. 𝗌(𝑛) = 𝑛+1 ∀𝑛∈ℕ and e 2. 𝗌(𝑛 + 𝑚) = 𝑛 + 𝗌(𝑚) ∀𝑛,𝑚∈ℕ. f The quantity 𝑛 + 𝑚 is called the sum of 𝑛 and 𝑚. Lemma 2.1. 8 Let 𝑛 + 𝑚 represent the sum of 𝑛 and 𝑚. ∀𝑛,𝑚,𝑘∈ℕ l 𝗌(𝑚) + 𝑛 = 𝑚 + 𝗌(𝑛) e 𝗌(𝑚 + 𝑛) = 𝗌(𝑚) + 𝑛 ∀𝑛,𝑚∈ℕ m 𝗌(𝑛) = 1+𝑛 ∀𝑛∈ℕ ✎PROOF: 1. Proof that 𝗌(𝑚) + 𝑛 = 𝑚 + 𝗌(𝑛): (a) Let 𝑀 be the set of values for which 𝗌(𝑚) + 𝑛 = 𝑚 + 𝗌(𝑛). (b) Proof that 1 ∈ 𝑀: 𝗌(𝑚) + 1 = 𝗌[𝗌(𝑚)]

by definition of addition over ℕ page 15

= 𝗌[𝑚 + 1]


= 𝑚 + 𝗌(1)


(c) Proof that 𝑚 ∈ 𝑀 ⟹ 𝗌(𝑚) ∈ 𝑀: 𝗌(𝑚) + 𝗌(𝑛) = 𝗌(𝑚) + [𝑛 + 1]


= [𝗌(𝑚) + 𝑛] + 1

by associative property

= [𝑚 + 𝗌(𝑛)] + 1

by left hypothesis

= 𝑚 + [𝗌(𝑛) + 1]


= 𝑚 + 𝗌(𝗌(𝑛))


(d) Therefore, by the axiom of induction (page 14), 𝑀 = ℕ (all of ℕ has the property of 𝑀). 2. Proof that 𝗌(𝑚) + 𝑛 = 𝗌(𝑚 + 𝑛): 𝗌(𝑚) + 𝑛 = 𝑚 + 𝗌(𝑛)

6

7 8

by previous lemma (2)

= 𝑚 + [𝑛 + 1]


= [𝑚 + 𝑛] + 1


= 𝗌(𝑚 + 𝑛)


quote: 📘 Frege (1903) translation: 📘 Frege (1952) page 234, 📘 Frege (1997) page 280 image: http://en.wikipedia.org/wiki/Image:Frege.jpg 📘 Dedekind (1888a), 📘 Dedekind (1888b) page 97, 📘 Landau (1966) page 4 📘 Dedekind (1888b) pages 97–99


Daniel J. Greenhoe


version 0.30

page 16


3. Proof that 1 + 𝑛 = 𝗌(𝑛): (a) Let 𝑀 be the set of values for which 1 + 𝑛 = 𝗌(𝑛). (b) Proof that 1 ∈ 𝑀: 1 + 1 = 𝗌(1)


(c) Proof that 𝑚 ∈ 𝑀 ⟹ 𝗌(𝑚) ∈ 𝑀: 1 + 𝗌(𝑛) = 1 + [𝑛 + 1]


= [1 + 𝑛] + 1


= 𝗌(𝑛) + 1

by left hypothesis

= 𝗌(𝗌(𝑛))


(d) Therefore, by the axiom of induction (page 14), 𝑀 = ℕ (all of ℕ has the property of 𝑀).

✏ Theorem 2.1 (Associative property of addition). 9 Let 𝑛 + 𝑚 represent the sum of 𝑛 and 𝑚. t h (𝑛 + 𝑚) + 𝑘 = 𝑛 + (𝑚 + 𝑘) ∀𝑛,𝑚,𝑘∈ℕ (associative) m ✎PROOF: 1. Let 𝑀 be the set of values for which (ℕ, +) is associative. 2. Proof that 1 ∈ 𝑀: (𝑛 + 𝑚) + 1 = 𝗌(𝑛 + 𝑚)

by Definition 2.1 page 15

= 𝑛 + 𝗌(𝑚)


= 𝑛 + (𝑚 + 1)


3. Proof that 𝑘 ∈ 𝑀 ⟹ 𝗌(𝑘) ∈ 𝑀: [𝑛 + 𝑚] + 𝗌(𝑘) = 𝗌([𝑛 + 𝑚] + 𝑘)


= 𝗌(𝑛 + [𝑚 + 𝑘])

by left hypothesis

= 𝑛 + [𝗌(𝑚 + 𝑘)]


= 𝑛 + [𝑚 + 𝗌(𝑘)]


4. Therefore, by the axiom of induction (page 14), 𝑀 = ℕ (all of ℕ is associative under +).

✏ Theorem 2.2 (Commutative property of addition). t h 𝑛+𝑚 = 𝑚+𝑛 ∀𝑛,𝑚∈ℕ (commutative) m

10

Let 𝑛 + 𝑚 represent the sum of 𝑛 and 𝑚.

✎PROOF: 9 10

📘 Dedekind (1888b) pages 97–99, 📘 Landau (1966) pages 4–5, 📘 Thurston (1956) page 53 📘 Dedekind (1888b) pages 97–99, 📘 Landau (1966) pages 4–5, 📘 Thurston (1956) page 53

version 0.30


Daniel J. Greenhoe


page 17


1. Let 𝑀 be the set of values for which (ℕ, +) is commutative. 2. Proof that 1 ∈ 𝑀: 1 + 𝑛 = 𝗌(𝑛) =𝑛+1

by Lemma 2.1 page 15 by definition of addition over ℕ page 15

3. Proof that 𝑚 ∈ 𝑀 ⟹ 𝗌(𝑚) ∈ 𝑀: 𝑛 + 𝗌(𝑚) = 𝑛 + [𝑚 + 1]


= [𝑛 + 𝑚] + 1


= [𝑚 + 𝑛] + 1

by left hypothesis

= 𝑚 + [𝑛 + 1]


= 𝑚 + [1 + 𝑛]

by Lemma 2.1 page 15

= [𝑚 + 1] + 𝑛


= 𝗌(𝑚) + 𝑛


4. Therefore, by the axiom of induction (page 14), 𝑀 = ℕ (all of ℕ is commutative under +).

✏

2.1.3 Multiplication Definition 2.2 (Multiplication over ℕ). 11 Let 𝗌(𝑛) be the successor of 𝑛. Let ⋅ ∶ ℕ × ℕ → ℕ be an operator that satisfies 1. 𝑛 ⋅ 1 = 𝑛 ∀𝑛∈ℕ and d e 2. 𝑛 ⋅ 𝗌(𝑚) = 𝑛 ⋅ 𝑚 + 𝑛 ∀𝑛,𝑚∈ℕ f The quantity 𝑛 ⋅ 𝑚 is called the product of 𝑛 and 𝑚. The operation + is called addition. Lemma 2.2. 12 l 𝗌(𝑛) 𝑚 = 𝑛𝑚 + 𝑚 e 1⋅𝑛 = 𝑛 m

∀𝑛,𝑚∈ℕ ∀𝑛∈ℕ

✎PROOF: 1. Proof that 𝗌(𝑛) 𝑚 = 𝑛𝑚 + 𝑚: (a) Let 𝑀 be the set of values for which 𝗌(𝑛) 𝑚 = 𝑛𝑚 + 𝑚. (b) Proof that 1 ∈ 𝑀: 𝗌(𝑛) ⋅ 1 = 𝗌(𝑛)

11 12

by definition of multiplication over ℕ page 17

=𝑛+1


=𝑛⋅1+1


📘 Dedekind (1888b) page 101, 📘 Landau (1966) page 14 📘 Dedekind (1888b) pages 101–102


Daniel J. Greenhoe


version 0.30

page 18


(c) Proof that 𝑚 ∈ 𝑀 ⟹ 𝗌(𝑛) 𝗌(𝑚) = 𝑛𝗌(𝑚) + 𝗌(𝑚): 𝗌(𝑛) 𝗌(𝑚) = 𝗌(𝑛) 𝑚 + 𝗌(𝑛)


= [𝑛𝑚 + 𝑚] + 𝗌(𝑛)

by left hypothesis

= 𝑛𝑚 + [𝑚 + 𝗌(𝑛)]

by associative property of + over ℕ page 16

= 𝑛𝑚 + [𝗌(𝑛) + 𝑚]

by commutative property of + over ℕ page 16

= 𝑛𝑚 + [𝑛 + 𝗌(𝑚)]


= [𝑛𝑚 + 𝑛] + 𝗌(𝑚)

by commutative property of + over ℕ page 16

= 𝑛𝗌(𝑚) + 𝗌(𝑚)


(d) Therefore, by the axiom of induction (page 14), 𝗌(𝑛) 𝑚 = 𝑛𝑚 + 𝑚. 2. Proof that 1 ⋅ 𝑛 = 𝑛: (a) Let 𝑀 be the set of values for which 1 ⋅ 𝑛 = 𝑛. (b) Proof that 1 ∈ 𝑀: 1⋅1=1


(c) Proof that 𝑛 ∈ 𝑀 ⟹ 1 ⋅ 𝗌(𝑛) = 𝗌(𝑛): by definition of addition over ℕ page 15

1 ⋅ 𝗌(𝑛) = 1 ⋅ 𝑛 + 1 =𝑛+1

by left hypothesis

= 𝗌(𝑛)


(d) Therefore, by the axiom of induction (page 14), 1 ⋅ 𝑛 = 𝑛.

✏ Theorem 2.3 (Commutative property of multiplication). 𝑚. t h 𝑛𝑚 = 𝑚𝑛 ∀𝑛,𝑚∈ℕ (commutative) m

13

Let 𝑛 ⋅ 𝑚 represent the product of 𝑛 and

✎PROOF: 1. Let 𝑀 be the set of values for which 𝑛𝑚 = 𝑚𝑛. 2. Proof that 1 ∈ 𝑀: 𝑛⋅1=𝑛 =1⋅𝑛

by definition of multiplication over ℕ page 17 by Lemma 2.2 page 17

3. Proof that 𝑚 ∈ 𝑀 ⟹ 𝗌(𝑚) ∈ 𝑀: 𝑛 ⋅ 𝗌(𝑚) = 𝑛𝑚 + 𝑛


= 𝑚𝑛 + 𝑛

by left hypothesis

= 𝗌(𝑚) 𝑛


4. Therefore, by the axiom of induction (page 14), 𝑀 = ℕ (𝑛𝑚 = 𝑚𝑛 13

∀𝑚 ∈ ℕ).

📘 Dedekind (1888b) page 102, 📘 Landau (1966) page 15

version 0.30


Daniel J. Greenhoe


page 19


✏ Theorem 2.4 (Distributive propterties). the product of 𝑛 and 𝑚. t 𝑛(𝑚 + 𝑘) = 𝑛𝑚 + 𝑛𝑘 ∀𝑛,𝑚,𝑘∈ℕ h (𝑛 + 𝑚)𝑘 = 𝑛𝑘 + 𝑚𝑘 ∀𝑛,𝑚,𝑘∈ℕ m

14

Let 𝑛 + 𝑚 represent the sum of 𝑛 and 𝑚 and 𝑛 ⋅ 𝑚 represent (left distributive) (right distributive)

✎PROOF: 1. Proof that ℕ is left distributive: (a) Let 𝑀 be the set of values for which (ℕ, +, ⋅) is distributive. (b) Proof that 1 ∈ 𝑀: 𝑛(𝑚 + 1) = 𝑛𝗌(𝑚)

by definition of + (Definition 2.1 page 15)

= 𝑛𝑚 + 𝑛

by definition of ⋅ (Definition 2.2 page 17)

= 𝑛𝑚 + 𝑛 ⋅ 1


(c) Proof that 𝑘 ∈ 𝑀 ⟹ 𝗌(𝑘) ∈ 𝑀: 𝑛(𝑚 + 𝗌(𝑘)) = 𝑛𝗌(𝑚 + 𝑘)

by definition of + (Definition 2.1 page 15)

= 𝑛(𝑚 + 𝑘) + 𝑛


= (𝑛𝑚 + 𝑛𝑘) + 𝑛

by left hypothesis

= 𝑛𝑚 + (𝑛𝑘 + 𝑛)

by associative property of (ℕ, +) (Theorem 2.1 page 16)

= 𝑛𝑚 + 𝑛𝗌(𝑘)


(d) Therefore, by the axiom of induction (page 14), 𝑀 = ℕ (all of (ℕ, +, ⋅) is distributive). 2. Proof that ℕ is right distributive: by commutative property of multiplication over ℕ (page 18)

(𝑛 + 𝑚)𝑘 = 𝑘(𝑛 + 𝑚) = 𝑘𝑛 + 𝑘𝑚

by left distributive property of ℕ

= 𝑛𝑘 + 𝑚𝑘

by commutative property of multiplication over ℕ (page 18)

✏ Theorem 2.5 (Associative propterty of multiplication). t h (𝑛𝑚)𝑘 = 𝑛(𝑚𝑘) ∀𝑛,𝑚,𝑘∈ℕ (associative) m

15

Let 𝑛 ⋅ 𝑚 represent the product of 𝑛 and 𝑚.

✎PROOF: 1. Let 𝑀 be the set of values for which 𝑛(𝑚𝑘) = (𝑛𝑚)𝑘). 2. Proof that 1 ∈ 𝑀: 𝑛(𝑚 ⋅ 1) = 𝑛𝑚


= (𝑛𝑚) ⋅ 1 14 15


📘 Dedekind (1888b) pages 102–103, 📘 Landau (1966) page 16 📘 Dedekind (1888b) page 103, 📘 Landau (1966) page 16


Daniel J. Greenhoe


version 0.30

page 20


3. Proof that 𝑘 ∈ 𝑀 ⟹ 𝗌(𝑘) ∈ 𝑀: 𝑛[𝑚𝗌(𝑘)] = 𝑛[𝑚(𝑘 + 1)]


= 𝑛[𝑚𝑘 + 𝑚)]

by left distributive property page 19

= 𝑛(𝑚𝑘) + 𝑛𝑚


= (𝑛𝑚)𝑘 + 𝑛𝑚

by left hypothesis

= (𝑛𝑚)𝑘 + (𝑛𝑚) ⋅ 1


= (𝑛𝑚)(𝑘 + 1)


= (𝑛𝑚)𝗌(𝑘)


4. Therefore, by the axiom of induction (page 14), 𝑀 = ℕ (𝑛(𝑚𝑘) = (𝑛𝑚)𝑘

∀𝑘 ∈ ℕ).

✏ Definition 2.3 (exponents on ℕ). 16 Let 𝗌(𝑛) be the successor of 𝑛. Let ∶ ℕ → ℕ be an operator that satisfies 1 d 1. 𝑎 = 𝑎 ∀𝑎,𝑛∈ℕ and e 𝗌(𝑛) 𝑛 2. 𝑎 = 𝑎 𝑎 ∀𝑎,𝑛∈ℕ f The quantity 𝑛 is called the exponent of 𝑎𝑛 . The quantity 𝑎 is called the base of 𝑎𝑛 . Theorem 2.6. 17 (𝑛+𝑚) 𝑛 𝑚 t 𝑎 𝑚 = (𝑎 ) (𝑎 ) 𝑛 (𝑛𝑚) h 𝑎 = 𝑎 m ( )𝑛 (𝑎𝑏) = (𝑎𝑛 ) (𝑏𝑛 )

∀𝑛,𝑚,𝑎∈ℕ ∀𝑛,𝑚,𝑎∈ℕ ∀𝑛,𝑎,𝑏∈ℕ

✎PROOF: 1. Proof that 𝑎𝑛+𝑚 = 𝑎𝑛 𝑎𝑚 : (a) Let 𝑀 be the set of values for which 𝑎𝑛+𝑚 = 𝑎𝑛 𝑎𝑚 . (b) Proof that 1 ∈ 𝑀: 𝑎𝑛+1 = 𝑎𝗌(𝑛)


𝑛

=𝑎 𝑎

by definition of exponents over ℕ page 20

= 𝑎 𝑛 𝑎1


(c) Proof that 𝑚 ∈ 𝑀 ⟹ 𝗌(𝑚) ∈ 𝑀: 𝑎𝑛+𝗌(𝑚) = 𝑎𝑛+[𝑚+1]


=𝑎

[𝑛+𝑚]+1


=𝑎

𝗌(𝑛+𝑚)


=𝑎

𝑛+𝑚


𝑎

𝑛 𝑚

= [𝑎 𝑎 ]𝑎 = 𝑎𝑛 [𝑎𝑚 𝑎] 𝑛 𝗌(𝑚)

=𝑎 𝑎

by left hypothesis by Theorem 2.5 page 19 by definition of exponents over ℕ page 20

(d) Therefore, by the axiom of induction (page 14), 𝑀 = ℕ. 16 17

📘 Dedekind (1888b) page 104 📘 Dedekind (1888b) pages 104–105

version 0.30


Daniel J. Greenhoe


page 21


2. Proof that (𝑎𝑛 )𝑚 = 𝑎𝑛𝑚 : (a) Let 𝑀 be the set of values for which (𝑎𝑛 )𝑚 = 𝑎𝑛𝑚 . (b) Proof that 1 ∈ 𝑀: (𝑎𝑛 )1 = (𝑎𝑛 )1 = 𝑎𝗌(𝑛)


𝑛

=𝑎 𝑎


= 𝑎 𝑛 𝑎1


(c) Proof that 𝑚 ∈ 𝑀 ⟹ 𝗌(𝑚) ∈ 𝑀: (𝑎𝑛 )𝗌 (𝑚) = (𝑎𝑛 )𝑚 (𝑎𝑛 ) = 𝑎𝑛𝑚 𝑎𝑛

by left hypothesis

=𝑎

𝑛𝑚+𝑛

by previous property

=𝑎

𝑛𝗌(𝑚)


(d) Therefore, by the axiom of induction (page 14), 𝑀 = ℕ. 3. Proof that (𝑎𝑏)𝑛 = 𝑎𝑛 𝑏𝑛 : (a) Let 𝑀 be the set of values for which (𝑎𝑛 )𝑚 = 𝑎𝑛𝑚 . (b) Proof that 1 ∈ 𝑀: (𝑎𝑏)1 = 𝑎𝑏


= (𝑎1 ) (𝑏1 )


(c) Proof that 𝑚 ∈ 𝑀 ⟹ 𝗌(𝑚) ∈ 𝑀: (𝑎𝑏)𝗌 (𝑛) = (𝑎𝑏)𝑛 (𝑎𝑏) 𝑛 𝑛

= (𝑎 𝑏 ) (𝑎𝑏) 𝑛

𝑛

= 𝑎 (𝑏 𝑎)𝑏 𝑛

𝑛

= 𝑎 (𝑎𝑏 )𝑏 𝑛

by left hypothesis by associative property of ⋅ over ℕ (page 19) by commutative property of ⋅ over ℕ (page 18)

𝑛

by associative property of ⋅ over ℕ (page 19)

𝗌


= (𝑎 𝑎) (𝑏 𝑏) 𝗌


= 𝑎 (𝑛) 𝑏 (𝑛)

(d) Therefore, by the axiom of induction (page 14), 𝑀 = ℕ.

✏

2.1.4 Order Definition 2.4. 18 Let ℕ be the set of natural numbers. d 𝑚 ≤ 𝑛 if e f there exists 𝑘 ∈ ℕ such that 𝑛 = 𝑚 + 𝑘 or 𝑚 = 𝑛 Theorem 2.7. t h (ℕ, ≤) is a totally ordered set. m 18

📘 Landau (1966) page 9


Daniel J. Greenhoe


version 0.30

page 22

2.2

2.2. WHOLE NUMBERS

Whole Numbers

Definition 2.5. Let ℕ be the set of natural numbers. d The set of whole numbers 𝕎 is defined as 𝕎 ≜ ℕ ∪ 0 where e f 0 is the element that satisfies the condition 𝗌(0) = 1. Theorem 2.8 (Division). 19 t h ∀𝑛, 𝑚 ∈ 𝕎 ∃𝑟, 𝑞 ∈ 𝕎 m

such that

𝑛 = 𝑞𝑚 + 𝑟 and 𝑟 < 𝑞𝑚

✎PROOF: 1. Let 𝑀 be defined as 𝑀 ≜ {𝑛 ∈ ℤ|∀𝑚 ∈ ℤ, ∃𝑞, 𝑟 ∈ 𝕎

such that

𝑛 = 𝑞𝑚 + 𝑟} .

2. Proof that 0 ∈ 𝑀: 0 = 0 ⋅ 𝑚 + 𝑟 3. Proof that 𝑛 ∈ 𝑀 ⟹ 𝗌(𝑛) ∈ 𝑀: 𝗌(𝑛) = 𝗌(𝑞𝑚 + 𝑟) = 𝑞𝑚 + 𝗌(𝑟) =

𝑞𝑚 + 𝗌(𝑟) for 𝗌(𝑟) < 𝑞𝑚 { 𝑞𝑚 + 𝑚 for 𝗌(𝑟) = 𝑚

=

𝑞𝑚 + 𝗌(𝑟) for 𝗌(𝑟) < 𝑞𝑚 { 𝗌(𝑞)𝑚 for 𝗌(𝑟) = 𝑚

left hypothesis by Definition 2.1 page 15


✏

2.3

Integers

Definition 2.6. d e The negative number −𝑛 is the number that satisfies 𝑛 + (−𝑛) = 0 forall 𝑛 ∈ ℕ. f Definition 2.7. d The set of integers ℤ is defined as 20 e ℤ ≜ 𝕎 ∪ {−𝑛|𝑛 ∈ ℕ} f

19

📘 Amann and Escher (2005) page 35 note about ℤ: “Z” stands for the German word zahlen which means numbers. Reference: 📘 Stillwell (2002) page 404 20

version 0.30


Daniel J. Greenhoe


page 23


2.4 Rational Numbers Definition 2.8. d The fractions 𝑚𝑛 and 𝑞𝑝 are equivalent if 𝑚𝑞 = 𝑛𝑝. e f The equivalence class reprented by the member 𝑚𝑛 is denoted [ 𝑚𝑛 ] ≜ { 𝑞𝑝 | 𝑞𝑝 = 𝑚𝑛 }. Definition 2.9. d 𝑚 𝑝 ≤ if e 𝑞 f 𝑛

𝑚𝑞 ⋜ 𝑛𝑝

A rational number is any number 𝑞 that is a ratio of two integers, the denominator not being 0. This is formally defined next. Definition 2.10. The set of rational numbers ℚ is defined as21 d 𝑛 e ℚ ≜ {[ ]|𝑛 ∈ ℤ and 𝑚 ∈ ℕ} . 𝑚 f A rational number is any element of ℚ. Definition 2.11. 22 Let ℚ be the set of rational numbers (Definition 2.10 page 23). Let ⊕ be the operation of addition on the set of intergers ℤ. Let ⊗ be the operation of multiplication on ℤ. Let the addition operator + and multiplication operator × be defined as follows: 𝑛 ⎧ 𝑟 + 𝑠 ≜ (𝑛𝑟 ⊗ 𝑑𝑠 ) ⊕ (𝑛𝑠 ⊗ 𝑑𝑟 ) d 𝑟 ≜ 𝑟 𝑛𝑟 ,𝑑𝑟 ∈ℤ and ⎫ ⎪ e ⎪ 𝑑𝑟 𝑑𝑟 ⊗ 𝑑𝑠 ⟹ f 𝑛𝑠 ⎬ ⎨ 𝑛𝑟 ⊗ 𝑛 𝑠 𝑠 ≜ 𝑛𝑠 ,𝑑𝑠 ∈ℤ ⎪ ⎪ 𝑟×𝑠 ≜ 𝑑𝑠 ⎭ 𝑑𝑟 ⊗ 𝑑𝑠 ⎩

2.5 Real numbers ❝Sp ter ging Kronecker noch weiter, in-

dem er die Existenz irrationaler Zahlen leugnete; so sagte er mir in seiner lebhaften und zu Paradoxen geneigten Art einmal: “Was n utzt uns Ihre schöne Untersuchung über die Zahl �? Wozu das Nachdenken über solche Probleme, wenne es doch gar keine irrationalen Zahlen gibt?”❞

❝Later,

Kronecker went even further and denied the existence of irrational numbers; thus, he once said to me in his lively and paradoxical way, “Of what use to us are your beautiful researches about the number 𝜋 ? Why consider such problems when in fact there are no irrational numbers?” ❞

Lindemann's recollection, published more than ten years after Leopold Kronecker's (pictured) death, of an encounter with Kronecker (1823–1891), German mathematician, after Lindemann had proved that 𝜋 is a transcendental number. 23

21

note about ℚ: “Q” stands for the English word “quotient”. Reference: 📘 Pugh (2002) page 2 22 📘 Strichartz (1995) page 19, 📘 Hobson (1926), pages 15–17 23 quote: 📘 Poincaré et al. (1904) page 246 ⟨note 3⟩ translation: 📘 Edwards (2005) page 203 image: http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Kronecker.html 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 24

2.5. REAL NUMBERS

❝I

found myself for the first time obliged to lecture upon the elements of the differential calculus and felt more keenly than ever before the lack of a really scientific foundation for arithmetic.…For myself this feeling of dissatisfaction was so overpowering that I made the fixed resolve to keep meditating on the question till I should find a purely arithmetic and perfectly rigorous foundation for the principles of infinitesimal analysis.…It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. I succeeded Nov. 24, 1858,…❞ Richard Dedekind (1831–1915), German mathematician 24

2.5.1

Construction

Definition 2.12 (Dedekind cuts). 25 A set 𝐿 is a cut if 1. 𝐿 ⊊ ℚ d e 2. 𝐿 ≠ ∅ f 3. 𝑟 ∈ 𝐿 and 𝑞 ∈ ℚ and 𝑞 < 𝑟 ⟹ 4. ∀𝑟 ∈ 𝐿, ∃𝑞 ∈ 𝐿 such that 𝑟 < 𝑞

(𝐿 is a proper subset of ℚ)

and

(𝐿 is non-empty)

and

𝑞∈𝐿

and (𝐿 is open)

Lemma 2.3. 26 Let (ℚ, ≤) be the ordered set of rationals. Let 𝐿 be a cut on ℚ. l 𝑝 ∈ 𝐿 𝑎𝑛𝑑 𝑞 ∉ 𝐿 ⟹ 𝑝 < 𝑞 e m 𝑞 ∉ 𝐿 𝑎𝑛𝑑 𝑞 < 𝑟 ⟹ 𝑟 ∉ 𝐿

✎PROOF: 𝑝≥𝑞 ⟹ 𝑞∈𝐿

by (3) in Definition 2.12 page 24

⟹ contradiction of left hypothesis 𝑟∈𝐿 ⟹ 𝑞∈𝐿

by (3) in Definition 2.12 page 24

⟹ contradiction of left hypothesis

✏ Definition 2.13. 27 The set of real numbers ℝ is the set of all cuts on ℚ. That is, d e ℝ ≜ {𝐿|𝐿 is a cut} f Any element of ℝ (any cut) is a real number. 24 25 26 27

quote: 📘 Dedekind (1872b) page 1 image: http://turnbull.mcs.st-and.ac.uk/history/PictDisplay/Dedekind.html 📘 Landau (1966) page 43, 📘 Rudin (1976) page 17 📘 Rudin (1976) page 17 📘 Pugh (2002) page 12

version 0.30


Daniel J. Greenhoe


page 25


2.5.2 Order structure Definition 2.14. 28 Let (ℚ, ⋜) be the ordered set of rational numbers. d e 𝑥 ≤ 𝑦 if 𝑥 ⊆ 𝑦 for 𝑥, 𝑦 ∈ ℝ (𝑥 and 𝑦 are cuts on ℚ) f Theorem 2.9. Let ≤ be the relation defined in Definition 2.14 (page 25). (ℝ, ≤) is a totally ordered set. In particular, 1. 𝑥 ≤ 𝑥 ∀𝑥∈ℝ (reflexive) t h 2. 𝑥 ≤ 𝑦 and 𝑦 ≤ 𝑧 ⟹ 𝑥 ≤ 𝑧 ∀𝑥,𝑦,𝑧∈ℝ (transitive) m 3. 𝑥 ≤ 𝑦 and 𝑦 ≤ 𝑥 ⟹ 𝑥 = 𝑦 ∀𝑥,𝑦∈ℝ (antisymmetric) 4. 𝑥, 𝑦 ∈ ℝ ⟹ 𝑥 ≤ 𝑦 or 𝑦 ≤ 𝑥 ∀𝑥,𝑦∈ℝ (comparable).

and and

partial order

and

Theorem 2.10. 29 t h The set ℝ has the least upper bound property. 30 m

2.5.3 Arithmetic Definition 2.15 (real addition). d e 𝑥 + 𝑦 = {𝑎 + 𝑏|𝑎 ∈ 𝑥, 𝑏 ∈ 𝑦} f

∀𝑥, 𝑦 ∈ ℝ

2.5.4 Least upper bound properties Theorem 2.11. 31 Let 𝑎, 𝑏, 𝜖 ∈ ℝ. t 𝑎 ≤ 𝑏 + 𝜖 ∀𝜖 > 0 ⟹ 𝑎 ≤ 𝑏 h m 𝑎 < 𝑏 + 𝜖 ∀𝜖 > 0 ⟹ 𝑎 ≤ 𝑏

✎PROOF:

1. Choose 𝜖 =

𝑎−𝑏 . 2

28

📘 Pugh (2002) page 13 📘 Pugh (2002) page 13, 📘 Rudin (1976) page 18 30 least upper bound property: Definition 3.23 (page 58) 31 📘 Apostol (1975) page 3 29


Daniel J. Greenhoe


version 0.30

page 26

2.5. REAL NUMBERS

2. 𝑎−𝑏 2 𝑎+𝑏 = 2 𝑎+𝑎 < 2 =𝑎

𝑏𝑏+𝜖 ⟹ contradiction of left hypotheses ⟹ 𝑎≤𝑏

✏ Theorem 2.12. t h 𝑎 < sup 𝑆 m

32

Let 𝑆 ∈ 𝟚ℝ be a subset of the set of real numbers ℝ. ⟹

∃𝑥 ∈ 𝑆

such that

𝑎 < 𝑥 ≤ sup 𝑆

✎PROOF: ¬(∃𝑥 ∈ 𝑆

such that

𝑎 < 𝑥 ≤ sup 𝑆) ⟹ ∀𝑥 ∈ 𝑆, 𝑎 < 𝑥 ≤ sup 𝑆 ⟹ ∀𝑥 ∈ 𝑆, 𝑥 ≤ 𝑎 < sup 𝑆 ⟹ 𝑎 is a least upper bound for 𝑆 ⟹ 𝑎 = sup 𝑆 ⟹ contradiction of left hypothesis ⟹ ∃𝑥 ∈ 𝑆

such that

𝑎 < 𝑥 ≤ sup 𝑆

✏ Theorem 2.13. 33 Let 𝐴 ∈ 𝟚ℝ and 𝐵 ∈ 𝟚ℝ be subsets of the set of real numbers ℝ. Define 𝐴 + 𝐵 ≜ {𝑥 + 𝑦|𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵}. t 𝐶 = 𝐴 + 𝐵 and ⎫ sup 𝐶 exists and ⎪ h sup 𝐴 exists and ⎬ ⟹ { sup 𝐶 = sup 𝐴 + sup 𝐵 m sup 𝐵 exists ⎪ ⎭ ✎PROOF: 1. Proof that sup 𝐶 exists and sup 𝐶 ≤ sup 𝐴 + sup 𝐵: 𝑧 = 𝑥 + 𝑦 ≤ sup 𝐴 + sup 𝐵

⟹ sup 𝐶 exists and sup 𝐶 ≤ sup 𝐴 + sup 𝐵

2. Proof that sup 𝐴 + sup 𝐵 ≤ sup 𝐶: ∃𝑥, 𝑦

such that

(sup 𝐴 − 𝜖) + (sup 𝐵 − 𝜖) < 𝑥 + 𝑦 ≤ sup 𝐴 + sup 𝐵

⟹ sup 𝐴 + sup 𝐵 − 2𝜖 < 𝑥 + 𝑦 ≤ sup 𝐶 ⟹ sup 𝐴 + sup 𝐵 ≤ sup 𝐶


✏ 32 33

📘 Apostol (1975) page 9 📘 Apostol (1975) page 10

version 0.30


Daniel J. Greenhoe


page 27


Theorem 2.14. Let 𝑆 ∈ 𝟚ℝ and 𝑇 ∈ 𝟚ℝ be subsets of the set of real numbers ℝ. t 𝑠 ≤ 𝑡 ∀𝑠 ∈ 𝑆, 𝑡 ∈ 𝑇 and sup 𝑆 exists and h ⟹ } { sup 𝑆 ≤ sup 𝑇 m sup 𝑇 exists ✎PROOF: 𝑠 ≤ 𝑡 ≤ sup 𝑇 ⟹ sup 𝑆 exists and sup 𝑆 ≤ sup 𝑇

✏

2.5.5 Completeness Theorem 2.15. ✎PROOF:

34

The set of rational numbers ℚ is not complete in the set of real numbers ℝ.

The proof is by an example of the number √2, which we will show is not a rational number.

1. Proof that [𝑛2 is even] ⟹ [𝑛 is even]: 𝑛2 is even ⟹ ∃𝑚 ∈ ℤ

such that

𝑛2 = 2𝑚2

⟹ 𝑛 = √2𝑚2 ⟹ ∃𝑝 ∈ ℤ

such that

𝑛=

2 ⋅ 2⏟ ⋅ 𝑝2 √ 𝑚2

⟹ 𝑛 = 2𝑝 ⟹ 𝑛 is even 2. Proof that √2 ∉ ℚ: √2 ∈ ℚ ⟹ ∃𝑛 ∈ ℤ, 𝑚 ∈ ℤ⧵{0}, not both even

such that

√2 = 𝑛 𝑚

⟹ 𝑛2 = 2𝑚2 ⟹ 𝑛2 is even ⟹ 𝑛 is even ⟹ 𝑛2 is divisible by 4 ⟹ 2𝑚2 is divisible by 4 ⟹ 𝑚2 is even ⟹ 𝑚 is even ⟹ both 𝑛 and 𝑚 are even ⟹ √2 ∉ ℚ

✏ Theorem 2.16 (The Archimedean Property / Eudoxus axiom). t h ∃𝑛 ∈ ℕ such that 𝑛𝑥 > 𝑦 ∀𝑥, 𝑦 ∈ ℝ+ m 34 35

35

📘 Rudin (1976), page 2 📘 Aliprantis and Burkinshaw (1998), page 17 📘 Carothers (2000) page 5


Daniel J. Greenhoe


version 0.30

page 28

2.5. REAL NUMBERS

✎PROOF: 1. Proof that if sup 𝐴 of a set 𝐴 exists, then ∀𝜖 > 0, ∃𝑥 ¬(∃𝑥 such that

such that

sup 𝐴 − 𝜖 < 𝑥 ≤ sup 𝐴) ⟹ ∄𝑥

sup 𝐴 − 𝜖 < 𝑥 ≤ sup 𝐴:

such that

sup 𝐴 − 𝜖 < 𝑥

⟹ 𝑥 ≤ sup 𝐴 − 𝜖

∀𝑥 ∈ 𝐴

⟹ (sup 𝐴 − 𝜖) is an upper bound of 𝐴 ⟹ sup 𝐴 ≤ sup 𝐴 − 𝜖 ⟹ impossibility ⟹ ∃𝑥

such that

(sup 𝐴 − 𝜖) < 𝑥 ≤ sup 𝐴

2. Proof that ℕ is unbounded above: ¬(ℕ is unbounded above) ⟹ ℕ is bounded above ⟹ ∃𝑥 ∈ ℝ such that 𝑛 ≤ 𝑥 ⟹ ∃𝑠 ∈ ℝ

such that

∀𝑛 ∈ ℕ

𝑠 = sup ℕ

⟹ ∃𝑛

such that

𝑠−1 𝑦

✏ Theorem 2.17. 36 Let ℝ be the set of real numbers and ℚ the set of rational numbers. t 𝑥, 𝑦 ∈ ℝ and h ⟹ ∃𝑟 ∈ ℚ such that 𝑥 < 𝑟 < 𝑦 } m 𝑥 1 by the Archimedean property page 27

⟹ 𝑞𝑦 − 𝑞𝑥 > 1 ⟹ ∃𝑝 ∈ ℕ such that 𝑞𝑦 < 𝑝 < 𝑞𝑥 𝑝 ⟹ 𝑦< 0, 𝑚 ∈ ℤ

(b) By Theorem 2.17 (page 28), there exists 𝑥 ∈ ℝ between 𝑥𝑚 and sup ⦅𝑥𝑛 ⦆ such that 𝑥𝑚 − 𝜖 < 𝑥 ≤ sup ⦅𝑥𝑛 ⦆

∀𝜖 > 0, 𝑚 ∈ ℤ

(c) 𝑥𝑚 − 𝑥 < 𝜖 2. Proof for monotonic decreasing sequences:

✏

2.5.6 Normed algebra structure Definition 2.16. 38 ℝ d The absolute value |⋅| ∈ ℝ is defined as e 𝑥 𝑓 𝑜𝑟 𝑥 ≥ 0 |𝑥| ≜ f { −𝑥 𝑓 𝑜𝑟 𝑥 < 0 Lemma 2.4. l e |𝑥| ≤ 𝑎 m

39

⟺

−𝑎 ≤ 𝑥 ≤ 𝑎

∀𝑥∈ℝ

✎PROOF: 1. Proof that |𝑥| ≤ 𝑎 ⟹ −𝑎 ≤ 𝑥 ≤ 𝑎 |𝑥| ≤ 𝑎 ⟹

𝑥 𝑓 𝑜𝑟 𝑥 ≥ 0 ≤𝑎 { −𝑥 𝑓 𝑜𝑟 𝑥 < 0 }

⟹

𝑥 ≤ 𝑎 𝑓 𝑜𝑟 𝑥 ≥ 0 { (−𝑥) ≤ 𝑎 𝑓 𝑜𝑟 𝑥 < 0 }

⟹

𝑥 ≤ 𝑎 𝑓 𝑜𝑟 𝑥 ≥ 0 { (𝑥 ≥ −𝑎) 𝑓 𝑜𝑟 𝑥 < 0 }


⟹ −𝑎 ≤ 𝑥 ≤ 𝑎 37 38 39

📘 Carothers (2000) page 📘 Apostol (1975) page 13 📘 Apostol (1975) page 13


Daniel J. Greenhoe


version 0.30

page 30

2.5. REAL NUMBERS

2. Proof that |𝑥| ≤ 𝑎 ⟸ −𝑎 ≤ 𝑥 ≤ 𝑎 |𝑥| = ≤

𝑥 𝑓 𝑜𝑟 𝑥 ≥ 0 { −𝑥 𝑓 𝑜𝑟 𝑥 < 0 }


𝑎 𝑓 𝑜𝑟 𝑥 ≥ 0 { 𝑎 𝑓 𝑜𝑟 𝑥 < 0 }

by left hypothesis

=𝑎

✏ ℝ

Theorem 2.19 (normed algebra properties). 40 Let |⋅| ∈ ℝ⊢ be the absolute value function. The pair (ℝ, |⋅|) is a normed algebra. In particular 1. ∀𝑥∈ℝ (non-negative) |𝑥| ≥ 0 t h 2. ∀𝑥∈ℝ (nondegenerate) |𝑥| = 0 ⟺ 𝑥 = 0 m 3. ∀𝑥,𝑦∈ℝ (homogeneous / multiplicative condition) |𝑥𝑦| = |𝑥| |𝑦| 4. |𝑥 + 𝑦| ≤ |𝑥| + |𝑦| ∀𝑥,𝑦∈ℝ (subadditive / triangle inequality) ✎PROOF: 1. Proof that |𝑥| ≥ 0: true by Definition 2.16 page 29. 2. Proof that |𝑥| = 0 ⟺ 𝑥 = 0: true by Definition 2.16 page 29. 3. Proof that |𝑥𝑦| = |𝑥||𝑦|: |𝑥𝑦| = =

𝑥𝑦 𝑓 𝑜𝑟 𝑥𝑦 ≥ 0 ( −𝑥𝑦 𝑓 𝑜𝑟 𝑥𝑦 < 0 )

by definition of |⋅| page 29

𝑥 𝑓 𝑜𝑟 𝑥 ≥ 0 𝑦 𝑓 𝑜𝑟 𝑦 ≥ 0 ( −𝑥 𝑓 𝑜𝑟 𝑥 < 0 )( −𝑦 𝑓 𝑜𝑟 𝑦 < 0 )

= |𝑥| |𝑦|

by definition of |⋅| page 29

4. Proof that |𝑥 + 𝑦| ≤ |𝑥| + |𝑦|: (a) Start with these inequalities: −|𝑥| ≤ 𝑥 ≤ |(|𝑥) −|𝑦| ≤ 𝑦 ≤ |(|𝑦) (b) Add the above two equations to get the following: −(|𝑥| + |𝑦|) ≤ 𝑥 + 𝑦 ≤ (|𝑥| + |𝑦|) (c) Then by Lemma 2.4 (page 29), |𝑥 + 𝑦| ≤ |𝑥| + |𝑦|

✏ 40

📘 Apostol (1975) page 13

version 0.30


Daniel J. Greenhoe


page 31


2.6 Complex numbers ❝Weil

nun alle möglichen Zahlen, die man sich nur immer vorstellen mag, entweder größer oder kleiner als 0, oder etwa 0 selbst sind, so ist klar, daß die Quadratwurzeln von Negativzahlen nicht einmal zu den möglichen Zahlen gerechnet werden können. Folglich müssen wir sagen, daß dies unmögliche Zahlen sind. Und dieser Umstand leitet uns auf den Begriß von solchen Zahlen, welche ihrer Natur nach unmöglich sind, und gewöhnlich imaginäre oder eingebildete Zahlen genannt werden, weil sie bloßin der Einbildung vorhanden sind.❞

❝And, since all numbers which it is pos-

sible to conceive, are either greater or less than 0, or are 0 itself, it is evident that we cannot rank the square root of a negative number amongst possible numbers, and we must therefore say that it is an impossible quantity. In this manner we are led to the idea of numbers, which from their nature are impossible; and therefore they are usually called imaginary quantities, because they exist merely in the imagination.❞

Leonhard Euler (1707–1783), mathematician

41

2.6.1 Definitions Definition 2.17. 42 The set of complex numbers ℂ is defined as d e ℂ ≜ {(𝑎, 𝑏) |𝑎, 𝑏 ∈ ℝ} (the set of all ordered pairs of real numbers) f A complex number is any element of ℂ. Addition and multiplication over ℂ are defined next. Definition 2.18. 43 Let ℂ be the set of complex numbers. d Addition and multiplication on ℂ are defined as follows: e ∀(𝑥𝑟 ,𝑥𝑖 ),(𝑦𝑟 ,𝑦𝑖 )∈ℂ (𝑥𝑟 , 𝑥𝑖 ) + (𝑦𝑟 , 𝑦𝑖 ) ≜ (𝑥𝑟 + 𝑦𝑟 , 𝑥𝑖 + 𝑦𝑖 ) f (𝑥𝑟 , 𝑥𝑖 ) ⋅ (𝑦𝑟 , 𝑦𝑖 ) ≜ (𝑥𝑟 𝑦𝑟 − 𝑥𝑖 𝑦𝑖 , 𝑥𝑟 𝑦𝑖 + 𝑥𝑖 𝑦𝑟 ) ∀(𝑥𝑟 ,𝑥𝑖 ),(𝑦𝑟 ,𝑦𝑖 )∈ℂ Definition 2.19. 44 Let ℂ be the set of complex numbers. d The imaginary number 𝑖 in ℂ is defined as e 𝑖 ≜ (0, 1) f Proposition 2.1. p r 𝑖2 = −1 p 41

42 43 44 45

45

Let ℂ be the set of complex numbers and 𝑖 the imaginary number.

quote: 📘 Euler (1770a) page 60 translation: 📘 Euler (1770b) page 43 image: http://en.wikipedia.org/wiki/File:Leonhard_Euler.jpg, public domain 📘 Landau (1966) page 92, 📘 Rudin (1976) page 12 📘 Landau (1966) pages 93–96, 📘 Thurston (1956) page 121, 📘 Bottazzini (1986) page 180, 📘 Hamilton (1837) pages 8 📘 Landau (1966) page 133, 📘 Euler (1777) page 184, 📘 Cardano (1545a), 📘 Cardano (1545b) 📘 Landau (1966) page 133


Daniel J. Greenhoe


version 0.30

page 32

2.6. COMPLEX NUMBERS

✎PROOF: 𝑖2 = (0, 1) (0, 1)


= (0 ⋅ 0 − 1 ⋅ 1, 0 ⋅ 1 + 1 ⋅ 0)


= (−1, 0) = −1

✏ Theorem 2.20. 46 t h (𝑎, 𝑏) = 𝑎 + 𝑖𝑏 m

∀(𝑎,𝑏)∈ℂ

(rectangular coordinates)

✎PROOF: (𝑎, 𝑏) = 𝑎(1, 0) + 𝑏 (0, 1) = 𝑎1 + 𝑏𝑖


= 𝑎 + 𝑖𝑏

✏

2.6.2

Field structure

Theorem 2.21 (Additive abelian group). 47 (ℂ, +) is a commutative group under addition; in particular 1. (𝑥 + 𝑦) + 𝑧 = 𝑥 + (𝑦 + 𝑧) ∀𝑥,𝑦,𝑧∈ℂ (associative) t h 2. 𝑥 + (0, 0) = (0, 0) + 𝑥 = 𝑥 ∀𝑥∈ℂ ((0, 0) is the additive identity element) m 3. − (𝑎, 𝑏) = (−𝑎, −𝑏) ∀(𝑎,𝑏)∈ℂ ((−𝑎, −𝑏) is the additive inverse of (𝑎, 𝑏)) 4. 𝑥 + 𝑦 = 𝑦 + 𝑥 ∀𝑥,𝑦∈ℂ (commutative) ✎PROOF: 1. Proof that (ℂ, +) is associative: (𝑥 + 𝑦) + 𝑧 = [(𝑥𝑟 , 𝑥𝑖 ) + (𝑦𝑟 , 𝑦𝑖 )] + (𝑧𝑟 , 𝑧𝑖 ) = (𝑥𝑟 + 𝑦𝑟 , 𝑥𝑖 + 𝑦𝑖 ) + (𝑧𝑟 , 𝑧𝑖 )

by definition of complex addition (page 31)

= ([𝑥𝑟 + 𝑦𝑟 ] + 𝑧𝑟 , [𝑥𝑖 + 𝑦𝑖 ] + 𝑧𝑖 )


= (𝑥𝑟 + [𝑦𝑟 + 𝑧𝑟 ], 𝑥𝑖 + [𝑦𝑖 + 𝑧𝑖 ]) = (𝑥𝑟 , 𝑥𝑖 ) + (𝑦𝑟 + 𝑧𝑟 , 𝑦𝑖 + 𝑧𝑖 )


= 𝑥 + (𝑦 + 𝑧) 2. Proof that (0, 0) ∈ ℂ is the additive identity element: (𝑎, 𝑏) + (0, 0) = (𝑎 + 0, 𝑏 + 0)


= (𝑎, 𝑏) (0, 0) + (𝑎, 𝑏) = (0 + 𝑎, 0 + 𝑏)


= (𝑎, 𝑏) 46 47

📘 Landau (1966) page 133 📘 Landau (1966) pages 93–94

version 0.30


Daniel J. Greenhoe


page 33


3. Proof that (−𝑎, −𝑏) ∈ ℂ is the inverse element of (𝑎, 𝑏): (𝑎, 𝑏) + (−𝑎, −𝑏) = (𝑎 − 𝑎, 𝑏 − 𝑏)


= (0, 0) (−𝑎, −𝑏) + (𝑎, 𝑏) = (−𝑎 + 𝑎, −𝑏 + 𝑏)


= (0, 0) 4. Proof that (ℂ, +) is commutative: 𝑥 + 𝑦 = (𝑥𝑟 , 𝑥𝑖 ) + (𝑦𝑟 , 𝑦𝑖 ) by definition of complex addition (page 31)

= (𝑥𝑟 + 𝑦𝑟 , 𝑥𝑖 + 𝑦𝑖 ) = (𝑦𝑟 + 𝑥𝑟 , 𝑦𝑖 + 𝑥𝑖 ) = (𝑦𝑟 , 𝑦𝑖 ) + (𝑥𝑟 , 𝑥𝑖 )


=𝑦+𝑥

✏ Theorem 2.22 (Multiplicative abelian group). 48 (ℂ, ⋅) is a commutative group under multiplication; in particular 1. (𝑥𝑦)𝑧 = 𝑥(𝑦𝑧) ∀𝑥,𝑦,𝑧∈ℂ (associative) t 2. 𝑥 (1, 0) = (1, 0) 𝑥 = 𝑥 ∀𝑥∈ℂ ((1, 0) is the multiplicative identity element) h −1 m 1 1 (𝑎, −𝑏) is the multiplicative inverse of (𝑎, 𝑏)) 3. (𝑎, 𝑏) = 2 2 (𝑎, −𝑏) ∀(𝑎,𝑏)∈ℂ ( 𝑎2 +𝑏 2 𝑎 +𝑏 4. 𝑥𝑦 = 𝑦𝑥 ∀𝑥,𝑦∈ℂ (commutative) ✎PROOF: 1. Proof that (ℂ, ×) is associative: (𝑥𝑦)𝑧 = [(𝑥𝑟 , 𝑥𝑖 ) (𝑦𝑟 , 𝑦𝑖 )] (𝑧𝑟 , 𝑧𝑖 ) = (⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑥𝑟 𝑦𝑟 − 𝑥𝑖 𝑦𝑖 , 𝑥𝑟 𝑦𝑖 + 𝑥𝑖 𝑦𝑟 )(⏟ 𝑧𝑟 , 𝑧𝑖 )


𝑧

𝑥𝑦

= ((𝑥𝑟 𝑦𝑟 − 𝑥𝑖 𝑦𝑖 )𝑧𝑟 − (𝑥𝑟 𝑦𝑖 + 𝑥𝑖 𝑦𝑟 )𝑧𝑖 , (𝑥𝑟 𝑦𝑟 − 𝑥𝑖 𝑦𝑖 )𝑧𝑖 + (𝑥𝑟 𝑦𝑖 + 𝑥𝑖 𝑦𝑟 )𝑧𝑟 )


= (𝑥𝑟 𝑦𝑟 𝑧𝑟 − 𝑥𝑖 𝑦𝑖 𝑧𝑟 − 𝑥𝑟 𝑦𝑖 𝑧𝑖 − 𝑥𝑖 𝑦𝑟 𝑧𝑖 , 𝑥𝑟 𝑦𝑟 𝑧𝑖 − 𝑥𝑖 𝑦𝑖 𝑧𝑖 + 𝑥𝑟 𝑦𝑖 𝑧𝑟 + 𝑥𝑖 𝑦𝑟 𝑧𝑟 ) = (𝑥𝑟 (𝑦𝑟 𝑧𝑟 − 𝑦𝑖 𝑧𝑖 ) − 𝑥𝑖 (𝑦𝑟 𝑧𝑖 + 𝑦𝑖 𝑧𝑟 ), 𝑥𝑟 (𝑦𝑟 𝑧𝑖 + 𝑦𝑖 𝑧𝑟 ) + 𝑥𝑖 (𝑦𝑟 𝑧𝑟 − 𝑦𝑖 𝑧𝑖 )) = (⏟ 𝑥𝑟 , 𝑥𝑖 )(⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑦𝑟 𝑧𝑟 − 𝑦𝑖 𝑧𝑖 , 𝑦𝑟 𝑧𝑖 + 𝑦𝑖 𝑧𝑟 ) 𝑥

𝑦𝑧

= (𝑥𝑟 , 𝑥𝑖 ) [(𝑦𝑟 , 𝑦𝑖 ) (𝑧𝑟 , 𝑧𝑖 )]


= 𝑥(𝑦𝑧) 2. Proof that (1, 0) ∈ ℂ is the additive identity element: (𝑎, 𝑏) (1, 0) = (𝑎 ⋅ 1 − 𝑏 ⋅ 0, 𝑎 ⋅ 0 + 𝑏 ⋅ 1)

by definition of complex multiplication (page 31)

= (𝑎, 𝑏) (1, 0) (𝑎, 𝑏) = (1 ⋅ 𝑎 − 0 ⋅ 𝑏, 1 ⋅ 𝑏 + 0 ⋅ 𝑎)


= (𝑎, 𝑏) 48

📘 Landau (1966) pages 96–98


Daniel J. Greenhoe


version 0.30

page 34


3. Proof that (𝑎, 𝑏)

[ 𝑎2

1 𝑎2 +𝑏2

(𝑎, −𝑏) ∈ ℂ is the inverse element of (𝑎, 𝑏):

1 1 (𝑎, −𝑏) = 2 (𝑎𝑎 + 𝑏𝑏, −𝑎𝑏 + 𝑏𝑎) by definition of complex multiplication (page 31) ] 𝑎 + 𝑏2 [ 𝑎2 + 𝑏2 1 2 2 = 2 (𝑎 + 𝑏 , 0 ) 𝑎 + 𝑏2 𝑎2 + 𝑏 2 0 = , ( 𝑎2 + 𝑏2 𝑎2 + 𝑏2 ) = (1, 0)

1 1 (𝑎, −𝑏) (𝑎, 𝑏) = 2 (𝑎𝑎 + 𝑏𝑏, 𝑎𝑏 − 𝑏𝑎) 2 ] +𝑏 𝑎 + 𝑏2 1 2 2 = 2 (𝑎 + 𝑏 , 0 ) 𝑎 + 𝑏2 𝑎2 + 𝑏 2 0 = , ( 𝑎2 + 𝑏2 𝑎2 + 𝑏2 )


= (1, 0) 4. Proof that (ℂ, ×) is commutative: 𝑥𝑦 = (𝑥𝑟 , 𝑥𝑖 ) (𝑦𝑟 , 𝑦𝑖 ) = (𝑥𝑟 𝑦𝑟 − 𝑥𝑖 𝑦𝑖 , 𝑥𝑟 𝑦𝑖 + 𝑥𝑖 𝑦𝑟 )


= (𝑦𝑟 𝑥𝑟 − 𝑦𝑖 𝑥𝑖 , 𝑦𝑖 𝑥𝑟 + 𝑦𝑟 𝑥𝑖 ) = (𝑦𝑟 , 𝑦𝑖 ) (𝑥𝑟 , 𝑥𝑖 ) = 𝑦𝑥 5. Proof that (ℂ, ×) forms a multiplicative group: (a) Proof that (1, 0) is the multiplicative identity element:

(b) Proof that

1 𝑎2 +𝑏2

(𝑎, 𝑏) (1, 0) = (𝑎1 − 𝑏0, 𝑎0 + 𝑏1)

= (𝑎, 𝑏)

(1, 0) (𝑎, 𝑏) = (1𝑎 − 0𝑏, 0𝑎 + 1𝑏)

= (𝑎, 𝑏)

(𝑎, −𝑏) is the multiplicative inverse element of (𝑎, 𝑏):

1 1 (𝑎, −𝑏) = 2 (𝑎2 + 𝑏2 , −𝑎𝑏 + 𝑏𝑎) 2 ] 𝑎 + 𝑏2 +𝑏 1 1 (𝑎, −𝑏) (𝑎, 𝑏) = 2 (𝑎2 + 𝑏2 , 𝑎𝑏 − 𝑏𝑎) [ 𝑎2 + 𝑏 2 ] 𝑎 + 𝑏2

(𝑎, 𝑏)

[ 𝑎2

= (1, 0) = (1, 0)

(c) Proof that (ℂ, ×) is closed: (𝑎, 𝑏) (𝑐, 𝑑) = (𝑎𝑐 − 𝑏𝑑, 𝑎𝑑 + 𝑏𝑐) ∈ ℂ.

✏ Proposition 2.2 (Distributive property). p 𝑧(𝑥 + 𝑦) = 𝑧𝑥 + 𝑧𝑦 ∀𝑥,𝑦,𝑧∈ℂ r ∀𝑥,𝑦,𝑧∈ℂ p (𝑥 + 𝑦)𝑧 = 𝑥𝑧 + 𝑦𝑧 49

49 (left distributive) (right distributive)

📘 Landau (1966) page 99

version 0.30


Daniel J. Greenhoe


page 35


✎PROOF: 𝑧(𝑥 + 𝑦) = (𝑧𝑟 , 𝑧𝑖 ) [(𝑥𝑟 , 𝑥𝑖 ) + (𝑦𝑟 , 𝑦𝑖 )] = (𝑧𝑟 , 𝑧𝑖 ) (𝑥𝑟 + 𝑦𝑟 , 𝑥𝑖 + 𝑦𝑖 )

by def. of complex add. (Definition 2.18 page 31)

= (𝑧𝑟 [𝑥𝑟 + 𝑦𝑟 ] − 𝑧𝑖 [𝑥𝑖 + 𝑦𝑖 ], 𝑧𝑖 [𝑥𝑟 + 𝑦𝑟 ] + 𝑧𝑟 [𝑥𝑖 + 𝑦𝑖 ])

by def. of complex mult.

(Definition 2.18 page 31)

= (𝑧𝑟 𝑥𝑟 − 𝑧𝑖 𝑥𝑖 + 𝑧𝑟 𝑦𝑟 − 𝑧𝑖 𝑦𝑖 , 𝑧𝑖 𝑥𝑟 + 𝑧𝑟 𝑥𝑖 + 𝑧𝑖 𝑦𝑟 + 𝑧𝑟 𝑦𝑖 ) = (𝑧𝑟 𝑥𝑟 − 𝑧𝑖 𝑥𝑖 , 𝑧𝑖 𝑥𝑟 + 𝑧𝑟 𝑥𝑖 ) + (𝑧𝑟 𝑦𝑟 − 𝑧𝑖 𝑦𝑖 , 𝑧𝑖 𝑦𝑟 + 𝑧𝑟 𝑦𝑖 ) by def. of complex add. (Definition 2.18 page 31) = (𝑧𝑟 , 𝑧𝑖 ) (𝑥𝑟 , 𝑥𝑖 ) + (𝑧𝑟 , 𝑧𝑖 ) (𝑦𝑟 , 𝑦𝑖 )

by def. of complex mult.


= 𝑧𝑥 + 𝑧𝑦 (𝑥 + 𝑦)𝑧 = 𝑧(𝑥 + 𝑦)

by commutative prop. of Theorem 2.22 page 33

= 𝑧𝑥 + 𝑧𝑦

by previous result

= 𝑥𝑧 + 𝑦𝑧

by commutative prop. of Theorem 2.22 page 33

✏ Theorem 2.23. t h (ℂ, +, ×) is a field. m ✎PROOF: 1. By Theorem 2.21 (page 32), (ℂ, +, ×) is an additive group. 2. By Theorem 2.22 (page 33), (ℂ, +, ×) is a multiplicative group. 3. By Proposition 2.2 (page 34), (ℂ, +, ×) is distributive. 4. These properties collectively imply that (ℂ, +, ×) is a field.

✏ Proposition 2.3. 50 p r (0, 0) (𝑎, 𝑏) = (𝑎, 𝑏) (0, 0) = (0, 0) p

∀(𝑎,𝑏)∈ℂ

✎PROOF: (0, 0) (𝑎, 𝑏) = (0 ⋅ 𝑎 − 0 ⋅ 𝑏, 0 ⋅ 𝑏 + 0 ⋅ 𝑎) = (0, 0)

by Definition 2.18 page 31 by Definition 2.18 page 31

(𝑎, 𝑏) (0, 0) = (𝑎 ⋅ 0 − 𝑏 ⋅ 0, 𝑏 ⋅ 0 + 𝑎 ⋅ 0) = (0, 0)


✏

2.6.3 Normed algebra structure Definition 2.20. 51 d The absolute value |⋅| ∈ ℝℂ is defined as e f |(𝑎, 𝑏)| ≜ √𝑎2 + 𝑏2 ∀(𝑎,𝑏)∈ℂ 50 51

📘 Apostol (1975) page 16 📘 Landau (1966) page 108


Daniel J. Greenhoe


version 0.30

page 36

2.6. COMPLEX NUMBERS ℂ

Theorem 2.24 (normed algebra properties). 52 Let |⋅| ∈ ℝ⊢ be the absolute value function. The pair (ℂ, |⋅|) is a normed algebra. In particular 1. ∀𝑥∈ℂ (non-negative) |𝑥| ≥ 0 t h 2. ∀𝑥∈ℂ (nondegenerate) |𝑥| = 0 ⟺ 𝑥 = 0 m 3. ∀𝑥,𝑦∈ℂ (homogeneous / multiplicative condition) |𝑥𝑦| = |𝑥| |𝑦| 4. |𝑥 + 𝑦| ≤ |𝑥| + |𝑦| ∀𝑥,𝑦∈ℂ (subadditive / triangle inequality)

✎PROOF:

1. Proof that |𝑥| ≥ 0: |𝑥|2 = 𝑎2 + 𝑏2

by definition of |𝑥|

≥0 2. Proof that |𝑥| = 0 ⟺ 𝑥 = 0: |𝑥| = 0 ⟹ |𝑥|2 = 0 ⟹ |𝑥| = 0 ⟹ 𝑎 2 + 𝑏2 = 0 ⟹ 𝑎=𝑏=0 ⟹ (𝑎, 𝑏) = 0 ⟹ 𝑥=0 𝑥 = 0 ⟹ (𝑎, 𝑏) = 0 ⟹ 𝑎=𝑏=0 ⟹ 𝑎 2 + 𝑏2 = 0 ⟹ |𝑥|2 = 0 ⟹ |𝑥| = 0 3. Proof that |𝑥𝑦| = |𝑥||𝑦|: |𝑥𝑦|2 = |(𝑎, 𝑏) (𝑐, 𝑑)|2

by definition of |𝑥| 2


= |(𝑎𝑐 − 𝑏𝑑, 𝑎𝑑 + 𝑏𝑐)|

= (𝑎𝑐 − 𝑏𝑑)2 + (𝑎𝑑 + 𝑏𝑐)2 2

2

by definition of |𝑥| 2

2

= (𝑎𝑐) + (𝑏𝑑) − 2(𝑎𝑐)(𝑏𝑑) + (𝑎𝑑) + (𝑏𝑐) + 2(𝑎𝑑)(𝑏𝑐) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (𝑎𝑐 − 𝑏𝑑)2

(𝑎𝑑 + 𝑏𝑐)2

= (𝑎𝑐)2 + (𝑎𝑑)2 + (𝑏𝑐)2 + (𝑏𝑑)2 = (𝑎2 + 𝑏2 )(𝑐 2 + 𝑑 2 ) = |𝑥|2 |𝑦|2

52


📘 Landau (1966) pages 108–110,

version 0.30


Daniel J. Greenhoe


page 37


4. Proof that |𝑥 + 𝑦| ≤ |𝑥| + |𝑦|: |𝑥 + 𝑦|2 = |(𝑎, 𝑏) + (𝑐, 𝑑)|2 = |(𝑎 + 𝑐, 𝑏 + 𝑑)|

by definition of 𝑥, 𝑦

2

2


= (𝑎 + 𝑐) + (𝑏 + 𝑑)

2


2 2 =𝑎 + 𝑐 2 + 2𝑎𝑐 + 𝑏⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ + 𝑑 2 + 2𝑏𝑑 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (𝑎 + 𝑐)2

(𝑏 + 𝑑)2

2 2 = 𝑎⏟ + 𝑏2 + 𝑐⏟ + 𝑑 2 + 2𝑎𝑐 + 2𝑏𝑑 ⏟⏟⏟⏟⏟ |𝑥|2

|𝑦|2

2ℜ(𝑥𝑦∗ )

= |𝑥|2 + |𝑦|2 + 2ℜ(𝑥𝑦∗ ) ≤ |𝑥|2 + |𝑦|2 + 2|𝑥𝑦∗ | 2

≤ |𝑥| + |𝑦|


2

by |⋅| ≥ 0 property (1)

✏ ⊢

Proposition 2.4. 53 Let |⋅| ∈ ℂℝ be the absolute value function. p 𝑥 |𝑥| r = ∀𝑥, 𝑦 ∈ ℂ p | 𝑦 | |𝑦| ✎PROOF: 𝑥 |𝑥| |𝑦 𝑦 | = |𝑦| |𝑦| |𝑦| | 𝑥𝑦 | = |𝑦| 𝑥 = |𝑦|

by homogeneous property of Theorem 2.24 page 36

✏

2.6.4 Star-algebra structure Definition 2.21. 54 Let (𝑎, 𝑏) ∈ ℂ. The conjugate operator ∗ ∶ ℂ → ℂ is defined as d e (𝑎, 𝑏)∗ ≜ (𝑎, −𝑏) ∀ (𝑎, 𝑏) ∈ ℂ f Theorem 2.25. 55 t h 𝑥 = 𝑥∗ ⟺ m

𝑥∈ℝ

✎PROOF: 53 54 55

📘 Apostol (1975) page 18 📘 Landau (1966) page 106 📘 Berberian (1961) page 25


Daniel J. Greenhoe


version 0.30

page 38


1. Proof that 𝑥 = 𝑥∗ ⟹ 𝑥 ∈ ℝ 𝑥 = (𝑎, 𝑏) ⟹ 𝑥∗ = (𝑎, 𝑏)∗ ∗

by definition of 𝑥

⟹ 𝑥 = (𝑎, −𝑏)


⟹ 𝑏 = −𝑏

by left hypothesis

⟹ 𝑏=0

because 𝑏 ∈ ℝ

⟹ 𝑥 = (𝑎, 0) ⟹ 𝑥∈ℝ


2. Proof that 𝑥 = 𝑥∗ ⟸ 𝑥 ∈ ℝ 𝑥 = (𝑎, 𝑏)


= (𝑎, 0)

by right hypothesis

= (𝑎, −0) = (𝑎, 0)∗ = (𝑎, 𝑏) =𝑥


∗

by right hypothesis

∗


✏ Theorem 2.26 (∗-algebra properties). 56 Let ∗ ∶ ℂ → ℂ be the conjugate operator. The pair (ℂ, ∗) is a ∗-algebra57 . In particular, t 𝑥∗∗ = 𝑥 ∀𝑥∈ℂ (involutary) h ∗ ∗ ∗ (𝑥 + 𝑦) = 𝑥 + 𝑦 ∀𝑥,𝑦∈ℂ (distributive) m ∗ ∗ ∗ (𝑥𝑦) = 𝑦 𝑥 ∀𝑥,𝑦∈ℂ (antiautomorphic) ✎PROOF: 𝑥∗∗ = (𝑥∗ )∗ = [(𝑥𝑟 , 𝑥𝑖 )∗ ]∗

where 𝑥 ≜ (𝑥𝑟 , 𝑥𝑖 )

∗

= (𝑥𝑟 , −𝑥𝑖 )

by definition of conjugate (page 37)

= (𝑥𝑟 , 𝑥𝑖 )


=𝑥

by the definition 𝑥 ≜ (𝑥𝑟 , 𝑥𝑖 )

(𝑥 + 𝑦)∗ = [(𝑥𝑟 , 𝑥𝑖 ) + (𝑦𝑟 , 𝑦𝑖 )]∗ = (𝑥𝑟 + 𝑦𝑟 , 𝑥𝑖 + 𝑦𝑖 )

where 𝑥 ≜ (𝑥𝑟 , 𝑥𝑖 ) and 𝑦 ≜ (𝑦𝑟 , 𝑦𝑖 )

∗


= (𝑥𝑟 + 𝑦𝑟 , −𝑥𝑖 − 𝑦𝑖 )

by definition of complex conjugate (page 37) by definition of complex addition (page 31)

= (𝑥𝑟 , −𝑥𝑖 ) + (𝑦𝑟 , −𝑦𝑖 ) ∗

∗

= (𝑥𝑟 , 𝑥𝑖 ) + (𝑦𝑟 , 𝑦𝑖 ) = 𝑥 ∗ + 𝑦∗

by definition of complex conjugate (page 37) by the definitions 𝑥 ≜ (𝑥𝑟 , 𝑥𝑖 ) and 𝑦 ≜ (𝑦𝑟 , 𝑦𝑖 )

(𝑥𝑦)∗ = [(𝑥𝑟 , 𝑥𝑖 ) (𝑦𝑟 , 𝑦𝑖 )]∗

where 𝑥 ≜ (𝑥𝑟 , 𝑥𝑖 ) and 𝑦 ≜ (𝑦𝑟 , 𝑦𝑖 ) ∗

= (𝑥𝑟 𝑦𝑟 − 𝑥𝑖 𝑦𝑖 , 𝑥𝑟 𝑦𝑖 + 𝑥𝑖 𝑦𝑟 )


= (𝑥𝑟 𝑦𝑟 − 𝑥𝑖 𝑦𝑖 , −𝑥𝑟 𝑦𝑖 − 𝑥𝑖 𝑦𝑟 )

by definition of complex conjugate (page 37)

= (𝑥𝑟 , −𝑥𝑖 ) (𝑦𝑟 , −𝑦𝑖 )


= (𝑦𝑟 , −𝑦𝑖 ) (𝑥𝑟 , −𝑥𝑖 )


∗

∗

= (𝑦𝑟 , 𝑦𝑖 ) (𝑥𝑟 , 𝑥𝑖 ) = 𝑦∗ 𝑥∗ . 56 57

by definition of complex conjugate (page 37) by the definitions 𝑥 ≜ (𝑥𝑟 , 𝑥𝑖 ) and 𝑦 ≜ (𝑦𝑟 , 𝑦𝑖 )

📘 Landau (1966) pages 106–107 ∗-algebra: Definition B.3 page 272

version 0.30


Daniel J. Greenhoe


page 39


✏ Theorem 2.27. 58 Let ∗ ∶ ℂ → ℂ be the conjugate operator. t ∀𝑥∈ℂ (conjugate of additive inverse) (−𝑥)∗ = −(𝑥∗ ) h −1 ∗ ∗ −1 ∀𝑥,𝑦∈ℂ (conjugate of multiplicative inverse) m (𝑥 ) = (𝑥 ) ✎PROOF: (−𝑥)∗ = [− (𝑥𝑟 , 𝑥𝑖 )]∗ = (−𝑥𝑟 , −𝑥𝑖 )∗

where 𝑥 ≜ (𝑥𝑟 , 𝑥𝑖 ) by definition of conjugate (page 37)

= (−𝑥𝑟 , 𝑥𝑖 ) = − (𝑥𝑟 , −𝑥𝑖 ) = −[(𝑥𝑟 , 𝑥𝑖 )∗ ]


∗

= − (𝑥 )

by the definition 𝑥 ≜ (𝑥𝑟 , 𝑥𝑖 )

∗

−1 −1 (𝑥 ) = ((𝑥𝑟 , 𝑥𝑖 ) )

∗

where 𝑥 ≜ (𝑥𝑟 , 𝑥𝑖 ) ∗

1 = (𝑥𝑟 , −𝑥𝑖 ) 2 ( 𝑥𝑟 + 𝑥2𝑖 )


∗

1 ∗ (𝑥𝑟 , −𝑥𝑖 ) 2 2 ( 𝑥𝑟 + 𝑥𝑖 ) 1 = 2 (𝑥𝑟 , 𝑥𝑖 ) 𝑥𝑟 + 𝑥2𝑖

=

by previous property by definition of conjugate (page 37)

= (𝑥𝑟 , −𝑥𝑖 )−1


= ((𝑥𝑟 , 𝑥𝑖 )∗ )−1


∗ −1

= (𝑥 )

by definition 𝑥 ≜ (𝑥𝑟 , 𝑥𝑖 )

✏ Corollary 2.1. c o ( 𝑥𝑦 )∗ = r

59 𝑥∗ 𝑦∗

Let ∗ ∶ ℂ → ℂ be the conjugate operator. ∀𝑥,𝑦∈ℂ

✎PROOF: ∗

𝑥 1 = 𝑥 (𝑦) ( 𝑦) = [𝑥∗ ]

∗

∗

1 [( 𝑦 ) ]


∗

= [𝑥∗ ][(𝑦−1 ) ] = [𝑥∗ ][(𝑦∗ )−1 ] =


𝑥∗ 𝑦∗

✏ 58 59

📘 Landau (1966) page 107 📘 Landau (1966) page 107


Daniel J. Greenhoe


version 0.30

page 40

2.6.5


C-star algebraic structure

Theorem 2.28. 60 Let |⋅| ∶ ℂ → ℝ⊢ be the absolute value function on ℂ. t 1. |𝑥∗ | = |𝑥| ∀𝑥∈ℂ h 2 ∗ ∀𝑥∈ℂ m 2. |𝑥| = 𝑥𝑥

✎PROOF: ∗

∗

|𝑥 | = |(𝑎, 𝑏) | = |(𝑎, −𝑏)| = √𝑎2 + (−𝑏)2

by definition of 𝑥 by Definition 2.21 page 37 by definition of |𝑥|

= √𝑎2 + 𝑏2 = |(𝑎, 𝑏)|


= |𝑥|


|𝑥|2 = |(𝑎, 𝑏)|2


= 𝑎 2 + 𝑏2 2


2

= (𝑎 + 𝑏 , 0) = (𝑎2 + 𝑏2 , 𝑏𝑎 − 𝑎𝑏) = (𝑎, 𝑏) (𝑎, −𝑏) = (𝑎, 𝑏) (𝑎, 𝑏)∗ = 𝑥𝑥

∗

by Definition 2.20 page 35 by definition of 𝑥

✏

2.6.6

Hermitian structure

The pair (ℂ, ∗) is a special case of a star-algebra (∗-algebra). In a star-algebra, the real and imaginary components of an element 𝑥 are defined as follows:61 1 ℜ𝑥 ≜ (𝑥 + 𝑥∗ ) 2

ℑ𝑥 ≜

1 𝑥 − 𝑥 ∗ ). 2𝑖 (

Theorem 2.29. 62 Let 𝑥 ≜ (𝑎, 𝑏) ∈ ℂ. Then t ℜ (𝑎, 𝑏) = 𝑎 ∀(𝑎,𝑏)∈ℂ h ∀(𝑎,𝑏)∈ℂ m ℑ (𝑎, 𝑏) = 𝑏 60

📘 Rudin (1976) page 14 ℜ and ℑ: Definition B.5 (page 273) 62 📘 Munkres (2000), page 87 61

version 0.30


Daniel J. Greenhoe


page 41


✎PROOF: 1 ℜ (𝑎, 𝑏) = ((𝑎, 𝑏) + (𝑎, 𝑏)∗ ) 2 1 = ((𝑎, 𝑏) + (𝑎, −𝑏)) 2 1 = (𝑎 + 𝑎, 𝑏 − 𝑏) 2 1 = (2𝑎, 0) 2 =𝑎 1 ℑ (𝑎, 𝑏) = ((𝑎, 𝑏) − (𝑎, 𝑏)∗ ) 2 1 = ((𝑎, 𝑏) − (𝑎, −𝑏)) 2 1 = (𝑎 − 𝑎, 𝑏 + 𝑏) 2 1 = (0, 2𝑏) 2 =𝑏

by Definition B.5 page 273 by Definition 2.21 page 37 by Definition 2.18 page 31

by Definition B.5 page 273 by Definition 2.21 page 37 by Definition 2.18 page 31

✏ Theorem 2.30. 63 Let |⋅| ∶ ℂ → ℝ⊢ be the absolute value function on ℂ. t 1. |ℜ𝑥| ≤ |𝑥| ∀𝑥∈ℂ h ∀𝑥∈ℂ m 2. |ℑ𝑥| ≤ |𝑥| ✎PROOF: |ℜ(𝑥)| = |ℜ (𝑎, 𝑏)| = |𝑎|

by definition of 𝑥 by Theorem 2.29 page 40

= √𝑎2 ≤ √𝑎2 + 𝑏2 = |(𝑎, 𝑏)|


= |𝑥|


|ℑ(𝑥)| = |ℑ (𝑎, 𝑏)|


= |𝑏|


= √𝑏2 ≤ √𝑎2 + 𝑏2 = |(𝑎, 𝑏)|


= |𝑥|


✏

2.7 Literature 📚 Literature survey: 63

📘 Rudin (1976), page 14


Daniel J. Greenhoe


version 0.30

page 42

2.7. LITERATURE

1. Classic/standard texts: 📘 Dedekind (1872a) 📘 Dedekind (1888a) 📘 Landau (1966) 📘 Thurston (1956) 📘 Birkhoff and MacLane (1996) 2. English translations and reprints of Literature item 1: 📘 Dedekind (1872b) 📘 Dedekind (1872c) 📘 Dedekind (1872d) 📘 Dedekind (1888b) 📘 Dedekind (1888c) 📘 Thurston (2007)

📚

version 0.30


Daniel J. Greenhoe


Part II Order Structures

CHAPTER 3 ORDER

Equivalence relations (Definition 1.9 page 7) require symmetry (𝑥 ≖ 𝑦 ⟺ 𝑦 ≖ 𝑥). However another very important type of relation, the order relation, actually requires anti-symmetry. This chapter presents some useful structures regarding order relations. Ordering relations on a set allow us to compare some pairs of elements in a set and determine whether or not one element is less than another. In this case, we say that those two elements are comparable; otherwise, they are incomparable. A set together with an order relation is called an ordered set, a partially ordered set, or a poset (Definition 3.2 page 46).

3.1 Preordered sets Definition 3.1. 1 Let 𝑋 be a set. A relation ⊑ is a preorder relation on 𝑋 if d 1. 𝑥 ⊑ 𝑥 ∀𝑥∈𝑋 e 2. 𝑥 ⊑ 𝑦 and 𝑦 ⊑ 𝑧 ⟹ 𝑥 ⊑ 𝑧 ∀𝑥,𝑦,𝑧∈𝑋 f A preordered set is the pair (𝑋, ⊑).

(reflexive)

and

(transitive)

Example 3.1. 2 e ⊑ is a preorder relation on the set of positive integers ℕ if x 𝑛⊑𝑚 ⟺ (𝑝 is a prime factor of 𝑛 ⟹ 𝑝 is a prime factor of 𝑚)

1 2

📘 Schröder (2003) page 115, 📘 Brown and Watson (1991), page 317 📘 Shen and Vereshchagin (2002) page 43

page 46

3.2

3.2. ORDER RELATIONS

Order relations

Definition 3.2. 3 Let 𝑋 be a set.Let 𝟚𝑋𝑋 be the set of all relations on 𝑋. A relation ≤ is an order relation in 𝟚𝑋𝑋 if 1. 𝑥 ≤ 𝑥 ∀𝑥∈𝑋 (reflexive) and preorder 2. 𝑥 ≤ 𝑦 and 𝑦 ≤ 𝑧 ⟹ 𝑥 ≤ 𝑧 ∀𝑥,𝑦,𝑧∈𝑋 (transitive) and 3. 𝑥 ≤ 𝑦 and 𝑦 ≤ 𝑥 ⟹ 𝑥 = 𝑦 ∀𝑥,𝑦∈𝑋 (anti-symmetric) d e An ordered set is the pair (𝑋, ≤). The set 𝑋 is called the base set of (𝑋, ≤). If 𝑥 ≤ 𝑦 or 𝑦 ≤ 𝑥, then f elements 𝑥 and 𝑦 are said to be comparable, denoted 𝑥 ∼ 𝑦. Otherwise they are incomparable, denoted 𝑥||𝑦. The relation ⪇ is the relation ≤ ⧵ = (“less than but not equal to”), where⧵is the set difference operator, and = is the equality relation. An order relation is also called a partial order relation. An ordered set is also called a partially ordered set or poset.

The familiar relations ≥, (next) can be defined in terms of the order relation ≤ (Definition 3.2—previous). Definition 3.3. 4 Let (𝑋, ≤) be an ordered set. The relations ≥, ∈ 𝟚𝑋𝑋 are defined as follows: 𝑥≥𝑦

def

𝑦≤𝑥

∀𝑥,𝑦∈𝑋

𝑥 ≤ 𝑦 and 𝑥 ≠ 𝑦

∀𝑥,𝑦∈𝑋

𝑥 ⪈ 𝑦 ⟺ 𝑥 ≥ 𝑦 and 𝑥 ≠ 𝑦 The relation ≥ is called the dual of ≤.

∀𝑥,𝑦∈𝑋

d e f

𝑥⪇𝑦

⟺ def

⟺ def

Theorem 3.1. 5 Let 𝑋 be a set. t h (𝑋, ≤) is an ordered set m

⟺

(𝑋, ≥) is an ordered set

Example 3.2. order relation ≤ (integer less than or equal to) ≥ ⊇ ⊆ (subset) | (divides) ⟹ (implies) ⟸

e x

{𝑥, 𝑦, 𝑧} {𝑥, 𝑦}

{𝑥, 𝑧} {𝑦, 𝑧}

{𝑥}

{𝑧} ∅

dual order relation (integer greater than or equal to) (super set) (divided by) (implied by) {𝑥, 𝑦, 𝑧}

Example 3.3. The Hasse diagram to the left illustrates the ordered set (𝟚{𝑥,𝑦,𝑧} , ⊆) and the Hasse diagram to the right illustrates its dual (𝟚{𝑥,𝑦,𝑧} , ⊇).

{𝑦}

{𝑥, 𝑦}

{𝑥, 𝑧} {𝑦, 𝑧}

{𝑥}

{𝑧} ∅

{𝑦}

3

📘 MacLane and Birkhoff (1999) page 470, 📘 Beran (1985) page 1, 📃 Korselt (1894) page 156 ⟨I, II, (1)⟩, 📃 Dedekind (1900) page 373 ⟨I–III⟩ 4 📃 Peirce (1880b) page 2 5 📘 Grätzer (1998), page 3 version 0.30


Daniel J. Greenhoe


page 47

CHAPTER 3. ORDER

3.3 Linearly ordered sets In an ordered set we can say that some element is less than or equal to some other element. That is, we can say that these two elements are comparable—we can compare them to see which one is lesser or equal to the other. But it is very possible that there are two elements that are not comparable, or incomparable. That is, we cannot say that one element is less than the other—it is simply not possible to compare them because their ordered pair is not an element of the order relation. For example, in the ordered set (𝟚{𝑥,𝑦,𝑧} , ⊆) of Example 3.9, we can say that {𝑥} ⊆ {𝑥, 𝑧} (we can compare these two sets with respect to the order relation ⊆), but we cannot say {𝑦} ⊆ {𝑥, 𝑧}, nor can we say {𝑥, 𝑧} ⊆ {𝑦}. Rather, these two elements {𝑦} and {𝑥, 𝑧} are simply incomparable. However, there are some ordered sets in which every element is comparable with every other element; and in this special case we say that this ordered set is a totally ordered set or is linearly ordered (next definition). Definition 3.4. 6 A relation ≤ is a linear order relation on 𝑋 if 1. ≤ is an order relation (Definition 3.2 page 46) and d e 2. 𝑥 ≤ 𝑦 or 𝑦 ≤ 𝑥 ∀𝑥,𝑦∈𝑋 (comparable). f A linearly ordered set is the pair (𝑋, ≤). A linearly ordered set is also called a totally ordered set, a fully ordered set, and a chain. Definition 3.5 (poset product). 7 The product 𝑃 × 𝑄 of ordered pairs 𝑃 ≜ (𝑋, ⋜) and 𝑄 ≜ (𝑌 , ⊴) is the ordered pair (𝑋 × 𝑌 , ≤) d e where def f ⟺ 𝑥1 ⋜ 𝑥2 and 𝑦1 ⊴ 𝑦2 ∀𝑥1 ,𝑥2 ∈𝑋; 𝑦1 ,𝑦2 ∈𝑌 (𝑥1 , 𝑦1 ) ≤ (𝑥2 , 𝑦2 )

3.4 Representation Definition 3.6. 8 𝑦 covers 𝑥 in the ordered set (𝑋, ≤) if 1. 𝑥 ≤ 𝑦 d e (𝑧 = 𝑥 or 𝑧 = 𝑦) 2. (𝑥 ≤ 𝑧 ≤ 𝑦) ⟹ f The case in which 𝑦 covers 𝑥 is denoted 𝑥 ≺ 𝑦.

(𝑦 is greater than 𝑥)

and

(there is no element between 𝑥 and 𝑦).

Example 3.4. Let ({𝑥, 𝑦, 𝑧}, ≤) be an ordered set with cover relation ≺. 𝑦 covers 𝑥 ⎫ ⎧ ⎪ ⎪ e 𝑥 with a handful of “least upper bound like properties”, and then define an ordering relation ⋜ in terms of > (Definition 4.1 page 59). In fact, Theorem 4.1 (page 59) shows that under Definition 4.1, (𝑋, ⋜) is a partially ordered set and > is a least upper bound on that ordered set. The same development is performed with regards to a greatest lower bound ? with the result that (𝑋, ⋜) is a partially ordered set and ? is a greatest lower bound on that ordered set (Theorem 4.2 page 60). Definition 4.1. 1 Let >, ⋜∶ 𝑋 2 → 𝑋 be binary operators on a set 𝑋. The algebraic structure ( 𝑋, ⋜, >) is a join semilattice if d 1. 𝑥>𝑥 = 𝑥 ∀𝑥∈𝑋 (idempotent) e 2. 𝑥 > 𝑦 = 𝑦 > 𝑥 ∀𝑥,𝑦∈𝑋 (commutative) f 3. (𝑥 > 𝑦) > 𝑧 = 𝑥 > (𝑦 > 𝑧) ∀𝑥,𝑦,𝑧∈𝑋 (associative). Definition 4.2. 2 Let ?, ⋜∶ 𝑋 2 → 𝑋 be binary operators on a set 𝑋. The algebraic structure ( 𝑋, ⋜, ?) is a meet semilattice if d 1. 𝑥?𝑥 = 𝑥 ∀𝑥∈𝑋 (idempotent) e 2. 𝑥 ? 𝑦 = 𝑦 ? 𝑥 ∀𝑥,𝑦∈𝑋 (commutative) f 3. (𝑥 ? 𝑦) ? 𝑧 = 𝑥 ? (𝑦 ? 𝑧) ∀𝑥,𝑦,𝑧∈𝑋 (associative).

and and

and and

Theorem 4.1. 3 Let >, ⋜∶ 𝑋 2 → 𝑋 be binary operators over a set 𝑋. t ( 𝑋, ⋜, >) is a 1. (𝑋, ⋜) is a partially ordered set h ⟹ { } { join semilattice 2. 𝑥 > 𝑦 is a least upper bound of 𝑥 and 𝑦 m ✎PROOF:

and ∀𝑥, 𝑦 ∈ 𝑋.

}

In order for (𝑋, ≤) to be an ordered set, ≤ must be, according to Definition 3.2 (page 46), reflexive, antisymmetric, and transitive; 1 2 3

📘 MacLane and Birkhoff (1999) page 475, 📘 Birkhoff (1967) page 22 📘 MacLane and Birkhoff (1999) page 475 📘 MacLane and Birkhoff (1999) page 475

page 60

4.1. SEMI-LATTICES

Proof that ≤ is reflexive: 𝑥=𝑥>𝑥 ⟺𝑥≤𝑥

by idempotent hypothesis by definition of ≤

⟹ ≤ is reflexive Proof that ≤ is antisymmetric: 𝑥 ≤ 𝑦 and 𝑦 ≤ 𝑥 ⟺ 𝑥 > 𝑦 = 𝑦 and 𝑦 > 𝑥 = 𝑥 ⟹ 𝑥 > 𝑦 = 𝑦 and 𝑥 > 𝑦 = 𝑥

by definition of ≤ by commutative hypothesis

⟹ 𝑥=𝑦 ⟹ ≤ is antisymmetric Proof that ≤ is transitive: 𝑥 ≤ 𝑦 and 𝑦 ≤ 𝑧 ⟺ 𝑥 > 𝑦 = 𝑦 and 𝑦 > 𝑧 = 𝑧

by definition of ≤

⟹ (𝑥 > 𝑦) > 𝑧 = 𝑧 ⟺ 𝑥 > (𝑦 > 𝑧) = 𝑧

by associative hypothesis

⟹ 𝑥>𝑧=𝑧 ⟺ 𝑥≤𝑧 ⟺ ≤ is transitive Proof that 𝑥 > 𝑦 is a lub of 𝑥 and 𝑦: 𝑥>𝑦=𝑦 ⟺ 𝑥≤𝑦 ⟺ 𝑥∨𝑦=𝑦

by definition of ≤ by definition of ∨

⟹ 𝑥>𝑦=𝑥∨𝑦 ⟹ 𝑥 > 𝑦 is the lub of 𝑥 and 𝑦

✏ Theorem 4.2. 4 Let ?, ⋜∶ 𝑋 2 → 𝑋 be binary operators over a set 𝑋. t ( 𝑋, ⋜, ?) is a 1. (𝑋, ⋜) is a partially ordered set h ⟹ { 2. 𝑥 ? 𝑦 is a greatest lower bound of 𝑥 and 𝑦 m { meet semilattice }

and ∀𝑥, 𝑦 ∈ 𝑋.

}

✎PROOF:

In order for (𝑋, ≤) to be an ordered set, ≤ must be, according to Definition 3.2 (page 46), reflexive, antisymmetric, and transitive; Proof that ≤ is reflexive: 𝑥=𝑥?𝑥 ⟺𝑥≤𝑥

by idempotent hypothesis by definition of ≤

⟹ ≤ is reflexive Proof that ≤ is antisymmetric: 𝑥 ≤ 𝑦 and 𝑦 ≤ 𝑥 ⟺ 𝑥 ? 𝑦 = 𝑥 and 𝑦 ? 𝑥 = 𝑦 ⟹ 𝑥 ? 𝑦 = 𝑥 and 𝑥 ? 𝑦 = 𝑦

by definition of ≤ by commutative hypothesis

⟹ 𝑥=𝑦 ⟹ ≤ is antisymmetric 4

📘 MacLane and Birkhoff (1999) page 475

version 0.30


Daniel J. Greenhoe


page 61

CHAPTER 4. LATTICES

Proof that ≤ is transitive: 𝑥 ≤ 𝑦 and 𝑦 ≤ 𝑧 ⟺ 𝑥 ? 𝑦 = 𝑥 and 𝑦 ? 𝑧 = 𝑦


⟹ 𝑥 ? (𝑦 ? 𝑧) = 𝑥 ⟺ (𝑥 ? 𝑦) ? 𝑧 = 𝑥

by associative hypothesis

⟹ 𝑥?𝑧=𝑥 ⟺ 𝑥≤𝑧 ⟺ ≤ is transitive Proof that 𝑥 ? 𝑦 is a glb of 𝑥 and 𝑦: 𝑥?𝑦=𝑥 ⟺ 𝑥≤𝑦 ⟺ 𝑥∧𝑦=𝑥

by definition of ≤ by definition of ∧

⟹ 𝑥?𝑦=𝑥∧𝑦 ⟹ 𝑥 ? 𝑦 is the glb of 𝑥 and 𝑦

✏

4.2 Lattices An ordered set is a set (Definition 1.1 page 4) together with the additional structure of an ordering relation (Definition 3.2 page 46). However, this amount of structure tends to be insufficient to ensure “wellbehaved” mathematical systems. This situation is greatly remedied if every pair of elements in an ordered set (partially or linearly ordered) has both a least upper bound and a greatest lower bound (Definition 3.22 page 58) in the ordered set; in this case, that ordered set is a lattice (next definition). GianCarlo Rota (1932–1999) illustrates the advantage of lattices over simple ordered sets by pointing out that the ordered set of partitions of an integer “is fraught with pathological properties”, while the lattice of partitions of a set “remains to this day rich in pleasant surprises”.5 Further examples of lattices follow in Section 4.3 (page 66). Definition 4.3. 6 An algebraic structure 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) is a lattice if 1. (𝑋, ≤) is an ordered set and d 2. 𝑥, 𝑦 ∈ 𝑋 ⟹ 𝑥 ∨ 𝑦 ∈ 𝑋 and e 3. 𝑥, 𝑦 ∈ 𝑋 ⟹ 𝑥∧𝑦∈𝑋 f ∗ The algebraic structure 𝙇 ≜ ( 𝑋, >, ? ; ≥) is the dual lattice of 𝙇, where > and ? are determined by ≥. The lattice 𝙇 is linear if (𝑋, ≤) is a chain (Definition 3.4 page 47). Definition 4.3 (previous) characterizes lattices in terms of order properties. Under this definition, lattices have an equivalent characterization in terms of algebraic properties. In particular, all lattices have four basic algebraic properties: all lattices are idempotent, commutative, associative, and absorptive. Conversely, any structure that possesses these four properties is a lattice. These results are demonstrated by Theorem 4.3 (next). However, note that the four properties are not independent, as it is possible to prove that any structure 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) that is commutative, associative, and absorptive, is also idempotent (Theorem 4.8 page 68). Thus, when proving that 𝙇 is a lattice, it is only necessary to prove that it is commutative, associative, and absorptive. 5

📃 Rota (1997) page 1440 ⟨(illustration)⟩, 📃 Rota (1964) page 498 ⟨partitions of a set⟩ 📘 MacLane and Birkhoff (1999) page 473, 📘 Birkhoff (1948) page 16, 📃 Ore (1935), 📃 Birkhoff (1933a) page 442, 📘 Maeda and Maeda (1970), page 1 6


Daniel J. Greenhoe


version 0.30

page 62

4.2. LATTICES

Theorem 4.3. 7 ( 𝑋, ∨, ∧ ; ≤) is a lattice ⟺ 𝑥 ∨ 𝑥 = 𝑥 𝑥∧𝑥 = t ⎧ h ⎪ 𝑥∨𝑦 = 𝑦∨𝑥 𝑥∧𝑦 = m ⎨ (𝑥 ∨ 𝑦) ∨ 𝑧 = 𝑥 ∨ (𝑦 ∨ 𝑧) (𝑥 ∧ 𝑦) ∧ 𝑧 = ⎪ 𝑥 ∧ (𝑥 ∨ 𝑦) = ⎩ 𝑥 ∨ (𝑥 ∧ 𝑦) = 𝑥

𝑥 𝑦∧𝑥 𝑥 ∧ (𝑦 ∧ 𝑧) 𝑥

∀𝑥∈𝑋

(idempotent)

and

∀𝑥,𝑦∈𝑋

(commutative)

and

∀𝑥,𝑦,𝑧∈𝑋

(associative)

and

∀𝑥,𝑦∈𝑋

(absorptive).

⎫ ⎪ ⎬ ⎪ ⎭

✎PROOF: 1. Proof that ( 𝑋, ∨, ∧ ; ≤) is a lattice ⟹ 4 properties: These follow directly from the definitions of least upper bound ∨ and greatest lower bound ∧. For the absorptive property, 𝑥 ≤ 𝑦 ⟹ 𝑥 ∨ (𝑥 ∧ 𝑦) = 𝑥 ∨ 𝑥 = 𝑥 𝑦 ≤ 𝑥 ⟹ 𝑥 ∨ (𝑥 ∧ 𝑦) = 𝑥 ∨ 𝑦 = 𝑥 𝑥 ≤ 𝑦 ⟹ 𝑥 ∧ (𝑥 ∨ 𝑦) = 𝑥 ∧ 𝑦 = 𝑥 𝑦 ≤ 𝑥 ⟹ 𝑥 ∧ (𝑥 ∨ 𝑦) = 𝑥 ∧ 𝑥 = 𝑥 2. Proof that ( 𝑋, ∨, ∧ ; ≤) is a lattice ⟸ 4 properties: According to Definition 4.3 (page 61), in order for ( 𝑋, ∨, ∧ ; ≤) to be a lattice, ( 𝑋, ∨, ∧ ; ≤) must be an ordered set, 𝑥 ∨ 𝑦 must be the least upper bound for any 𝑥, 𝑦 ∈ 𝑋 and 𝑥 ∧ 𝑦 must be the greatest lower bound for any 𝑥, 𝑦 ∈ 𝑋. (a) By Theorem 4.1 (page 59), ( 𝑋, ∨, ∧ ; ≤) is an ordered set. (b) By Theorem 4.1 (page 59), 𝑥 ∨ 𝑦 is the least upper bound for any 𝑥, 𝑦 ∈ 𝑋. (c) Proof that 𝑥 ∧ 𝑦 is the greatest lower bound for any 𝑥, 𝑦 ∈ 𝑋: To prove this, we must show that 𝑥 ≤ 𝑦 ⟺ 𝑥 ∧ 𝑦 = 𝑥. Proof that 𝑥 ≤ 𝑦 ⟹ 𝑥 ∧ 𝑦 = 𝑥: 𝑥 = 𝑥 ∧ (𝑥 ∨ 𝑦) =𝑥∧𝑦

by absorptive hypothesis by 𝑥 ≤ 𝑦 hypothesis and definition of ≤

Proof that 𝑥 ≤ 𝑦 ⟸ 𝑥 ∧ 𝑦 = 𝑥: 𝑦 = 𝑦 ∨ (𝑦 ∧ 𝑥)

⟹

by absorptive hypothesis

= 𝑦 ∨ (𝑥 ∧ 𝑦)

by commutative hypothesis

=𝑦∨𝑥

by 𝑥 ∧ 𝑦 = 𝑥 hypothesis

=𝑥∨𝑦

by commutative hypothesis

𝑥≤𝑦


✏ Lemma 4.1. l e 𝑥≤𝑦 m

8

Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice (Definition 4.3 page 61). ⟺

𝑥=𝑥∧𝑦

∀𝑥,𝑦∈𝙇

7

📘 MacLane and Birkhoff (1999) pages 473–475 ⟨LEMMA 1, THEOREM 4⟩, 📘 Burris and Sankappanavar (1981) pages 4–7, 📘 Birkhoff (1938), pages 795–796, 📃 Ore (1935) page 409 ⟨(𝛼)⟩, 📃 Birkhoff (1933a) page 442, 📃 Dedekind (1900) pages 371–372 ⟨(1)–(4)⟩ 8 📘 Holland (1970), page ??? version 0.30


Daniel J. Greenhoe


page 63

CHAPTER 4. LATTICES

✎PROOF: 1. Proof for ⟹ case: by left hypothesis and definition of ∧ (Definition 3.22 page 58). 2. Proof for ⟸ case: by right hypothesis and definition of ∧ (Definition 3.22 page 58).

✏ The identities of Theorem 4.3 (page 62) occur in pairs that are duals of each other. That is, for each identity, if you swap the join and meet operations, you will have the other identity in the pair. Thus, the characterization of lattices provided by Theorem 4.3 (page 62) is called self-dual. And because of this, lattices support the principle of duality (next theorem). Theorem 4.4 (Principle of duality). 9 Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. 𝜙 is an identity on 𝙇 in terms ⟹ 𝐓𝜙 is also an identity on 𝙇 t {of the operations ∨ and ∧ } h m where the operator 𝐓 performs the following mapping on the operations of 𝜙: ∨ → ∧, ∧→∨ ✎PROOF:

For each of the identities in Theorem 4.3 (page 62), the operator 𝐓 produces another identity that is also in the set of identities: 𝐓(1𝑎) 𝐓(1𝑏) 𝐓(2𝑎) 𝐓(2𝑏)

= = = =

𝐓[𝑥 ∨ 𝑦 𝐓[𝑥 ∧ 𝑦 𝐓[𝑥 ∨ (𝑦 ∧ 𝑧) 𝐓[𝑥 ∧ (𝑦 ∨ 𝑧)

= = = =

𝑦 ∨ 𝑥] 𝑦 ∧ 𝑥] (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧)] (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)]

= = = =

[𝑥 ∧ 𝑦 [𝑥 ∨ 𝑦 [𝑥 ∧ (𝑦 ∨ 𝑧) [𝑥 ∨ (𝑦 ∧ 𝑧)

= = = =

𝑦 ∧ 𝑥] 𝑦 ∨ 𝑥] (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)] (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧)]

= = = =

(1𝑏) (1𝑎) (2𝑏) (2𝑎)

Therefore, if the statement 𝜙 is consistent with regards to the lattice 𝙇, then 𝐓𝜙 is also consistent with regards to the lattice 𝙇. ✏

Proposition 4.1 (Monotony laws). 10 Let ( 𝑋, ∨, ∧ ; ≤) be a lattice. p 𝑎 ≤ 𝑏 and 𝑎 ∧ 𝑥 ≤ 𝑏 ∧ 𝑦 and r ⟹ } { 𝑎 ∨ 𝑥 ≤ 𝑏 ∨ 𝑦. p 𝑥 ≤ 𝑦. ✎PROOF: 1.(𝑎 ∧ 𝑥) ≤ 𝑎

by definition of meet operation ∧ Definition 3.22 page 58

≤𝑏

by left hypothesis

2.(𝑎 ∧ 𝑥) ≤ 𝑥

by definition of meet operation ∧ Definition 3.22 page 58

≤𝑦

by left hypothesis

3.(𝑎 ∧ 𝑥) = (𝑎 ∧ 𝑥) ∧ (𝑎 ∧ 𝑥) ⏟ ⏟ ≤𝑏

≤𝑦

≤𝑏∧𝑦

by 1 and 2

4.(𝑎 ∨ 𝑥) = (𝑎 ∨ 𝑥) ∨ (𝑎 ∨ 𝑥) ⏟ ⏟ ≤𝑏

≤𝑏∨𝑦

by idempotent property Theorem 4.3 page 62

by idempotent property Theorem 4.3 page 62

≤𝑦

by 1 and 2

✏ 9 10

📘 Padmanabhan and Rudeanu (2008) pages 7–8, 📘 Beran (1985) pages 29–30 📘 Givant and Halmos (2009) page 39, 📃 Doner and Tarski (1969) pages 97–99


Daniel J. Greenhoe


version 0.30

page 64

4.2. LATTICES

Minimax inequality. Suppose we arrange a finite sequence of values into 𝑚 groups of 𝑛 elements per group. This could be represented as an 𝑚 × 𝑛 matrix. Suppose now we find the minimum value in each row, and the maximum value in each column. We can call the maximum of all the minimum row values the maximin, and the minimum of all the maximum column values the minimax. Now, which is greater, the maximin or the minimax? The minimax inequality demonstrates that the maximin is always less than or equal to the minimax. The minimax inequality is illustrated below and stated formerly in Theorem 4.5 (page 64). 𝑛

𝑚

⎫ ⎧ ⎪ ⋀ { 𝑥11 𝑥12 ⋯ 𝑥1𝑛 } ⎪ ⎪ ⎪ 1 ⎪ ⎪ 𝑛 ⎪ 𝑥 𝑥 ⋯ 𝑥 22 2𝑛 } ⎪ 𝑚 { 21 ⎪ ⎪ ⋀ 1 ⎬ ⋁⎨ 𝑛 1 ⎪ ⎪ ⋮ ⋱ ⋱ ⋮ } ⎪ ⎪ ⋀{ ⎪ ⎪ 1𝑛 ⎪ ⎪ ⎪ ⋀ { 𝑥𝑚1 𝑥𝑚2 ⋯ 𝑥𝑚𝑛 } ⎪ ⎩ 1 ⎭ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝑚

𝑚

𝑚

⎫ ⎧ ⎪ ⋁ ⋁ ⋁ ⋁ ⎪ ⎪ 1 1 1 1 ⎪ ⎪ ⎪ ⎪ 𝑥 𝑥 ⋯ 𝑥 𝑛 11 12 1𝑛 ⎪ ⎪ ⎪ ⋀ ⎨ 𝑥21 𝑥22 ⋯ 𝑥2𝑛 ⎬ 1 ⎪ ⎪ ⎪ ⎪ ⋱ ⋱ ⋮ ⎪ ⎪ ⋮ ⎪ ⎪ ⎪ ⎪ 𝑥 ⎩ 𝑚1 𝑥𝑚2 ⋯ 𝑥𝑚𝑛 ⎭ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

≤

minimax

maximin

Theorem 4.5 (Minimax inequality). 11 Let ( 𝑋, ∨, ∧ ; ≤) be a lattice. 𝑛 𝑚 𝑚 𝑛 t h 𝑥 ∀𝑥𝑖𝑗 ∈ 𝑋 𝑥 ≤ ⋀ ⋁ 𝑖𝑗 ⋁ ⋀ 𝑖𝑗 m 𝑗=1 𝑖=1 𝑖=1 𝑗=1 ⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟ minimax: smallest of the largest

maxmini: largest of the smallest

✎PROOF: 𝑛

𝑛

𝑥𝑖𝑘 (⋀ ) 𝑘=1 ⏟⏟⏟⏟⏟⏟⏟

≤ 𝑥𝑖𝑗 ≤

smallest for any given 𝑖 𝑚 𝑛

⟹

𝑥 ⋁ (⋀ 𝑖𝑘 ) 𝑖=1 𝑘=1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ largest amoung all 𝑖s of the smallest values 𝑛 𝑚

⟹

𝑛

≤

𝑥𝑘𝑗 ) (⋁ 𝑘=1 ⏟⏟⏟⏟⏟⏟⏟

∀𝑖, 𝑗

largest for any given 𝑗 𝑚

𝑥 ⋀ (⋁ 𝑘𝑗 ) 𝑗=1 𝑘=1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ smallest amoung all 𝑗s of the largest values 𝑚 𝑛

𝑥 𝑥 ≤ ⋀ (⋁ 𝑖𝑗 ) ⋁ (⋀ 𝑖𝑗 ) 𝑖=1 𝑗=1 𝑗=1 𝑖=1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ maxmini

(change of variables)

minimax

✏

Distributive inequalities. Special cases of the minimax inequality include three distributive inequalities (next theorem). If for some lattice any one of these inequalities is an equality, then all three are equalities (Theorem 6.1 page 90); and in this case, the lattice is a called a distributive lattice (Definition 6.2 page 89). 11

📘 Birkhoff (1948) pages 19–20

version 0.30


Daniel J. Greenhoe


page 65

CHAPTER 4. LATTICES

Theorem 4.6 (distributive inequalities). 12 ( 𝑋, ∨, ∧ ; ≤) is a lattice ⟹ for all x,y,z X t 𝑥 ∧ (𝑦 ∨ 𝑧) ≥ (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) h 𝑥 ∨ (𝑦 ∧ 𝑧) ≤ (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) m (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ∨ (𝑦 ∧ 𝑧) ≤ (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) ∧ (𝑦 ∨ 𝑧)

(join super-distributive)

and

(meet sub-distributive)

and

(median inequality).

✎PROOF: 1. Proof that ∧ sub-distributes over ∨: (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ≤ (𝑥 ∨ 𝑥) ∧ (𝑦 ∨ 𝑧) = 𝑥 ∧ (𝑦 ∨ 𝑧)

by minimax inequality (Theorem 4.5 page 64) by idempotent property of lattices (Theorem 4.3 page 62)

⎧ ⎪ ⎪ ⋀{ 𝑥 𝑦 } ⎫ ⎬ ⋁⎨ ⋀ 𝑥 𝑧 } ⎪ ⎪ { ⎩ ⎭

≤

⋁ ⎧ ⎪ 𝑥 ⋀⎨ ⎪ ⎩ 𝑥

⋁ 𝑦 𝑧

⎫ ⎪ ⎬ ⎪ ⎭

2. Proof that ∨ super-distributes over ∧: 𝑥 ∨ (𝑦 ∧ 𝑧) = (𝑥 ∧ 𝑥) ∨ (𝑦 ∧ 𝑧) ≤ (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧)

by idempotent property of lattices (Theorem 4.3 page 62) by minimax inequality (Theorem 4.5 page 64)

⎧ ⎪ ⎪ ⋀{ 𝑥 𝑥 } ⎫ ⎬ ⋁⎨ ⋀ ⎪ { 𝑦 𝑧 } ⎪ ⎩ ⎭

≤

⋁ ⎧ ⎪ 𝑥 ⋀⎨ ⎪ ⎩ 𝑦

⋁ 𝑥 𝑧

⎫ ⎪ ⎬ ⎪ ⎭

3. Proof that of median inequality: by minimax inequality (Theorem 4.5 page 64)

✏

Modular inequalities. Besides the distributive property, another consequence of the minimax inequality is the modularity inequality (next theorem). A lattice in which this inequality becomes equality is said to be modular (Definition 5.3 page 80). Theorem 4.7 (Modular inequality). 13 Let ( 𝑋, ∨, ∧ ; ≤) be a lattice (Definition 4.3 page 61). t h 𝑥≤𝑦 ⟹ 𝑥 ∨ (𝑦 ∧ 𝑧) ≤ 𝑦 ∧ (𝑥 ∨ 𝑧) m ✎PROOF: 𝑥 ∨ (𝑦 ∧ 𝑧) = (𝑥 ∧ 𝑥) ∨ (𝑦 ∧ 𝑧)

by absorptive property (Theorem 4.3 page 62)

≤ (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧)

by the minimax inequality (Theorem 4.5 page 64)

= 𝑦 ∧ (𝑥 ∨ 𝑧)

by left hypothesis

⎧ ⎪ ⋀{ 𝑥 𝑥 } ⎫ ⎪ ⎬ ⋁⎨ ⋀ ⎪ { 𝑦 𝑧 } ⎪ ⎩ ⎭

≤

⋁ ⎧ ⎪ 𝑥 ⋀⎨ ⎪ ⎩ 𝑦

⋁ 𝑥 𝑧

⎫ ⎪ ⎬ ⎪ ⎭

✏ 12

📘 Davey and Priestley (2002) page 85, 📘 Grätzer (2003) page 38, 📃 Birkhoff (1933a) page 444, 📃 Korselt (1894) page 157, 📘 Müller-Olm (1997) page 13 ⟨terminology⟩ 13 📘 Birkhoff (1948) page 19, 📘 Burris and Sankappanavar (1981) page 11, 📃 Dedekind (1900) page 374 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 66

4.3

4.3. EXAMPLES

Examples

Example 4.1. 14 the ordered set illustrated to the right is not a lattice because, for example, while 𝑥 and 𝑦 have upper bounds 𝑎, 𝑏, and 1, 𝑥 and 𝑦 have no least upper bound. Obviously 1 is not the least upper bound because 𝑎 ≤ 1 and 𝑏 ≤ 1. And neither 𝑎 nor 𝑏 is a least upper bound because 𝑎 ≰ 𝑏 and 𝑏 ≰ 𝑎; rather, 𝑎 and 𝑏 are incomparable (𝑎||𝑏). Note that if we remove either or both of the two lines crossing the center, the ordered set becomes a lattice.

𝑎 𝑥

Example 4.2 (Discrete lattice). Let 𝟚𝐴 be the power set of a set 𝐴, ⊆ the set inclusion relation, ∪ the set union operation, and ∩ the set intersection operation. Then the tupple ( 𝟚{𝑥,𝑦,𝑧} , ∪, ∩ ; ⊆) is a lattice.

{𝑥, 𝑦, 𝑧}

Examples of least upper bounds

Examples of greatest lower bounds

1 𝑏 𝑦 0

{𝑥, 𝑧}

{𝑥, 𝑦}

{𝑦, 𝑧}

{𝑥}

{𝑧}

{𝑥} ∪ {𝑧} {𝑥} ∩ {𝑧} = {𝑥, 𝑧} = ∅ {𝑥, 𝑦} ∪ {𝑦} {𝑥, 𝑦} ∩ {𝑦} = {𝑥, 𝑦} = {𝑦} {𝑥, 𝑧} ∪ {𝑦, 𝑧} = {𝑥, 𝑦, 𝑧} {𝑥, 𝑧} ∩ {𝑦, 𝑧} = {𝑧}

{𝑦}

∅

Example 4.3 (Integer factor lattice). 15 For any pair of natural numbers 𝑛, 𝑚 ∈ ℕ, let 𝑛|𝑚 represent the relation “𝑚 divides 𝑛”, lcm(𝑛, 𝑚) the least common multiple of 𝑛 and 𝑚, and gcd(𝑛, 𝑚) the greatest common divisor of 𝑛 and 𝑚. e x ( {1, 2, 3, 5, 6, 10, 15, 30}, gcd, lcm ; |) is a lattice.

30 6

10

15

2

3

5

1 4 3 2 1

Example 4.4 (Linear lattice). Let ≤ be the standard counting ordering relation on the set of integers; and for any pair of integers 𝑛, 𝑚 ∈ ℕ, let max(𝑛, 𝑚) be the maximum of 𝑛 and 𝑚, and min(𝑛, 𝑚) be the minimum of 𝑛 and 𝑚. Then the tupple ( {1, 2, 3, 4}, max, min ; ≤) is a lattice.

𝙑5

Example 4.5 (Subspace lattices). 16 Let ⦅𝙑𝑛 ⦆ be a sequence of subspaces, ⊆ be the set inclusion relation, + the subspace addition operator, and ∩ the set intersection operator. Then the tuple ( {𝙑𝑛 }, +, ∩ ; ⊆) is a lattice.

𝙑4 𝙑2 𝙑3

𝙑1 𝟬

Example 4.6 (Projection operator lattices). tion operators in a Hilbert space 𝑋. ( {𝐏𝑛 }, ∨, ∧ ; ≤) is a lattice e x

where 𝐏1 ≤ 𝐏2 𝐏1 ∨ 𝐏2 𝐏1 ∧ 𝐏2

def

⟺ = =

15 16 17 18

Let ⦅𝐏𝑛 ⦆ be a sequence of projec-

𝐏5 𝐏4 𝐏2

𝐏1 𝐏2 = 𝐏1 𝐏2 = 𝐏1 𝐏1 + 𝐏2 − 𝐏1 𝐏2 𝐏1 𝐏2

Example 4.7 (Lattice of a single topology). 14

17

18

𝐏3

𝐏1 0

Let 𝑋 be a set, 𝜏 a topology on 𝑋, ⊆ the set inclusion

📘 Oxley (2006) page 54, 📘 Farley (1997), page 3, 📘 Farley (1996), page 5, 📘 Birkhoff (1967) pages 15–16 📘 MacLane and Birkhoff (1999) page 484, 📘 Sheffer (1920) page 310 ⟨footnote 1⟩ 📘 Isham (1999) pages 21–22 📘 Isham (1999) pages 21–22, 📘 Dunford and Schwartz (1957), pages 481–482 📘 Burris and Sankappanavar (1981) page 9, 📘 Birkhoff (1936a) page 161

version 0.30


Daniel J. Greenhoe


page 67

CHAPTER 4. LATTICES

relation, ∪ the set union operator, and ∩ the set intersection operator. Then the tuple ( 𝜏, ∪, ∩ ; ⊆) is a lattice. 𝟚{𝑥,𝑦} ≜ { ∅, {𝑥, 𝑦}, {𝑥}, {𝑦}} Example 4.8 (Lattice of topologies). 19 Let 𝑋 be a set and {𝜏1 , 𝜏2 , 𝜏3 , …} all the possible topologies on 𝑋. Let ⊆ be the set inclusion relation, ∪ the set union operator, and ∩ the set { ∅, {𝑥, 𝑦}, {𝑥}} { ∅, {𝑥, 𝑦}, {𝑦}} intersection operator. Then the tuple ( {(𝑋, 𝜏𝑛 )}, ∪, ∩ ; ⊆) is { ∅, {𝑥, 𝑦}} a lattice.

Proposition 4.2. 20 Let 𝑋𝑛 be a finite set with order 𝑛 = | 𝑋𝑛 |. Let 𝐿𝑛 be the number of labeled lattices on 𝑋𝑛 , 𝑙𝑛 the number of unlabeled lattices, and 𝑝𝑛 the number of unlabeled posets. 𝑛 0 1 2 3 4 5 6 7 8 9 10 p 𝐿 1 1 2 6 36 380 6390 157962 5396888 243, 179, 064 13, 938, 711, 210 𝑛 r 𝑙 1 1 1 1 2 5 15 53 222 1078 5994 p 𝑛 𝑝𝑛 1 1 2 5 16 63 318 2045 16, 999 183, 231 2, 567, 284

Example 4.9 (lattices on 1–3 element sets). 21 There is only one unlabeled lattice for finite sets with 3 or fewer elements (Proposition 4.2 page 67). Thus, these lattices are all linearly ordered. These 3 lattices are illustrated to the right.

lattices on 1, 2, and 3 element sets

lattices on 4 element sets 22

Example 4.10 (lattices on a 4 element set). There are 2 unlabeled lattices on a 4 element set (Proposition 4.2 page 67). These are illustrated to the right.

lattices on 5 element sets 23

Example 4.11 (lattices on a 5 element set). There are 5 unlabeled lattices on a 5 element set (Proposition 4.2 page 67). These are illustrated to the right.

Example 4.12 (lattices on a 6 element set). 24 There are 15 unlabeled lattices on a 6 element set (Proposition 4.2 page 67). These are illustrated in the following table. Notice that the lattices in the second row are simply generated from the 5 element lattices (Example 4.11 page 67) with a “head” or “tail” added to each one. 19

📘 Isham (1999) page 44, 📘 Isham (1989), page 1515 Sloane (2014) ⟨http://oeis.org/A055512⟩, 💻 Sloane (2014) ⟨http://oeis.org/A006966⟩, 💻 Sloane (2014) ⟨http://oeis.org/A000112⟩, 📘 Heitzig and Reinhold (2002) 21 📘 Kyuno (1979), page 412, 📘 Stanley (1997), page 102 22 📘 Kyuno (1979), page 412, 📘 Stanley (1997), page 102 23 📘 Kyuno (1979), page 413, 📘 Stanley (1997), page 102 24 📘 Kyuno (1979), page 413, 📘 Stanley (1997), page 102 20

💻


Daniel J. Greenhoe


version 0.30

page 68

4.4. CHARACTERIZATIONS

lattices on 6 element sets self-dual

non-self dual

Example 4.13 (lattices on a 7 element set). 25 There are 53 unlabeled lattices on a 7 element set (Proposition 4.2 page 67). These are illustrated in Figure 4.1 (page 69). Example 4.14 (lattices on 8 element sets). There are 222 unlabeled lattices on a 8 element set (PropoSee Kyuno's paper26 for Hasse diagrams of all 222 lattices.

sition 4.2 page 67).

4.4

Characterizations

Theorem 4.3 (page 62)gave eight equations in three variables and two operators that are true of all lattices. But the converse is also true: that is, if the eight equations of Theorem 4.3 are true for all values of the underlying set, then that set together with the two operators are a lattice. That is, the eight equations in three variables of Theorem 4.3 characterize lattices, or serve as an equational basis for lattices.27 And this is not the only system of equations that characterize a lattice. There are other systems that use fewer equations in more variables. Here are some examples of lattice characterizations: 8 equations in 3 variables Theorem 4.3 page 62 6 equations in 3 variables Theorem 4.8 page 68 2 equations in 5 variables Theorem 4.9 page 70 1 equation in 8 variables with length 29 Theorem 4.10 page 70 1 equation in 7 variables with length 79 Theorem 4.10 page 70 Since these characterizations are equivalent to the definition of the lattice, we could in fact change things around and essentially make any of these characterizations into the definition, and make the definition into a theorem.28 Theorem 4.3 (page 62) gave 4 necessary and sufficient pairs of properties for a structure ( 𝑋, ∨, ∧ ; ≤) to be a lattice. However, these 4 pairs are actually overly sufficient (they are not independent), as demonstrated next. Theorem 4.8. 29 ( 𝑋, ∨, ∧ ; ≤) is a lattice ⟺ t ⎧ 𝑥∨𝑦 = 𝑦∨𝑥 𝑥∧𝑦 = 𝑦∧𝑥 h ⎪ m ⎨ (𝑥 ∨ 𝑦) ∨ 𝑧 = 𝑥 ∨ (𝑦 ∨ 𝑧) (𝑥 ∧ 𝑦) ∧ 𝑧 = 𝑥 ∧ (𝑦 ∧ 𝑧) ⎪ 𝑥 ∧ (𝑥 ∨ 𝑦) = 𝑥 ⎩ 𝑥 ∨ (𝑥 ∧ 𝑦) = 𝑥 25 26 27 28 29

∀𝑥,𝑦∈𝑋

(commutative)

and


(associative)

and

∀𝑥,𝑦∈𝑋

(absorptive)

⎫ ⎪ ⎬ ⎪ ⎭

📘 Kyuno (1979), pages 413–414 📘 Kyuno (1979), pages 415–421 📘 McKenzie (1970) page 24, 📘 Tarski (1966) 📘 Burris and Sankappanavar (1981) pages 6–7, 📘 Padmanabhan and Rudeanu (2008) page 8, 📘 Beran (1985) page 5, 📘 McKenzie (1970) page 24

version 0.30


Daniel J. Greenhoe


page 69

CHAPTER 4. LATTICES

self dual 7 element lattices (13 lattices)

non-self dual 7 element lattices — first half (20 lattices)

non-self dual 7 element lattices — duals of first half (20 lattices)

Figure 4.1: The 53 unlabeled lattices on a 7 element set (Example 4.13 page 68)


Daniel J. Greenhoe


version 0.30

page 70 ✎PROOF:

4.5. BOUNDED LATTICES

Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤).

1. Proof that 𝙇 is a lattice ⟹ 3 properties: by Theorem 4.3 page 62 2. Proof that 𝙇 is a lattice ⟸ 3 properties: (a) Proof that 3 properties ⟹ 𝙇 is idempotent: 𝑥 ∨ 𝑥 = 𝑥 ∨ [𝑥 ∧ (𝑥 ∨ 𝑦)]

by absorptive property

= 𝑥 ∨ [𝑥 ∧ 𝑧]

where 𝑧 ≜ 𝑥 ∨ 𝑦

=𝑥


𝑥 ∧ 𝑥 = 𝑥 ∧ [𝑥 ∨ (𝑥 ∧ 𝑦)]


= 𝑥 ∧ [𝑥 ∨ 𝑧]

where 𝑧 ≜ 𝑥 ∧ 𝑦

=𝑥


(b) By Theorem 4.3 page 62 and because 𝙇 is commutative, associative, absorptive, and idempotent with respect to ∨ and ∧, 𝙇 is a lattice.

✏ Theorem 4.9 (Lattice characterization in 2 equations and 5 variables). 30 Let 𝑋 be a set and ∨ and ∧ be two binary operators on 𝑋. t (𝑋, ≤, ∨, ∧) is a lattice if and only if h 𝑥 = (𝑥 ∧ 𝑦) ∨ 𝑥 ∀𝑥,𝑦∈𝑋 and m [(𝑥 ∧ 𝑦) ∧ 𝑧 ∨ 𝑢] ∨ 𝑤 = [(𝑦 ∧ 𝑧) ∧ 𝑥 ∨ 𝑤] ∨ (𝑦 ∨ 𝑢) ∧ 𝑢 ∀𝑥,𝑦,𝑧,𝑢,𝑤∈𝑋 Theorem 4.10 (Lattice characterizations in 1 equation). 31 Let 𝑋 be a set and ∨ and ∧ be two binary operators on 𝑋. The following four statements are all equivalent: 1. ( 𝑋, ∨, ∧ ; ≤) is a lattice 2. (((𝑦 ∨ 𝑥) ∧ 𝑥) ∨ (((𝑧 ∧ (𝑥 ∨ 𝑥)) ∨ (𝑢 ∧ 𝑥)) ∧ 𝑣)) ∧ (𝑤 ∨ ((𝑠 ∨ 𝑥) ∧ (𝑥 ∨ 𝑡))) = 𝑥 (1 equation, 8 variables, length 29) ∀𝑥,𝑦,𝑧,𝑢,𝑣,𝑤,𝑠,𝑡∈𝑋 3. (((𝑦 ∨ 𝑥) ∧ 𝑥) ∨ (((𝑧 ∧ (𝑥 ∨ 𝑥)) ∨ (𝑢 ∧ 𝑥)) ∧ 𝑣)) ∧ (((𝑤 ∨ 𝑥) ∧ (𝑠 ∨ 𝑥)) ∨ 𝑡) = 𝑥 t h ∀𝑥,𝑦,𝑧,𝑢,𝑣,𝑤,𝑠,𝑡∈𝑋 (1 equation, 8 variables, length 29) m 4. (((𝑥 ∧ 𝑦) ∨ (𝑦 ∧ (𝑥 ∨ 𝑦))) ∧ 𝑧) ∨ (((𝑥 ∧ (((𝑥1 ∧ 𝑦) ∨ (𝑦 ∧ 𝑥2 ))∨ 𝑦)) ∨ (((𝑦 ∧ (((𝑥1 ∨ (𝑦 ∨ 𝑥2 )) ∧ (𝑥3 ∨ 𝑦)) ∧ 𝑦)) ∨ (𝑢 ∧ (𝑦∨ (((𝑥1 ∨ (𝑦 ∨ 𝑥2 )) ∧ (𝑥3 ∨ 𝑦)) ∧ 𝑦)))) ∧ (𝑥 ∨ (((𝑥1 ∧ 𝑦) ∨ (𝑦∧ 𝑥2 )) ∨ 𝑦)))) ∧ (((𝑥 ∧ 𝑦) ∨ (𝑦 ∧ (𝑥 ∨ 𝑦))) ∨ 𝑧)) = 𝑦 (1 equation, 7 variables, length 79) ∀𝑥,𝑦,𝑧,𝑥1 ,𝑥2 ,𝑥3 ,𝑢∈𝑋

4.5

Bounded lattices

Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. By the definition of a lattice (Definition 4.3 page 61), the upper bound (𝑥 ∨ 𝑦) and lower bound (𝑥 ∧ 𝑦) of any two elements in 𝑋 is also in 𝑋. But what about the upper and lower bounds of the entire set 𝑋 (⋁ 𝑋 and ⋀ 𝑋)32 ? If both of these are in 𝑋, then the lattice 𝙇 is said 30

📘 Tamura (1975) page 137 📘 McCune et al. (2003b) page 2, 📘 McCune et al. (2003a), 📘 McCune and Padmanabhan (1996) page 144, http: //www.cs.unm.edu/%7Everoff/LT/ 32 ⋁ 𝑋: Definition 3.21 page 58, ⋀ 𝑋:Definition 3.22 (page 58) 31

version 0.30


Daniel J. Greenhoe


page 71

CHAPTER 4. LATTICES

to be bounded (next definition). All finite lattices are bounded (next proposition). However, not all lattices are bounded—for example, the lattice (ℤ, ≤) (the lattice of integers with the standard integer ordering relation) is unbounded. Proposition 4.4 (page 71) gives two properties of bounded lattices. Boundedness is one of the “classic 10” properties (Theorem 8.2 page 120) of Boolean algebras (Definition 8.1 ′ page 115). Conversely, a bounded and complemented lattice that satisfies the conditions 1 = 0 and Elkan's law is a Boolean algebra (Proposition 8.4 page 131). Definition 4.4. Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. Let ⋁ 𝑋 be the least upper bound of (𝑋, ≤) and let ⋀ 𝑋 be the greatest lower bound of (𝑋, ≤). 𝙇 is upper bounded if (⋁ 𝑋 ) ∈ 𝑋. d 𝙇 is lower bounded if (⋀ 𝑋 ) ∈ 𝑋. e 𝙇 is bounded if 𝙇 is both upper and lower bounded. f A bounded lattice is optionally denoted ( 𝑋, ∨, ∧, 0, 1 ; ≤), where 0 ≜ ⋀ 𝑋 and 1 ≜ ⋁ 𝑋. Proposition 4.3. Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. p r 𝙇 is finite ⟹ 𝙇 is bounded p Proposition 4.4. Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice with ⋁ 𝑋 ≜ 1 and ⋀ 𝑋 ≜ 0. ⎧ 𝑥 ∨ 1 = 1 ∀𝑥∈𝑋 (upper bounded) and p ⎪ 𝑥 ∧ 0 = 0 ∀𝑥∈𝑋 (lower bounded) and r 𝙇 is bounded ⟹ ⎨ 𝑥 ∨ 0 = 𝑥 ∀𝑥∈𝑋 (join-identity) and p ⎪ ⎩ 𝑥 ∧ 1 = 𝑥 ∀𝑥∈𝑋 (meet-identity) ✎PROOF: 𝑥 ∨ 1 = 𝑥 ∨ ( 𝑋) ⋁ = 𝑋 ⋁ =1

by definition of 1 (Definition 4.4 page 71)

𝑥 ∧ 0 = 𝑥 ∧ ( 𝑋) ⋀ = 𝑋 ⋀ =0


𝑥 =

⋁

because 𝑥 ∈ 𝑋 by definition of 1 (Definition 4.4 page 71)

because 𝑥 ∈ 𝑋 by definition of 0 (Definition 4.4 page 71)

{𝑥}

{𝑥, 0} ⋁ = 𝑥∨0

because {𝑥} ⊆ {0, 𝑥} and isotone property (Proposition 3.3 page 58)

= 𝑥 ∨ ( 𝑋) ⋀


≤ 𝑥 ∨ ( {𝑥}) ⋀

because {𝑥} ⊆ 𝑋 and isotone property (Proposition 3.3 page 58)

≤ 𝑥 ∨ ( {𝑥, 𝑥}) ⋀ = 𝑥 ∨ (𝑥 ∧ 𝑥)

by definition of {⋅}

= 𝑥

by absorptive property of lattices (Theorem 4.3 page 62)

= 𝑥 ∧ (𝑥 ∨ 𝑥)


≜ 𝑥 ∧ ( {𝑥, 𝑥}) ⋁

by definition of ∨ (Definition 3.21 page 58)

≜ 𝑥 ∧ ( {𝑥}) ⋁

by definition of set {⋅}

≤ 𝑥 ∧ ( 𝑋) ⋁

because {𝑥} ⊆ {𝑥, 1} and by isotone property of

≤



by definition of ∧ (Definition 3.22 page 58)

Daniel J. Greenhoe

⋀

(Proposition 3.3 page 58)


version 0.30

page 72

4.6. FUNCTIONS ON LATTICES

= 𝑥∧1 = ≤

⋀

⋀ = 𝑥


{𝑥, 1}


{𝑥}

because {𝑥} ⊆ {𝑥, 1} and by isotone property of

⋀

(Proposition 3.3 page 58)

✏ Definition 4.5. 33 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice (Definition 4.4 page 71). The height 𝗁(𝑥) of a point 𝑥 ∈ 𝙇 is the least upper bound of the lengths (Definition 3.12 page 54) of all d the chains that have 0 and in which 𝑥 is the least upper bound. The height 𝗁(𝙇) of the lattice e f 𝙇 is defined as 𝗁(𝙇) ≜ 𝗁(1) .

4.6

Functions on lattices

4.6.1

Isomorphisms

Lattices and ordered set (Definition 3.2 page 46) are examples of mathematical order structures. Often we are interested in similarities between two lattices 𝙇1 and 𝙇2 with respect to order. Similarities between lattices can be described by defining a function 𝜃 that maps from the first lattice to the second. The degree of similarity can be roughly described in terms of the mapping 𝜃 as follows: 1. If there exists a mapping that is bijective (Definition 15.11 page 252) then the number of elements in 𝙇1 and 𝙇2 is the same. However, their order structure may still be very different. 2. Lattices 𝙇1 and 𝙇2 are more similar if there exists a mapping that is bijective and order preserving (Definition 3.9 page 51). Despite having this property however, their order structure may still be remarkably different, as illustrated by Example 3.18 (page 53) and Example 3.19 (page 53). 3. Lattices 𝙇1 and 𝙇2 are essentially identical (except possibly for their labeling) if there exists a mapping 𝜃 that is not only bijective and order preserving , but whose inverse (Definition 15.2 page 241) is also bijective (Theorem 4.11 page 72). In this case, the lattices 𝙇1 and 𝙇2 are isomorphic and the mapping 𝜃 is an isomorphism. An isomorphism between 𝙇1 and 𝙇2 implies that the two lattices have an identical order structure. In particular, the isomorphism 𝜃 preserves joins and meets (next definition). Definition 4.6. Let 𝙇1 ≜ ( 𝑋, ∨, ∧ ; ≤) and 𝙇2 ≜ ( 𝑌 , >, ? ; ⋜) be lattices. 𝙇1 and 𝙇2 are algebraically isomorphic, or simply isomorphic, if there exists a function 𝜃 ∈ 𝑌 𝑋 such that 1. 𝜃(𝑥 ∨ 𝑦) = 𝜃(𝑥) > 𝜃(𝑦) ∀𝑥, 𝑦 ∈ 𝑋 (preserves joins) and d 2. 𝜃(𝑥 ∧ 𝑦) = 𝜃(𝑥) ? 𝜃(𝑦) ∀𝑥, 𝑦 ∈ 𝑋 (preserves meets). e f In this case, the function 𝜃 is said to be an isomorphism from 𝙇 to 𝙇 , and the isomorphic 1 2 relationship between 𝙇1 and 𝙇2 is denoted as 𝙇1 ≡ 𝙇 2 . Theorem 4.11. 34 Let ( 𝑋, ∨, ∧ ; ≤) and ( 𝑌 , >, ? ; ⋜) be lattices and 𝜃 ∈ 𝑌 𝑋 be a bijective function with inverse 𝜃 −1 ∈ 𝑋 𝑌 . Let ( 𝑋, ∨, ∧ ; ≤) ≡ ( 𝑌 , >, ? ; ⋜) represent the condition that the two lattices 33 34

📘 Birkhoff (1967) page 5 📘 Burris and Sankappanavar (2000), page 10

version 0.30


Daniel J. Greenhoe


page 73

CHAPTER 4. LATTICES

are isomorphic. t 𝑥1 ≤ 𝑥2 ⟹ 𝜃(𝑥1 ) ⋜ 𝜃(𝑥2 ) ∀𝑥1 ,𝑥2 ∈𝑋 h ( 𝑋, ∨, ∧ ; ≤) ≡ ( 𝑌 , >, ? ; ⋜) ⟺ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ −1 −1 𝑦1 ⋜ 𝑦2 ⟹ 𝜃 (𝑦1 ) ⋜ 𝜃 (𝑦2 ) ∀𝑦1 ,𝑦2 ∈𝑌 } m ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ isomorphic

𝜃 and 𝜃 −1 are order preserving with respect to ≤ and ⋜35

✎PROOF:

Let 𝜃 ∈ 𝑌 𝑋 be the isomorphism between lattices ( 𝑋, ∨, ∧ ; ≤) and ( 𝑌 , >, ? ; ⋜).

1. Proof that order preserving ⟹ preserves joins: (a) Proof that 𝜃(𝑥1 ∨ 𝑥2 ) ⧁ 𝜃(𝑥1 ) > 𝜃(𝑥2 ): i. Note that 𝑥1 ≤ 𝑥1 ∨ 𝑥2 𝑥2 ≤ 𝑥1 ∨ 𝑥2 . ii. Because 𝜃 is order preserving 𝜃(𝑥1 ) ⋜ 𝜃(𝑥1 ∨ 𝑥2 ) 𝜃(𝑥2 ) ⋜ 𝜃(𝑥1 ∨ 𝑥2 ). iii. We can then finish the proof of item (1a): 𝜃(𝑥1 ) > 𝜃(𝑥2 ) ⋜ 𝜃(𝑥 ⏟⏟⏟⏟⏟ 1 ∨ 𝑥2 ) ⏟⏟⏟⏟⏟ 1 ∨ 𝑥2 ) > 𝜃(𝑥 𝑥1 ≤ 𝑥1 ∨ 𝑥2

by order preserving hypothesis

𝑥2 ≤ 𝑥 1 ∨ 𝑥 2

= 𝜃(𝑥1 ∨ 𝑥2 )

by idempotent property page 62

(b) Proof that 𝜃(𝑥1 ∨ 𝑥2 ) ⋜ 𝜃(𝑥1 ) > 𝜃(𝑥2 ): i. Just as in item (1a), note that 𝜃 −1 (𝑦1 ) ∨ 𝜃 −1 (𝑦2 ) ≤ 𝜃 −1 (𝑦1 > 𝑦2 ): −1 −1 𝜃 −1 (𝑦1 ) ∨ 𝜃 −1 (𝑦2 ) ≤ 𝜃⏟⏟⏟⏟⏟⏟⏟⏟⏟ (𝑦1 > 𝑦2 ) ∨ 𝜃⏟⏟⏟⏟⏟⏟⏟⏟⏟ (𝑦1 > 𝑦2 ) 𝑦1 ⋜ 𝑦1 > 𝑦2


𝑦2 ⋜ 𝑦1 > 𝑦2

= 𝜃 −1 (𝑦1 > 𝑦2 )


ii. Because 𝜃 is order preserving 𝜃 [𝜃 −1 (𝑦1 ) ∨ 𝜃 −1 (𝑦2 )] ⋜ 𝜃𝜃 −1 (𝑦1 > 𝑦2 )

by item (1(b)i) page 73

= 𝑦 1 > 𝑦2

by definition of inverse function 𝜃 −1

iii. Let 𝑢1 ≜ 𝜃(𝑥1 ) and 𝑢2 ≜ 𝜃(𝑥2 ). iv. We can then finish the proof of item (1b): 𝜃(𝑥1 ∨ 𝑥2 ) = 𝜃 [𝜃 −1 𝜃(𝑥1 ) ∨ 𝜃 −1 𝜃(𝑥2 )]

by definition of inverse function 𝜃 −1

= 𝜃 [𝜃 −1 (𝑢1 ) ∨ 𝜃 −1 (𝑢2 )]

by definition of 𝑢1 , 𝑢2 , item (1(b)iii)

⋜ 𝑢 1 > 𝑢2

by item (1(b)ii)

= 𝜃(𝑥1 ) > 𝜃(𝑥2 )

by definition of 𝑢1 , 𝑢2 , item (1(b)iii)

(c) And so, combining item (1a) and item (1b), we have 𝜃(𝑥1 ∨ 𝑥2 ) ⧁ 𝜃(𝑥1 ) > 𝜃(𝑥2 ) 𝜃(𝑥1 ∨ 𝑥2 ) ⋜ 𝜃(𝑥1 ) > 𝜃(𝑥2 ) 35

(item (1a) page 73) (item (1b) page 73)

and

}

⟹

𝜃(𝑥1 ∨ 𝑥2 ) = 𝜃(𝑥1 ) > 𝜃(𝑥2 )

order preserving : Definition 3.9 page 51


Daniel J. Greenhoe


version 0.30

page 74

4.6. FUNCTIONS ON LATTICES

2. Proof that order preserving ⟹ preserves meets: (a) Proof that 𝜃(𝑥1 ∧ 𝑥2 ) ⋜ 𝜃(𝑥1 ) ? 𝜃(𝑥2 ): 𝜃(𝑥1 ) ? 𝜃(𝑥2 ) ⧁ 𝜃(𝑥 ⏟⏟⏟⏟⏟ 1 ∧ 𝑥2 ) ? 𝜃(𝑥 ⏟⏟⏟⏟⏟ 1 ∧ 𝑥2 ) 𝑥1 ≥ 𝑥1 ∧ 𝑥2


𝑥2 ≥ 𝑥 1 ∧ 𝑥 2

= 𝜃(𝑥1 ∧ 𝑥2 )


(b) Proof that 𝜃(𝑥1 ∧ 𝑥2 ) ⧁ 𝜃(𝑥1 ) ? 𝜃(𝑥2 ): i. Just as in item (2a), note that 𝜃 −1 (𝑦1 ) ∧ 𝜃 −1 (𝑦2 ) ≥ 𝜃 −1 (𝑦1 ? 𝑦2 ): −1 −1 𝜃 −1 (𝑦1 ) ∧ 𝜃 −1 (𝑦2 ) ≥ 𝜃⏟⏟⏟⏟⏟⏟⏟⏟⏟ (𝑦1 ? 𝑦2 ) ? 𝜃⏟⏟⏟⏟⏟⏟⏟⏟⏟ (𝑦1 ? 𝑦2 ) 𝑦1 ⧁ 𝑦1 ? 𝑦2


𝑦2 ⧁ 𝑦1 ? 𝑦2

= 𝜃 (𝑦1 ? 𝑦2 ) −1


ii. Because 𝜃 is order preserving 𝜃 [𝜃 −1 (𝑦1 ) ∧ 𝜃 −1 (𝑦2 )] ⧁ 𝜃𝜃 −1 (𝑦1 ? 𝑦2 )

by item (2(b)i)

= 𝑦 1 ? 𝑦2 iii. Let 𝑣1 ≜ 𝜃(𝑥1 ) and 𝑣2 ≜ 𝜃(𝑥2 ). iv. We can then finish the proof of item (2a): 𝜃(𝑥1 ∧ 𝑥2 ) = 𝜃 [𝜃 −1 𝜃(𝑥1 ) ∧ 𝜃 −1 𝜃(𝑥2 )] = 𝜃 [𝜃 −1 (𝑣1 ) ∧ 𝜃 −1 (𝑣2 )]

by item (2(b)iii)

⧁ 𝑣1 ? 𝑣2

by item (2(b)ii)

= 𝜃(𝑥1 ) ? 𝜃(𝑥2 )

by item (2(b)iii)

(c) And so, combining item (2a) and item (2b), we have 𝜃(𝑥1 ∧ 𝑥2 ) ⋜ 𝜃(𝑥1 ) ? 𝜃(𝑥2 ) 𝜃(𝑥1 ∧ 𝑥2 ) ⧁ 𝜃(𝑥1 ) ? 𝜃(𝑥2 )

(item (2a) page 74)

and

(item (2b) page 74)

}

𝜃(𝑥1 ∧ 𝑥2 ) = 𝜃(𝑥1 ) ? 𝜃(𝑥2 )

⟹

3. Proof that order preserving ⟸ isomorphic: 𝑥 ≤ 𝑦 ⟹ 𝜃(𝑦) = 𝜃(𝑥 ∨ 𝑦) = 𝜃(𝑥) > 𝜃(𝑦)

by right hypothesis

⟹ 𝜃(𝑥) ⋜ 𝜃(𝑦) 𝑥 ≤ 𝑦 ⟹ 𝜃(𝑥) = 𝜃(𝑥 ∧ 𝑦) = 𝜃(𝑥) ? 𝜃(𝑦)

by right hypothesis

⟹ 𝜃(𝑥) ⋜ 𝜃(𝑦)

✏ Example 4.15. Let 𝙇 ≡ 𝙈 represent the condition that a lattice 𝙇 and a lattice 𝙈 are isomorphic. {𝑥,𝑦,𝑧} , ∪, ∩ ; ⊆) ≡ ( {1, 2, 3, 5, 6, 10, 15, 30}, lcm, gcd ; |) (𝟚 e with isomorphism x 𝟙𝐴 (𝑧) 𝜃(𝐴) = 5 ⋅ 3𝟙𝐴 (𝑦) ⋅ 2𝟙𝐴 (𝑥) ∀𝐴∈𝟚{𝑎,𝑏,𝑐} Explicit cases are listed below and illustrated in Example 3.9 (page 49) and Example 3.10 (page 49). 𝜃 (∅) = 50 ⋅ 30 ⋅ 20 0

0

=2

𝜃 ({𝑥, 𝑧}) = 5 ⋅ 3 ⋅ 2

= 10

𝜃 ({𝑦}) = 50 ⋅ 31 ⋅ 20

=3

𝜃 ({𝑦, 𝑧}) = 51 ⋅ 31 ⋅ 20

= 15

𝜃 ({𝑥, 𝑦}) = 50 ⋅ 31 ⋅ 21

=6

𝜃 ({𝑥, 𝑦, 𝑧}) = 51 ⋅ 31 ⋅ 21

= 30

version 0.30


1

Daniel J. Greenhoe

0

=5

1

𝜃 ({𝑥}) = 5 ⋅ 3 ⋅ 2

1

𝜃 ({𝑧}) = 51 ⋅ 30 ⋅ 20

=1


page 75

CHAPTER 4. LATTICES

✎PROOF: 𝜃(𝐴 ∪ 𝐵) = 5𝟙𝐴∪𝐵 (𝑎) ⋅ 3𝟙𝐴∪𝐵 (𝑏) ⋅ 2𝟙𝐴∪𝐵 (𝑐) = 5𝟙𝐴 (𝑎)∨𝟙𝐵 (𝑎) ⋅ 3𝟙𝐴 (𝑏)∨𝟙𝐵 (𝑏) ⋅ 2𝟙𝐴 (𝑐)∨𝟙𝐵 (𝑐)


= lcm (5𝟙𝐴 (𝑎) , 5𝟙𝐵 (𝑎) ) ⋅ lcm (3𝟙𝐴 (𝑏) , 3𝟙𝐵 (𝑏) ) ⋅ lcm (2𝟙𝐴 (𝑐) , 2𝟙𝐵 (𝑐) ) = lcm (5𝟙𝐴 (𝑎) ⋅ 3𝟙𝐴 (𝑏) ⋅ 2𝟙𝐴 (𝑐) , 5𝟙𝐵 (𝑎) ⋅ 3𝟙𝐵 (𝑏) ⋅ 2𝟙𝐵 (𝑐) ) = lcm (𝜃(𝐴), 𝜃(𝐵)) 𝜃(𝐴 ∩ 𝐵) = 5𝟙𝐴∩𝐵 (𝑎) ⋅ 3𝟙𝐴∩𝐵 (𝑏) ⋅ 2𝟙𝐴∩𝐵 (𝑐) = 5𝟙𝐴 (𝑎)∧𝟙𝐵 (𝑎) ⋅ 3𝟙𝐴 (𝑏)∧𝟙𝐵 (𝑏) ⋅ 2𝟙𝐴 (𝑐)∧𝟙𝐵 (𝑐)


= gcd (5𝟙𝐴 (𝑎) , 5𝟙𝐵 (𝑎) ) ⋅ gcd (3𝟙𝐴 (𝑏) , 3𝟙𝐵 (𝑏) ) ⋅ gcd (2𝟙𝐴 (𝑐) , 2𝟙𝐵 (𝑐) ) = gcd (5𝟙𝐴 (𝑎) ⋅ 3𝟙𝐴 (𝑏) ⋅ 2𝟙𝐴 (𝑐) , 5𝟙𝐵 (𝑎) ⋅ 3𝟙𝐵 (𝑏) ⋅ 2𝟙𝐵 (𝑐) ) = gcd (𝜃(𝐴), 𝜃(𝐵))

✏

4.6.2 Metrics Definition 4.7. 36 Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. A function 𝗏 ∈ ℝ𝑋 is a subvaluation if 1. 𝗏(𝑥) ≥ 0 ∀𝑥∈𝑋 d e 2. 𝗏(𝑥 ∨ 𝑦) + 𝗏(𝑥 ∧ 𝑦) ≤ 𝗏(𝑥) + 𝗏(𝑦) ∀𝑥,𝑦∈𝑋 f A subvaluation 𝗏 is isotone if 𝑥 ≤ 𝑦 ⟹ 𝗏(𝑥) ≤ 𝗏(𝑦). A subvaluation 𝗏 is positive if 𝑥 < 𝑦 ⟹ 𝗏(𝑥) < 𝗏(𝑦). Definition 4.8. 37 Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. A function 𝗏 ∈ ℝ𝑋 is a valuation if d 1. 𝗏(𝑥) ≥ 0 ∀𝑥∈𝑋 e 2. 𝗏(𝑥 ∨ 𝑦) + 𝗏(𝑥 ∧ 𝑦) = 𝗏(𝑥) + 𝗏(𝑦) ∀𝑥,𝑦∈𝑋 f 3. 𝑥 ≤ 𝑦 ⟹ 𝗏(𝑥) ≤ 𝗏(𝑦) ∀𝑥,𝑦∈𝑋

and

and and (isotone).

Proposition 4.5 (lattice subvaluation metric). 38 Let 𝙇 be a lattice. p 𝖽(𝑥, 𝑦) = 2𝗏(𝑥 ∨ 𝑦) − 𝗏(𝑥) − 𝗏(𝑦) is a metr {𝗏 is a positive subvaluation on} ⟹ { } ric. 𝙇 p Proposition 4.6 (lattice valuation metric). 39 Let 𝙇 be a lattice. p 𝖽(𝑥, 𝑦) = 𝗏(𝑥) + 𝗏(𝑦) − 2𝗏(𝑥 ∧ 𝑦) is a metr {𝗏 is a positive valuation on 𝙇 } ⟹ { } ric. p 36

📘 Deza and Deza (2006) page 143 📘 Deza and Deza (2006) page 143, 📘 Istrǎţescu (1987) page 127 ⟨differs from Deza⟩, 📘 Birkhoff (1948) page 74 ⟨not compatible with Deza⟩ 38 📘 Deza and Deza (2006) page 143 39 📘 Deza and Deza (2006) page 143 37


Daniel J. Greenhoe


version 0.30

page 76

4.6.3

4.7. LITERATURE

Lattice products

Theorem 4.12 (lattice product). 40 Let (𝑋 × 𝑌 , ≤) be the poset product41 of (𝑋, ⋜) and (𝑌 , ⊴). t ( 𝑋, >, ? ; ⋜) 𝑖𝑠𝑎𝑙𝑎𝑡𝑡𝑖𝑐𝑒 and h ( 𝑋 × 𝑌 , ∨, ∧ ; ≤) is also a lattice ⟹ } m ( 𝑌 , ⊻, ⊼ ; ⊴) 𝑖𝑠𝑎𝑙𝑎𝑡𝑡𝑖𝑐𝑒

4.7

Literature

📚 Literature survey: 1. Early lattice theory concepts: 📃 Dedekind (1900) 📃 Ore (1935) 2. Garrett Birkhoff's contribution: (a) The modern concept of the lattice was introduced by Garrett Birkhoff in 1933: 📃 Birkhoff (1933a) 📃 Birkhoff (1933b) (b) However, Birkhoff came to realize that the concept of the lattice had actually already been published in 1900 by Richard Dedekind. Birkhoff later remarked in an interview “My ideas about lattice theory developed gradually …It was my father who, when he told Ore at Yale about what I was doing some time in 1933, found out from Ore that my lattices coincided with Dedekind's Dualgruppen …I was lucky to have gone beyond Dedekind before I discovered his work. It would have been quite discouraging if I had discovered all my results anticipated by Dedekind.”42 (c) Birkhoff wrote a book in 1940 called Lattice Theory. There are basically three editions: 📘 Birkhoff (1940) 📘 Birkhoff (1948) 📘 Birkhoff (1967) With regards to his Lattice Theory book and another book entitled A Survey of Modern Algebra written with Saunders MacLane, Birkhoff remarked, “Morse had told me that no one under 30 should write a book. So I thought it over and wrote two!”43 3. Standard text books of lattice theory: 📘 Birkhoff (1967) 📘 Grätzer (1998) 📘 Crawley and Dilworth (1973) 4. Characterizations / equational bases: (a) General discussion: 📘 Tarski (1966) 📘 Baker (1969) 📘 McKenzie (1970) 📘 McKenzie (1972) 📘 Pigozzi (1975) 📘 Taylor (1979) 📘 Taylor (2008) 📘 Jipsen and Rose (1992) pages 115–127 ⟨Chapter 5⟩ 📘 Padmanabhan and Rudeanu (2008) (b) Characterizations for lattices: 📘 Kalman (1968) 📘 Tamura (1975) 📘 Sobociński (1979) 40

📘 MacLane and Birkhoff (1967), page 489 poset product: Definition 3.5 page 47 42 📘 Albers and Alexanderson (1985), page 4 43 📘 Albers and Alexanderson (1985), page 4 41

version 0.30


Daniel J. Greenhoe


page 77

CHAPTER 4. LATTICES (c) Specific characterizations: 📘 Padmanabhan (1969) ⟨2 equations in 7 variables⟩ 📘 McCune and Padmanabhan (1996), page 144 ⟨1 equation, 7 variables, length 79⟩ 📘 McCune et al. (2003a) ⟨1 equation, 8 variables, length 29⟩ 📘 McCune et al. (2003b) ⟨1 equation, 8 variables, length 29⟩ 5. Historical sources of lattices concepts: (a) Peirce (1880) credits Boole and Jevons with the commutative property: 📘 Peirce (1880b), page 33 ⟨“(5)”⟩ (b) Peirce (1880) credits Boole and Jevons with the associative property: (c) Peirce (1880) credits Jevons (1864) with the idempotent property: 📘 Jevons (1864) page 41 𝐴 + 𝐴 = 𝐴 “Law of Unity” 𝐴𝐴 = 𝐴 “Law of Simplicity” 6. Lattice drawing program: Ralph Freese, http://www.math.hawaii.edu/~ralph/LatDraw/

📚


Daniel J. Greenhoe


version 0.30

page 78

version 0.30

4.7. LITERATURE


Daniel J. Greenhoe


CHAPTER 5 MODULAR LATTICES

5.1 Modular relation Definition 5.1. 1 Let ( 𝑋, ∨, ∧ ; ≤) be a lattice. Let 𝟚𝑋𝑋 be the set of all relations in 𝑋 2 . The modularity relation Ⓜ ∈ 𝟚𝑋𝑋 and the dual modularity relation Ⓜ∗ ∈ 𝟚𝑋𝑋 are defined as 𝑥Ⓜ𝑦

def

⟺

2 {(𝑥, 𝑦) ∈ 𝑋 |𝑎 ≤ 𝑦

⟹

𝑦 ∧ (𝑥 ∨ 𝑎) = (𝑦 ∧ 𝑥) ∨ 𝑎

∀𝑎 ∈ 𝑋 }

def

d e f

𝑥Ⓜ∗ 𝑦 ⟺ {(𝑥, 𝑦) ∈ 𝑋 2 |𝑎 ≥ 𝑦 ⟹ 𝑦 ∨ (𝑥 ∧ 𝑎) = (𝑦 ∨ 𝑥) ∧ 𝑎 ∀𝑎 ∈ 𝑋 } . A pair (𝑥, 𝑦) ∈ Ⓜ is alternatively denoted as (𝑥, 𝑦) Ⓜ, and is called a modular pair. A pair (𝑥, 𝑦) ∈ Ⓜ∗ is alternatively denoted as (𝑥, 𝑦) Ⓜ∗ , and is called a dual modular pair. A pair (𝑥, 𝑦) that is not a modular pair ((𝑥, 𝑦) ∉ Ⓜ) is denoted 𝑥Ⓜ𝑦. / A pair (𝑥, 𝑦) that is not a dual modular pair is ∗ denoted 𝑥Ⓜ/ 𝑦.

Proposition 5.1. p r {𝑥Ⓜ𝑦 ⟺ p

2

Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. 𝑥Ⓜ∗ 𝑦}

∀𝑥,𝑦∈𝑋

✎PROOF: 𝑥Ⓜ𝑦 ⟺ {𝑎 ≤ 𝑦

⟹

𝑦 ∧ (𝑥 ∨ 𝑎) = (𝑦 ∧ 𝑥) ∨ 𝑎 ∀𝑎 ∈ 𝑋} by definition of Ⓜ—page 79

⟺ {𝑎 ≥ 𝑦

⟹

𝑎 ∧ (𝑥 ∨ 𝑦) = (𝑎 ∧ 𝑥) ∨ 𝑦

∀𝑎 ∈ 𝑋} by definition of ≥—page 46

⟺ {𝑎 ≥ 𝑦

⟹

(𝑎 ∧ 𝑥) ∨ 𝑦 = 𝑎 ∧ (𝑥 ∨ 𝑦)

∀𝑎 ∈ 𝑋} by symmetric property of =—page 7

⟺ {𝑎 ≥ 𝑦

⟹

𝑦 ∨ (𝑥 ∧ 𝑎) = (𝑦 ∨ 𝑥) ∧ 𝑎 ∀𝑎 ∈ 𝑋} by commutative property of lattices—page 62

∗

by definition of Ⓜ∗ —Definition 5.1 page 79

⟺ 𝑥Ⓜ 𝑦

✏ Proposition 5.2. 1 2 3

3

Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice.

📘 Stern (1999) page 11, 📘 Maeda and Maeda (1970), page 1 ⟨Definition (1.1)⟩, 📘 Maeda (1966) page 248 📘 Maeda and Maeda (1970), page 1 ⟨Lemma (1.2)⟩ 📘 Maeda and Maeda (1970), page 1

page 80

5.2. SEMIMODULAR LATTICES

p 𝑥 ≤ 𝑦 or r } 𝑦≤𝑥 p ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

⎧ ⎪ ⎨ ⎪ ⎩

⟹

𝑥, 𝑦 are comparable

𝑥 𝑦 𝑥 𝑦

Ⓜ 𝑦 Ⓜ 𝑥 Ⓜ∗ 𝑦 Ⓜ∗ 𝑥.

and and and

✎PROOF: 𝑥 ≤ 𝑦 ⟹ {𝑎 ≤ 𝑦 ⟹ 𝑦 ∧ (𝑥 ∨ 𝑎) = 𝑥 ∨ 𝑎 = (𝑦 ∧ 𝑥) ∨ 𝑎 ⟺ 𝑥Ⓜ𝑦

∀𝑎 ∈ 𝑋}

by definition of Ⓜ (Definition 5.1 page 79)

𝑥 ≤ 𝑦 ⟹ {𝑎 ≤ 𝑥 ⟹ 𝑥 ∧ (𝑦 ∨ 𝑎) = 𝑥 = 𝑥 ∨ 𝑎 = (𝑥 ∧ 𝑦) ∨ 𝑎 ⟺ 𝑦Ⓜ𝑥

∀𝑎 ∈ 𝑋}

by definition of Ⓜ (Definition 5.1 page 79)

∗

because 𝑥 ≤ 𝑦 ⟹ 𝑥Ⓜ𝑦 and by Proposition 5.1 page 79

∗

because 𝑥 ≤ 𝑦 ⟹ 𝑦Ⓜ𝑥 and by Proposition 5.1 page 79

𝑥 ≤ 𝑦 ⟹ 𝑥Ⓜ 𝑦 𝑥 ≤ 𝑦 ⟹ 𝑦Ⓜ 𝑥

✏ Proposition 5.3. Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. p 𝑥 Ⓜ 𝑥 ∀𝑥∈𝑋 (Ⓜ is reflexive) r ∗ ∀𝑥∈𝑋 (Ⓜ∗ is reflexive) p 𝑥 Ⓜ 𝑥 ✎PROOF:

5.2

✏

Because 𝑥 ≤ 𝑥 and by Proposition 5.2 (page 79).

Semimodular lattices

Definition 5.2. 4 d A lattice ( 𝑋, ∨, ∧ ; ≤) is semimodular if e 𝑥Ⓜ𝑦 ⟹ 𝑦Ⓜ𝑥 f A semimodular lattice is also called M-symmetric.

5.3

Modular lattices

Modular lattices are a generalization of the distributive lattice in the sense that all distributive lattices are modular, but not equivalent because not all modular lattices are distributive (Theorem 6.5 page 105). Definition 5.3. 5 d A lattice ( 𝑋, ∨, ∧ ; ≤) is modular if e 𝑥Ⓜ𝑦 ∀𝑥, 𝑦 ∈ 𝑋. f

5.3.1

Characterizations

This section describes some characterizations of modular lattices— that is, sets of properties that are equivalent to the definition of modular lattices (Definition 5.3 page 80): 4 5

📘 Maeda and Maeda (1970), page 3 ⟨Definition (1.7)⟩ 📘 Birkhoff (1967) page 82, 📘 Maeda and Maeda (1970), page 3 ⟨Definition (1.7)⟩

version 0.30


Daniel J. Greenhoe


page 81

CHAPTER 5. MODULAR LATTICES

Ore 1935 (order characterization) Theorem 5.1 page 81 N5 lattice (order characterization) Theorem 5.2 page 82 Riecan 1957 (algebraic characterization) Theorem 5.3 page 84 Alternatively, any of the sets of properties listed in this section could be used as the definition of modular lattices and the definition would in turn become a theorem/proposition.

Order characterizations Theorem 5.1. 6 Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. {𝑥 ≤ 𝑦 ⟺ ⟹ 𝑥 ∨ (𝑧 ∧ 𝑦) = (𝑥 ∨ 𝑧) ∧ 𝑦} t 𝙇 is modular h ⟺ 𝑥 ∨ [(𝑥 ∨ 𝑦) ∧ 𝑧] = (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) m ⟺ 𝑥 ∧ [(𝑥 ∧ 𝑦) ∨ 𝑧] = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)

∀𝑥,𝑦,𝑧∈𝑋 ∀𝑥,𝑦,𝑧∈𝑋 ∀𝑥,𝑦,𝑧∈𝑋

✎PROOF: 1. Proof that 𝙇 is modular ⟺ {𝑥 ≤ 𝑦 {𝙇 is modular } ⟺ {𝑥 ≤ 𝑦 ⟺ {𝑎 ≤ 𝑦 ⟺ {𝑥Ⓜ𝑦

⟹

𝑥 ∨ (𝑧 ∧ 𝑦) = (𝑥 ∨ 𝑧) ∧ 𝑦}:

⟹

𝑦 ∧ (𝑧 ∨ 𝑥) = (𝑦 ∧ 𝑧) ∨ 𝑥

∀𝑥, 𝑦, 𝑧 ∈ 𝑋} by Definition 5.3 page 80

⟹

𝑦 ∧ (𝑥 ∨ 𝑎) = (𝑦 ∧ 𝑥) ∨ 𝑎

∀𝑥, 𝑦, 𝑎 ∈ 𝑋} by change of variables by Definition 5.1 page 79

∀𝑥,𝑦∈𝑋 }

2. Proof that 𝙇 is modular ⟺ 𝑥 ∨ [(𝑥 ∨ 𝑦) ∧ 𝑧] = (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧): (a) Proof that 𝙇 is modular ⟹ 𝑥 ∨ [(𝑥 ∨ 𝑦) ∧ 𝑧] = (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧): First note that 𝑥 ≤ 𝑥 ∨ 𝑦. 𝑥 ∨ [(𝑥 ∨ 𝑦) ∧ 𝑧] = 𝑥 ∨ (𝑢 ∧ 𝑧)|𝑢≜𝑥∨𝑦

by substitution 𝑢 ≜ 𝑥 ∨ 𝑦

= 𝑢 ∧ (𝑥 ∨ 𝑧)|𝑢≜𝑥∨𝑦

by modularity hypothesis

= (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧)

because 𝑢 ≜ 𝑥 ∨ 𝑦

(b) Proof that 𝙇 is modular ⟸ 𝑥 ∨ [(𝑥 ∨ 𝑦) ∧ 𝑧] = (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧): 𝑥 ≤ 𝑦 ⟹ 𝑥 ∨ (𝑦 ∧ 𝑧) = 𝑥 ∨ (𝑦 ∧ 𝑧)

by right hypothesis and 𝑥 ≤ 𝑦

= 𝑥 ∨ (𝑧 ∧ 𝑦)

by commutative property Theorem 4.3 page 62

= 𝑥 ∨ [𝑧 ∧ (𝑥 ∨ 𝑦)]

because 𝑥 ≤ 𝑦

= 𝑥 ∨ [(𝑥 ∨ 𝑦) ∧ 𝑧]


= (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧)

by right hypothesis

= 𝑦 ∧ (𝑥 ∨ 𝑧)


3. Proof that 𝙇 is modular ⟺ {𝑦 ≤ 𝑥 ⟹ 𝑥 ∧ (𝑦 ∨ 𝑧) = 𝑦 ∨ (𝑥 ∧ 𝑧)}: 𝙇 is modular ⟺ {𝑥 ≤ 𝑦 ⟹ 𝑥 ∨ (𝑦 ∧ 𝑧) = 𝑦 ∧ (𝑥 ∨ 𝑧)} by definition of modular page 80 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ modularity definition (Definition 5.3 page 80)

⟺ {𝑦 ≤ 𝑥 ⟹ 𝑦 ∨ (𝑥 ∧ 𝑧) = 𝑥 ∧ (𝑦 ∨ 𝑧)} by change of variables: 𝑥 ↔ 𝑦 ⟺ {𝑦 ≤ 𝑥 ⟹ 𝑥 ∧ (𝑦 ∨ 𝑧) = 𝑦 ∨ (𝑥 ∧ 𝑧)} by reflexive property of = Definition 1.9 page 7 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ dual of Definition 5.3

4. Proof that {𝑦 ≤ 𝑥 ⟹ 𝑥 ∧ (𝑦 ∨ 𝑧) = 𝑦 ∨ (𝑥 ∧ 𝑧)} ⟺ {𝑥 ∧ [(𝑥 ∧ 𝑦) ∨ 𝑧] = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)}: 6

📘 Padmanabhan and Rudeanu (2008) page 39, 📘 Ore (1935) page 413 ⟨(2)⟩


Daniel J. Greenhoe


version 0.30

page 82

5.3. MODULAR LATTICES

(a) Proof that {𝑦 ≤ 𝑥 ⟹ 𝑥 ∧ (𝑦 ∨ 𝑧) = 𝑦 ∨ (𝑥 ∧ 𝑧)} ⟹ {𝑥 ∧ [(𝑥 ∧ 𝑦) ∨ 𝑧] = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)}: First note that 𝑥 ∧ 𝑦 ≤ 𝑥. 𝑥 ∧ [(𝑥 ∧ 𝑦) ∨ 𝑧] = 𝑥 ∧ (𝑢 ∨ 𝑧)|𝑢≜𝑥∧𝑦

by substitution 𝑢 ≜ 𝑥 ∧ 𝑦

= 𝑢 ∨ (𝑥 ∧ 𝑧)|𝑢≜𝑥∧𝑦

by left hypothesis

= (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)

because 𝑢 ≜ 𝑥 ∧ 𝑦

(b) Proof that {𝑦 ≤ 𝑥 ⟹ 𝑥 ∧ (𝑦 ∨ 𝑧) = 𝑦 ∨ (𝑥 ∧ 𝑧)} ⟸ {𝑥 ∧ [(𝑥 ∧ 𝑦) ∨ 𝑧] = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)}: 𝑦 ≤ 𝑥 ⟹ 𝑥 ∧ (𝑦 ∨ 𝑧) = 𝑥 ∧ (𝑧 ∨ 𝑦)


= 𝑥 ∧ [𝑧 ∨ (𝑥 ∧ 𝑦)]

because 𝑦 ≤ 𝑥

= 𝑥 ∧ [(𝑥 ∧ 𝑦) ∨ 𝑧]


= (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)

by right hypothesis

= 𝑦 ∨ (𝑥 ∧ 𝑧)

because 𝑦 ≤ 𝑥

✏ Definition 5.4 (N5 lattice/pentagon). 7 d The N5 lattice is the ordered set ({0, 𝑎, 𝑏, 𝑝, 1}, ≤) with cover relation e ≺= {(0, 𝑎) , (𝑎, 𝑏) , (𝑏, 1) , (𝑝, 1) , (0, 𝑝)}. f The N5 lattice is also called the pentagon.

1 𝑏 𝑝 𝑎 0

Lemma 5.1. 8 l e The 𝑁5 lattice (pentagon lattice) is non-modular. m ✎PROOF: 𝑥 ≤ 𝑦 ⟹ 𝑦 ∧ (𝑧 ∨ 𝑥) = 𝑦 ∧ 𝑏 =𝑦

by Definition 3.21 page 58 (lub) by Definition 3.22 page 58 (glb)

≠𝑥 =𝑥∨𝑎

by Definition 3.21 page 58 (lub)

= (𝑦 ∧ 𝑧) ∨ 𝑥


✏ Theorem 5.2. 9 Let 𝙇 be a lattice (Definition 4.3 page 61). t h 𝙇 is modular ⟺ 𝙇 does not contain 𝑁5 as a sublattice. m ✎PROOF: 1. Proof that 𝙇 is modular ⟹ 𝙇 does not contain 𝑁5: This is because 𝑁5 is a non-modular lattice. Proof: Lemma 5.1 page 82 2. Proof that 𝙇 does not contain 𝑁5 ⟹ 𝙇 is modular: 7

📘 Beran (1985) pages 12–13, 📃 Dedekind (1900) pages 391–392 ⟨(44) and (45)⟩ 📘 Burris and Sankappanavar (1981) page 11 9 📘 Burris and Sankappanavar (1981) page 11, 📘 Grätzer (1971) page 70, 📃 Dedekind (1900) ⟨cf Stern 1999 page 8

10⟩ version 0.30


Daniel J. Greenhoe


page 83


(a) In what follows, we will prove the equivalent contrapositive statement: 𝑁5 ∈ 𝐿 ⟸ 𝐿 is not modular (every non-modular lattice must contain 𝑁5). (b) We will show that for any choice of 𝑥, 𝑦 ∈ 𝐿 such that 𝑥 ≤ 𝑦 and under the following definitions, all non-modular lattices contain the 𝑁5 lattice illustrated below: 𝑥∨𝑧

1 𝑦 𝑥

𝑧

𝑎 ≜ 𝑥 ∨ (𝑦 ∧ 𝑧)

0

𝑏 ≜ 𝑦 ∧ (𝑥 ∨ 𝑧)

𝑏 𝑎

𝑧 𝑦∧𝑧

(c) Proofs for comparable elements: 𝑏 = 𝑦 ∧ (𝑥 ∨ 𝑧) ≤𝑥∨𝑧 𝑎 = 𝑥 ∨ (𝑦 ∧ 𝑧)

by definition of 𝑏 in item (2b) by definition of ∧ page 58 by definition of 𝑎 in item (2b)

≤ 𝑦 ∧ (𝑥 ∨ 𝑧)

by modularity inequality Theorem 4.7

=𝑏

by definition of 𝑏 in item (2b)

𝑦 ∧ 𝑧 ≤ 𝑥 ∨ (𝑦 ∧ 𝑧) =𝑎 𝑧≤𝑥∨𝑧 𝑦∧𝑧≤𝑧

by definition of ∨ page 58 by definition of 𝑎 in item (2b) by definition of ∧ page 58 by definition of ∧ page 58

(d) Proofs for noncomparable elements: 𝑎 ∨ 𝑧 = [𝑥 ∨ (𝑦 ∧ 𝑧)] ∨ 𝑧

by definition of 𝑎

= 𝑧 ∨ [𝑥 ∨ (𝑦 ∧ 𝑧)]

by commutative property of lattices (page 62)

= [𝑧 ∨ 𝑥] ∨ (𝑦 ∧ 𝑧)

by associative property of lattices (page 62)

= [𝑥 ∨ 𝑧] ∨ (𝑦 ∧ 𝑧)


= 𝑥 ∨ [𝑧 ∨ (𝑦 ∧ 𝑧)]


=𝑥∨𝑧

by absorptive property of lattices (page 62)

𝑏 ∨ 𝑧 = (𝑏 ∨ 𝑎) ∨ 𝑧

by previous result

= 𝑏 ∨ (𝑎 ∨ 𝑧)


= 𝑏 ∨ (𝑥 ∨ 𝑧)

by previous result

=𝑥∨𝑧

by previous result

𝑎 ∧ 𝑧 = (𝑎 ∧ 𝑏) ∧ 𝑧

by previous result

= 𝑎 ∧ (𝑏 ∧ 𝑧)


= 𝑎 ∧ (𝑦 ∧ 𝑧)

by previous result

=𝑦∧𝑧

by previous result

𝑏 ∧ 𝑧 = [𝑦 ∧ (𝑥 ∨ 𝑧)] ∨ 𝑧


= 𝑧 ∧ [𝑦 ∧ (𝑥 ∨ 𝑧)]


= [𝑧 ∧ 𝑦] ∧ (𝑥 ∨ 𝑧)



Daniel J. Greenhoe


version 0.30

page 84

5.4. EXAMPLES

= [𝑦 ∧ 𝑧] ∧ (𝑥 ∨ 𝑧)


= 𝑦 ∧ [𝑧 ∧ (𝑥 ∨ 𝑧)]


=𝑦∧𝑧

by absorptive property of lattices (page 62)

(e) Thus, all non-modular lattices must contain an 𝑁5 sublattice. That is, 𝙇 is a non-modular lattice

⟹

𝙇 contains an 𝑁5 sublattice.

And this implies (by the contrapostive of the statement) 𝙇 does not contain an 𝑁5 sublattice

⟹

𝙇 is modular lattice.

✏

Algebraic characterizations Theorem 5.3. 10 Let 𝘼 ≜ ( 𝑋, ∨, ∧ ; ≤) be an algebraic structure. t (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) = [(𝑧 ∧ 𝑥) ∨ 𝑦] ∧ 𝑥 ∀𝑥,𝑦,𝑧∈𝑋 and h ⟺ } ∀𝑥,𝑦,𝑧∈𝑋 m { [𝑥 ∨ (𝑦 ∨ 𝑧)] ∧ 𝑧 = 𝑧

5.3.2

𝘼 is a { modular lattice }

Special cases

Theorem 5.4. 11 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice. and ⎧ 1. 𝙇 is complemented ⎫ 1. 𝙇 does not contain ⎧ t ⎪ 2. 𝙇 is atomic and ⎪ ⎪ h any N5 sublattice ⎬ ⟹ ⎨ 3. 𝙇 does not contain an N5 lattice m ⎨ ⎪ ⎪ ⎪ ⎩ 2. 𝙇 is modular with elements 0 and 1 ⎩ ⎭

5.4

and

⎫ ⎪ ⎬ ⎪ ⎭

Examples

Example 5.1. The lattice illustrated to the right is the N5 lattice (Definition 5.4 page 82). The N5 lattice has a total of 5 × 5 = 25 pairs of elements of the form (𝑥, 𝑦) where 𝑥, 𝑦 ∈ 𝑋. Of these 25, all are modular pairs except for the pair (𝑧, 𝑦). That is, 𝑧Ⓜ𝑦. / Therefore, the N5 lattice is non-semimodular (and non-modular).

1 𝑦 𝑥

𝑧 0

✎PROOF: 1. Five are of the form (𝑥, 𝑥) and are therefore modular pairs by the reflexive property and Proposition 5.3 page 80: 1Ⓜ1, 𝑦Ⓜ𝑦, 𝑥Ⓜ𝑥, 𝑧Ⓜ𝑧, 0Ⓜ0. 2. Of the remaining 20, 16 more are modular pairs simply because they are comparable and by Proposition 5.2 (page 79): 1Ⓜ𝑦 1Ⓜ𝑥 1Ⓜ0 𝑦Ⓜ𝑥 𝑦Ⓜ0 𝑥Ⓜ0 1Ⓜ𝑧 𝑧Ⓜ0 𝑦Ⓜ1 𝑥Ⓜ1 0Ⓜ1 𝑥Ⓜ𝑦 0Ⓜ𝑦 0Ⓜ𝑥 𝑧Ⓜ1 0Ⓜ𝑧 10

📘 Padmanabhan and Rudeanu (2008) pages 42–43, 📘 Riečan (1957) 📘 Saliǐ (1988) page 27, 📘 Dilworth (1982), pages 333–353 ⟨cf Stern 1999⟩, 📘 Stern (1999) page 11, 📘 McLaughlin (1956) 11

version 0.30


Daniel J. Greenhoe


page 85


3. Of the remaining 4, 3 are modular pairs and 1 is a nonmodular pair: 𝑦Ⓜ𝑧 𝑥Ⓜ𝑧 𝑧Ⓜ𝑦 / 𝑧Ⓜ𝑥 𝑥 ≤ 𝑦 ⟹ 𝑦 ∧ (𝑧 ∨ 𝑥) = 𝑦 ∧ 1

=𝑦

0 ≤ 𝑧 ⟹ 𝑧 ∧ (𝑦 ∨ 0) = 𝑧 ∧ 𝑦

≠𝑥

=0∨𝑥

= (𝑦 ∧ 𝑧) ∨ 𝑥

⟹ 𝑧Ⓜ𝑦 /

=0

=0∨0

= (𝑧 ∧ 𝑦) ∨ 0

⟹ 𝑦Ⓜ𝑧

0 ≤ 𝑧 ⟹ 𝑧 ∧ (𝑥 ∨ 0) = 𝑧 ∧ 𝑥

=0

=0∨0

= (𝑧 ∧ 𝑥) ∨ 0

⟹ 𝑥Ⓜ𝑧

0 ≤ 𝑥 ⟹ 𝑥 ∧ (𝑧 ∨ 0) = 𝑥 ∧ 𝑧

=0

=0∨0

= (𝑥 ∧ 𝑧) ∨ 0

⟹ 𝑧Ⓜ𝑥

✏ Example 5.2. Of the non-comparable pairs in the lattice illustrated to the right, the following are modular pairs: 𝑥Ⓜ𝑦, 𝑦Ⓜ𝑥, 𝑥Ⓜ𝑎, 𝑎Ⓜ𝑥, 𝑦Ⓜ𝑎, 𝑎Ⓜ𝑦, 𝑏Ⓜ𝑥, 𝑏Ⓜ𝑦 and the remaining non-comparable pairs are non-modular: 𝑥Ⓜ𝑏, / 𝑦Ⓜ𝑏. / Therefore, although the Hasse diagram shown is horizontally and vertically symmetric, the lattice itself is not M-symmetric (not semimodular), and thus also not modular and not distributive. ✎PROOF:

1 𝑏 𝑎 𝑦

𝑥 0

𝑦(𝑥 + 0) = 𝑦𝑥

= 𝑦𝑥 + 0

⟹ 𝑥Ⓜ𝑦

𝑥(𝑦 + 0) = 𝑥𝑦

= 𝑥𝑦 + 0

⟹ 𝑦Ⓜ𝑥

𝑎(𝑥 + 0) = 𝑎𝑥

= 𝑎𝑥 + 0

⟹ 𝑥Ⓜ𝑎

𝑥(𝑎 + 0) = 𝑥𝑎

= 𝑥𝑎 + 0

⟹ 𝑎Ⓜ𝑥

𝑎(𝑦 + 0) = 𝑎𝑦

= 𝑎𝑦 + 0

⟹ 𝑦Ⓜ𝑎

𝑦(𝑎 + 0) = 𝑦𝑎

= 𝑦𝑎 + 0

⟹ 𝑎Ⓜ𝑦

𝑏(𝑥 + 𝑎) = 𝑏1

=𝑏

𝑥(𝑏 + 0) = 𝑥𝑏

= 𝑥𝑏 + 0

𝑏(𝑦 + 𝑎) = 𝑏1

=𝑏

𝑦(𝑏 + 0) = 𝑦𝑏

= 𝑦𝑏 + 0

≠𝑎

=0+𝑎

= 𝑏𝑥 + 𝑎

⟹ 𝑥Ⓜ𝑏 / ⟹ 𝑏Ⓜ𝑥

≠𝑎

=0+𝑎

= 𝑏𝑦 + 𝑎

⟹ 𝑦Ⓜ𝑏 / ⟹ 𝑏Ⓜ𝑦

✏ 𝑎

1 𝑦

𝑥

0

𝑧

𝑥

Example 5.3. The lattices illustrated to the right and left are duals of each other. Both are non-modular and both are nonsemimodular.

1

𝑧

𝑦 𝑎 0

✎PROOF: Left hand side lattice: 𝑎(𝑧 + 𝑥) = 𝑎1

=𝑎

≠𝑥

=0+𝑥

= 𝑎𝑧 + 𝑥

⟹ 𝑧Ⓜ𝑎 /

𝑧(𝑎 + 0) = 𝑧𝑎

= 𝑧𝑎 + 0

⟹ 𝑎Ⓜ𝑧

𝑧(𝑥 + 0) = 𝑧𝑥

= 𝑧𝑥 + 0

⟹ 𝑥Ⓜ𝑧

𝑥(𝑧 + 𝑎) = 𝑥1

=𝑥

Right hand side lattice:

≠𝑎

=0+𝑎

= 𝑥𝑧 + 𝑎

⟹ 𝑧Ⓜ𝑥 /

✏ 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 86 1

𝑎 𝑥

0

𝑏 𝑦

5.4. EXAMPLES

Example 5.4. The lattice illustrated to the left is modular. The lattice illustrated to the right is non-modular and nonsemimodular.

𝑎 𝑥

1

0

𝑏 𝑦

✎PROOF: 1. Proof that the left hand side is modular: because it does not contain the N5 lattice and by Theorem 5.2 (page 82). 2. Proof that the right hand side is non-modular and non-semimodular: 𝑥(𝑏 + 𝑦) = 𝑥𝑏

=0

𝑏(𝑥 + 𝑦) = 𝑏1

=𝑏

𝑦(𝑎 + 𝑥) = 𝑦𝑎

=0

𝑎(𝑦 + 𝑥) = 𝑎1

=𝑎

≠𝑦 ≠𝑥

=0+𝑦

= 𝑥𝑏 + 𝑦

⟹ 𝑏Ⓜ𝑥

=0+𝑦

= 𝑏𝑥 + 𝑦

⟹ 𝑥Ⓜ𝑏 /

=0+𝑥

= 𝑦𝑎 + 𝑥

⟹ 𝑎Ⓜ𝑦

=0+𝑥

= 𝑎𝑦 + 𝑥

⟹ 𝑦Ⓜ𝑎 /

✏ Proposition 5.4. 12 Let 𝑋𝑛 be a finite set with order 𝑛 = | 𝑋𝑛 |. Let 𝑙𝑛 be the number of unlabeled lattices on 𝑋𝑛 , 𝑑𝑛 the number of unlabeled distributive lattices on 𝑋𝑛 , and 𝑚𝑛 the number of unlabeled modular lattices on 𝑋𝑛 . 0 1 2 3 4 5 6 7 8 9 10 11 12 13 p 𝑛 r 𝑙𝑛 1 1 1 1 2 5 15 53 222 1078 5994 37622 262, 776 2, 018, 305 p 𝑚 1 1 1 1 2 4 8 16 34 72 157 343 𝑛 Example 5.5 (modularity in 5 element sets). There are a total of five unlabeled lattices on a five element set (Proposition 4.2 page 67); and of these five, four are modular, and three of the five are distributive (Example 6.2 page 107). modular lattices on 5 element sets e x

Example 5.6 (modularity in 6 element sets). There are a total of 15 unlabeled lattices on a six element set (Proposition 4.2 page 67 and Example 4.12 page 67); and of these 15, eight are modular, and five of the eight are distributive (Proposition 6.3 page 107). There are no six element non-modular lattices that are also semimodular. modular and distributive lattices modular but not distributive

e x

non-semimodular lattices (and non-modular and non-distributive)

12

𝑙𝑛 : 💻 Sloane (2014) ⟨http://oeis.org/A006966⟩ | 𝑚𝑛 : 💻 Sloane (2014) ⟨http://oeis.org/A006981⟩ | 𝑙𝑛 : 📘 Heitzig and Reinhold (2002) version 0.30


Daniel J. Greenhoe


page 87

CHAPTER 5. MODULAR LATTICES 1

𝑎

𝑏

𝑦 𝑥

0

𝑧

Example 5.7. The lattices illustrated to the left and right are duals of each other. Both are non-modular. The left hand side lattice is also nonsemimodular, however the right hand side lattice is semimodular.13

1

𝑎

𝑐

𝑏 𝑥

0

𝑦

✎PROOF: Proof for lattice on left hand side: 𝑦(𝑎 + 0) = 𝑦𝑎

= 𝑦𝑎 + 0

⟹ 𝑎Ⓜ𝑦

𝑎(𝑦 + 𝑥) = 𝑎𝑎

=𝑎

=𝑦+𝑥

= 𝑎𝑦 + 𝑥

⟹ 𝑦Ⓜ𝑎

𝑏(𝑎 + 𝑧) = 𝑏1

=𝑏

=𝑦+𝑧

= 𝑏𝑎 + 𝑧

⟹ 𝑎Ⓜ𝑏

𝑎(𝑏 + 𝑥) = 𝑎1

=𝑎

=𝑦+𝑥

= 𝑎𝑏 + 𝑥

⟹ 𝑏Ⓜ𝑎

𝑏(𝑥 + 𝑧) = 𝑏1

=𝑏

≠𝑧

=0+𝑧

=

= 𝑥𝑏 + 0

⟹ 𝑏Ⓜ𝑥

=0+𝑦

= 𝑐𝑥 + 𝑦

⟹ 𝑥Ⓜ𝑐

𝑥(𝑏 + 0) = 𝑥𝑏

𝑏𝑥 + 𝑧

⟹ 𝑥Ⓜ𝑏 /

Proof for lattice on right hand side: 𝑐(𝑥 + 𝑦) = 𝑐𝑏

=𝑦

𝑥(𝑐 + 0) = 𝑥𝑐

= 𝑥𝑐 + 0

𝑏(𝑎 + 𝑥) = 𝑏𝑎

=𝑥

=𝑥+𝑥

= 𝑏𝑎 + 𝑥

𝑏(𝑎 + 𝑦) = 𝑏1

=𝑏

=𝑥+𝑦

= 𝑏𝑎 + 𝑦

⟹ 𝑎Ⓜ𝑏

𝑎(𝑏 + 𝑥) = 𝑎𝑏

=1

=1+𝑥

= 𝑎𝑏 + 𝑥

⟹ 𝑏Ⓜ𝑎

𝑐(𝑎 + 𝑦) = 𝑐1

=𝑐

≠𝑦

=0+𝑦

= 𝑐𝑎 + 𝑦

⟹ 𝑎Ⓜ𝑐 /

𝑎(𝑐 + 𝑥) = 𝑎1

=𝑎

≠𝑥

=0+𝑥

= 𝑎𝑐 + 𝑥

⟹ 𝑐 Ⓜ𝑎 /

𝑐(𝑥 + 𝑦) = 𝑐𝑏

=𝑦

=0+𝑦

= 𝑐𝑥 + 𝑦

⟹ 𝑥Ⓜ𝑐

𝑥(𝑐 + 0) = 𝑥𝑐

= 𝑥𝑐 + 0

⟹ 𝑐Ⓜ𝑥 and

⟹ 𝑐Ⓜ𝑥

⋮

✏ Example 5.8 (modular lattices on 7 element sets). There are a total of 53 unlabeled lattices on a seven element set (Example 4.13 page 68). Of these 53, 16 are modular, and 8 of these 16 are distributive (Proposition 6.3 page 107). modular (and distributive) lattices on 7 element sets

modular but non-distributive lattices on 7 element sets

e x

13

📘 Maeda and Maeda (1970), page 5 ⟨Exercise 1.1⟩


Daniel J. Greenhoe


version 0.30

page 88

version 0.30

5.4. EXAMPLES


Daniel J. Greenhoe


CHAPTER 6 DISTRIBUTIVE LATTICES

6.1 Distributivity relation Definition 6.1. 1 Let ( 𝑋, ∨, ∧ ; ≤) be a lattice (Definition 4.3 page 61). Let 𝟚𝑋𝑋𝑋 be the set of all relations in 𝑋 3 . The distributivity relation Ⓓ ∈ 𝟚𝑋𝑋𝑋 and the dual distributivity relation Ⓓ∗ ∈ 𝟚𝑋𝑋𝑋 are defined as Ⓓ ≜ {(𝑥, 𝑦, 𝑧) ∈ 𝑋 3 |𝑥 ∧ (𝑦 ∨ 𝑧) = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)} (each (𝑥, 𝑦, 𝑧) is disjunctive distributive) and d e Ⓓ∗ ≜ {(𝑥, 𝑦, 𝑧) ∈ 𝑋 3 |𝑥 ∨ (𝑦 ∧ 𝑧) = (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧)} (each (𝑥, 𝑦, 𝑧) is conjunctive distributive). f A triple (𝑥, 𝑦, 𝑧) ∈ Ⓓ is alternatively denoted as (𝑥, 𝑦, 𝑧) Ⓓ, and is called a distributive triple. A triple (𝑥, 𝑦, 𝑧) ∈ Ⓓ∗ is alternatively denoted as (𝑥, 𝑦, 𝑧) Ⓓ∗ , and is called a dual distributive triple.

6.2 Distributive Lattices 6.2.1 Definition This section introduces distributive lattices. Theorem 4.6 (page 65) demonstrates that all lattices ( 𝑋, ∨, ∧ ; ≤) satisfy the following distributive inequalities: 𝑥 ∧ (𝑦 ∨ 𝑧) ≥ (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) 𝑥 ∨ (𝑦 ∧ 𝑧) ≤ (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ∨ (𝑦 ∧ 𝑧) ≤ (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) ∧ (𝑦 ∨ 𝑧)


(join super-distributive)

and


(meet sub distributive).

and


(median inequality).

Theorem 6.1 (page 90) demonstrates that when one of these inequalities is equality, then all three of them are equalities. And in this case, the lattice is defined to be distributive (next definition). 1

📘 Maeda and Maeda (1970), page 15 ⟨Definition 4.1⟩, 📃 Foulis (1962) page 67, 📘 von Neumann (1960), page 32 ⟨Definition 5.1⟩, 📃 Davis (1955) page 314 ⟨disjunctive distributive and conjunctive distributive f.⟩

page 90

6.2. DISTRIBUTIVE LATTICES

Definition 6.2. 2 d A lattice ( 𝑋, ∨, ∧ ; ≤) is distributive if e (𝑥, 𝑦, 𝑧) ∈ Ⓓ ∀𝑥, 𝑦, 𝑧 ∈ 𝑋 f Are all lattices distributive? The answer is “no”. Lemma 6.1 (page 92) and Lemma 6.2 (page 93) demonstrate two lattices that are not distributive: the N5 lattice (Definition 5.4 page 82) and the M3 lattice (Definition 6.3 page 93).

6.2.2

Characterizations

This section describes some characterizations (equational bases) of distributive lattices both in terms of lattices (order characterizations) and in terms of abstract algebraic structures (algebraic characterizations). Order characterizations (first assuming a structure is a lattice): Median property 1894 Theorem 6.1 page 90 Birkhoff distributivity criterion 1934 Theorem 6.2 page 94 Cancellation property 1934 Theorem 6.3 page 97 Algebraic characterizations (first assuming nothing): Birkhoff 1946 Proposition 6.1 page 100 Birkhoff 1948 Proposition 6.2 page 100 Sholander 1951 Theorem 6.4 page 100 Alternatively, any of the sets of properties listed in this section could be used as the definition of distributive lattices and the definition would in turn become a theorem/proposition. In addition, if a lattice is uniquely complemented and satisfies any one of a number of Huntington properties, then it is also distributive (Theorem 7.2 page 111), and hence also a Boolean algebra (Definition 8.1 page 115).

Order characterizations By the definition given in Definition 6.2 (page 90), a lattice is distributive if the meet operation ∧ distributes over the join operation ∨. And in view that the properties of lattices are self-dual, it is perhaps not surprising that the dual of the identity of Definition 6.2 is also true for any distributive lattice— that is, the join operation ∨ distributes over the meet operation ∧ (next theorem). But besides these two identities that are duals of each other, there is another identity that is not only equivalent to the first two, but is a dual of itself. This is called the median property,3 and is given by (3) in Theorem 6.1 (next theorem). Theorem 6.1.

4

Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice (Definition 4.3 page 61).

2

📘 Burris and Sankappanavar (1981) page 10, 📘 Birkhoff (1948) page 133, 📃 Ore (1935) page 414 ⟨arithmetic axiom⟩, 📃 Birkhoff (1933a) page 453, 📘 Balbes and Dwinger (1975) page 48 ⟨Definition II.5.1⟩ 3 median property: see also Literature item 5 page 112 4 📃 Dilworth (1984) page 237, 📘 Burris and Sankappanavar (1981) page 10, 📃 Ore (1935) page 416 ⟨(7),(8), Theorem 3⟩, 📃 Ore (1940) ⟨cf Gratzer 2003 page 159⟩, 📘 Schröder (1890) page 286 ⟨cf Birkhoff(1948)p.133⟩, 📃 Korselt (1894) ⟨cf Birkhoff(1948)p.133⟩ version 0.30


Daniel J. Greenhoe


page 91

CHAPTER 6. DISTRIBUTIVE LATTICES

t h m

𝙇 is distributive (Definition 6.2 page 90) ⟺ 𝑥 ∧ (𝑦 ∨ 𝑧) = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ∀𝑥,𝑦,𝑧∈𝑋 ⟺ 𝑥 ∨ (𝑦 ∧ 𝑧) = (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) ∀𝑥,𝑦,𝑧∈𝑋 ⟺ (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) ∧ (𝑦 ∨ 𝑧) = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ∨ (𝑦 ∧ 𝑧)

(disjunctive distributive) (conjunctive distributive)


(median property)

✎PROOF:

Let the join operation ∨ be represented by +, the meet operation ∧ be represented by juxtaposition, and let meet take algebraic precedence over join (+). 1. Proof that distributive ⟺ disjunctive distributive: {𝙇 is distributive } ⟺ {𝑥 ∧ (𝑦 ∨ 𝑧) = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ⟺ {(𝑥, 𝑦, 𝑧) ∈ Ⓓ ∀𝑥, 𝑦, 𝑧 ∈ 𝑋}

∀𝑥, 𝑦, 𝑧 ∈ 𝑋}


2. Proof that disjunctive distributive ⟹ conjunctive distributive: 𝑥 + (𝑦𝑧) = [⏟⏟⏟⏟⏟ 𝑥 + (𝑥𝑦)] + (𝑦𝑧)

by absorptive property of lattices page 62

expand 𝑥 wrt 𝑦

= 𝑥 + [(𝑥𝑦) + (𝑦𝑧)]

by associative property of lattices page 62

= 𝑥 + [(𝑦𝑥) + (𝑦𝑧)]

by commutative property of lattices page 62

= 𝑥 + [𝑦(𝑥 + 𝑧)]

by left hypothesis

𝑥(𝑥 + 𝑧)] + [𝑦(𝑥 + 𝑧)] = [⏟⏟⏟⏟⏟


expand 𝑥 wrt 𝑧

= [(𝑥 + 𝑧)𝑥] + [(𝑥 + 𝑧)𝑦]


= (𝑥 + 𝑧)(𝑥 + 𝑦)

by left hypothesis

= (𝑥 + 𝑦)(𝑥 + 𝑧)


3. Proof that conjunctive distributive ⟹ disjunctive distributive: 𝑥(𝑥 + 𝑦)] (𝑦 + 𝑧) 𝑥(𝑦 + 𝑧) = [⏟⏟⏟⏟⏟


expand 𝑥 wrt 𝑦

= 𝑥[(𝑥 + 𝑦)(𝑦 + 𝑧)]

by associative property of lattices page 62

= 𝑥[(𝑦 + 𝑥)(𝑦 + 𝑧)]


= 𝑥[𝑦 + (𝑥𝑧)]

by right hypothesis

𝑥 + (𝑥𝑧)] [𝑦 + (𝑥𝑧)] = [⏟⏟⏟⏟⏟


expand 𝑥 wrt 𝑧

= [(𝑥𝑧) + 𝑥][(𝑥𝑧) + 𝑦]


= (𝑥𝑧) + (𝑥𝑦)

by left hypothesis

= (𝑥𝑦) + (𝑥𝑧)


4. Proof that disjunctive distributive ⟹ median property: (𝑥 + 𝑦)(𝑥 + 𝑧)(𝑦 + 𝑧) = (𝑥 + 𝑦)[(𝑥 + 𝑧)𝑦 + (𝑥 + 𝑧)𝑧]

by disjunctive distributive hypothesis

= (𝑥 + 𝑦)[𝑦(𝑥 + 𝑧) + 𝑧(𝑥 + 𝑧)]

by commutative property (Theorem 4.3 page 62)

= (𝑥 + 𝑦)(𝑦𝑥 + 𝑦𝑧 + 𝑧𝑥 + 𝑧𝑧)


= (𝑥 + 𝑦)(𝑥𝑦 + 𝑥𝑧 + 𝑦𝑧 + 𝑧)


= (𝑥 + 𝑦)𝑥𝑦 + (𝑥 + 𝑦)𝑥𝑧 + (𝑥 + 𝑦)𝑦𝑧 + (𝑥 + 𝑦)𝑧



Daniel J. Greenhoe


version 0.30

page 92


= 𝑥𝑦(𝑥 + 𝑦) + 𝑥𝑧(𝑥 + 𝑦) + 𝑦𝑧(𝑥 + 𝑦) + 𝑧(𝑥 + 𝑦)


= 𝑥𝑦𝑥 + 𝑥𝑦𝑦 + 𝑥𝑧𝑥 + 𝑥𝑧𝑦 + 𝑦𝑧𝑥 + 𝑦𝑧𝑦 + 𝑧𝑥 + 𝑧𝑦


= 𝑥𝑦 + 𝑥𝑦 + 𝑥𝑧 + 𝑥𝑦𝑧 + 𝑥𝑦𝑧 + 𝑦𝑧 + 𝑥𝑧 + 𝑦𝑧


= 𝑥𝑦 + 𝑥𝑦𝑧 + 𝑥𝑧 + 𝑦𝑧

by idempotent property (Theorem 4.3 page 62)

= (𝑥𝑦)(𝑥𝑦) + 𝑥𝑦𝑧 + 𝑥𝑧 + 𝑦𝑧


= (𝑥𝑦)(𝑥𝑦 + 𝑧) + 𝑥𝑧 + 𝑦𝑧


= 𝑥𝑦 + 𝑥𝑧 + 𝑦𝑧


5. Proof that median property ⟹ disjunctive distributive: (a) Proof that 𝙇 is modular: 𝑦 ≤ 𝑥 ⟹ 𝑥(𝑦 + 𝑧) = 𝑥(𝑥 + 𝑧)(𝑦 + 𝑧)


= (𝑥 + 𝑦)(𝑥 + 𝑧)(𝑦 + 𝑧)

by 𝑦 ≤ 𝑥 hypothesis

= 𝑥𝑦 + 𝑥𝑧 + 𝑦𝑧

by median property hypothesis

= 𝑦 + 𝑥𝑧 + 𝑦𝑧


= 𝑦 + 𝑥𝑧


⟹ 𝙇 is modular (b) Proof that 𝑎 + 𝑎𝑏 = 𝑎: 𝑎𝑏 ≤ 𝑎

by definition of ∧ Definition 3.22 page 58

⟹ 𝑎 + 𝑎𝑏 = 𝑎

by definition of ∨ Definition 3.21 page 58

(c) Proof that median property ⟹ disjunctive distributive: 𝑥(𝑦 + 𝑧) = 𝑥𝑥(𝑦 + 𝑧)


= 𝑥(𝑥 + 𝑦)𝑥(𝑥 + 𝑧)(𝑦 + 𝑧) ⏟⏟⏟ ⏟⏟⏟ 𝑥


𝑥

= 𝑥[(𝑥 + 𝑦)(𝑥 + 𝑧)(𝑦 + 𝑧)]


= 𝑥(𝑥𝑦 + 𝑥𝑧 + 𝑦𝑧) ⏟

by median property hypothesis

𝑧′

= 𝑥(𝑥𝑦) + 𝑥(𝑥𝑧 + ⏟ 𝑦𝑧)

by item (5a) and by Theorem 5.1 page 81

″

𝑧

= 𝑥(𝑥𝑦) + 𝑥(𝑥𝑧) + 𝑥(𝑦𝑧)

by item (5a) and by Theorem 5.1 page 81

= 𝑥𝑦 + 𝑥𝑧 + 𝑥𝑦𝑧


= 𝑥𝑦 + 𝑥𝑧

by item (5b)

✏ Lemma 6.1. 5 l e The 𝑁5 lattice is non-distributive m 5

📘 Burris and Sankappanavar (1981) page 11

version 0.30


Daniel J. Greenhoe


page 93


✎PROOF: 𝑦 ∧ (𝑥 ∨ 𝑧) = 𝑦 ∧ 𝑏


=𝑦

by Definition 3.22 page 58 (glb)

=𝑦∨𝑎


= 𝑦 ∨ (𝑦 ∧ 𝑧)


≠ 𝑥 ∨ (𝑦 ∧ 𝑧)

because 𝑥 ≠ 𝑦

= (𝑦 ∧ 𝑥) ∨ (𝑦 ∧ 𝑧)


✏ Definition 6.3 (M3 lattice/diamond). 6 d The M3 lattice is the ordered set ({0, 𝑝, 𝑞, 𝑟, 1}, ≤) with covering relation e ≺= {(𝑝, 1) , (𝑞, 1) , (𝑟, 1) , (0, 𝑝) , (0, 𝑞) , (0, 𝑟)}. f The M3 lattice is also called the diamond, and is illustrated by the Hasse diagram to the right.

1 𝑟 𝑝

𝑞 0

Remark 6.1. The M3 lattice is isomorphic to the lattices (𝒫({𝑥, 𝑦, 𝑧}), ≤) 7 where 𝒫({𝑥, 𝑦, 𝑧}) is the set of partitions on {𝑥, 𝑦, 𝑧} and with ≤ defined as in Proposition 14.8 (page 217) (ℛ({𝑥, 𝑦}), ⊆) where ℛ({𝑥, 𝑦}) is the set of rings of sets on {𝑥, 𝑦} (𝒜({𝑥, 𝑦, 𝑧}), ⊆) where 𝒜({𝑥, 𝑦, 𝑧}) is the set of algebras of sets on {𝑥, 𝑦, 𝑧}. See Example 14.11 (page 217), Example 14.7 (page 215), Example 14.16 (page 228), and Figure 14.7 (page 230). Lemma 6.2. 8 l 𝙇 is an M3 lattice e ⟹ { m { (Definition 6.3 page 93) }

1. 2.

𝙇 is not distributive 𝙇 is modular

(Definition 6.2 page 90) (Definition 5.3 page 80)

and

}

✎PROOF: 1. Proof that 𝑀3 is non-distributive: 𝑥 ∧ (𝑎 ∨ 𝑐) = 𝑥 ∧ 𝑦

by def. of l.u.b. page 58

=𝑥

by def. of g.l.b. page 58

≠𝑏 =𝑏∨𝑏

by Theorem 4.3 page 62 (idempotent property)

= (𝑥 ∧ 𝑎) ∨ (𝑥 ∧ 𝑐) ⏟ ⏟ 𝑏

by def. of g.l.b. page 58

𝑏

2. Proof that 𝑀3 is modular: (proof by exhaustion) = 𝑦 ∧ (𝑥 ∨ 𝑎) 𝑥 ∨ (𝑦 ∧ 𝑏) = 𝑥 ∨ 𝑏 𝑥 ∨ (𝑦 ∧ 𝑎) = 𝑥 ∨ 𝑎 =𝑦

= 𝑦 ∧ (𝑥 ∨ 𝑏) 𝑥 ∨ (𝑦 ∧ 𝑐) = 𝑥 ∨ 𝑐

=𝑥

=𝑦

=𝑦∧𝑥

=𝑦∧𝑦

=𝑦∧𝑦 6

📘 Beran (1985) pages 12–13, 📘 Korselt (1894) page 157 ⟨𝑝1 ≡ 𝑥, 𝑝2 ≡ 𝑦, 𝑝3 ≡ 𝑧, 𝑔 ≡ 1, 0 ≡ 0⟩ 📘 Saliǐ (1988) page 22 8 📘 Birkhoff (1948) page 6, 📘 Burris and Sankappanavar (1981) page 11, 📘 Korselt (1894) page 157 ⟨cf Salii1988 p. 7

37⟩ 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 94


= 𝑦 ∧ (𝑥 ∨ 𝑐)

𝑏 ∨ (𝑥 ∧ 𝑎) = 𝑏 ∨ 𝑏 𝑏 ∨ (𝑦 ∧ 𝑎) = 𝑏 ∨ 𝑎

𝑎 ∨ (𝑦 ∧ 𝑥) = 𝑎 ∨ 𝑥

=𝑏

=𝑎

=𝑥∧𝑎

=𝑦

=𝑦∧𝑎

= 𝑥 ∧ (𝑏 ∨ 𝑎)

=𝑦∧𝑦

= 𝑦 ∧ (𝑏 ∨ 𝑎)

= 𝑦 ∧ (𝑎 ∨ 𝑥)

𝑏 ∨ (𝑥 ∧ 𝑐) = 𝑏 ∨ 𝑏

𝑏 ∨ (𝑦 ∧ 𝑥) = 𝑏 ∨ 𝑥

𝑎 ∨ (𝑦 ∧ 𝑏) = 𝑎 ∨ 𝑏

=𝑏

=𝑥

=𝑥∧𝑐

=𝑎

=𝑦∧𝑥

= 𝑥 ∧ (𝑏 ∨ 𝑐)

=𝑦∧𝑎

= 𝑦 ∧ (𝑏 ∨ 𝑥)

= 𝑦 ∧ (𝑎 ∨ 𝑏)

𝑏 ∨ (𝑥 ∧ 𝑦) = 𝑏 ∨ 𝑥

𝑏 ∨ (𝑦 ∧ 𝑐) = 𝑏 ∨ 𝑐

𝑎 ∨ (𝑦 ∧ 𝑐) = 𝑎 ∨ 𝑐

=𝑥

=𝑐

=𝑥∧𝑦

=𝑦

=𝑦∧𝑐

= 𝑥 ∧ (𝑏 ∨ 𝑦)

=𝑦∧𝑦

= 𝑦 ∧ (𝑏 ∨ 𝑐)

= 𝑦 ∧ (𝑎 ∨ 𝑐)

𝑏 ∨ (𝑐 ∧ 𝑥) = 𝑏 ∨ 𝑏 𝑏 ∨ (𝑎 ∧ 𝑥) = 𝑏 ∨ 𝑏

𝑐 ∨ (𝑦 ∧ 𝑎) = 𝑐 ∨ 𝑎

=𝑏

=𝑏

=𝑐∧𝑥

=𝑦

=𝑎∧𝑥

= 𝑐 ∧ (𝑏 ∨ 𝑥)

=𝑦∧𝑦

= 𝑎 ∧ (𝑏 ∨ 𝑥)

= 𝑦 ∧ (𝑐 ∨ 𝑎)

𝑏 ∨ (𝑐 ∧ 𝑦) = 𝑏 ∨ 𝑐

𝑏 ∨ (𝑎 ∧ 𝑦) = 𝑏 ∨ 𝑎

𝑐 ∨ (𝑦 ∧ 𝑥) = 𝑐 ∨ 𝑥

=𝑐

=𝑎

=𝑐∧𝑦

=𝑦

=𝑎∧𝑦

= 𝑐 ∧ (𝑏 ∨ 𝑦)

=𝑦∧𝑦

= 𝑎 ∧ (𝑏 ∨ 𝑦)

= 𝑦 ∧ (𝑐 ∨ 𝑥)

𝑏 ∨ (𝑐 ∧ 𝑎) = 𝑏 ∨ 𝑏

𝑏 ∨ (𝑎 ∧ 𝑐) = 𝑏 ∨ 𝑏

𝑐 ∨ (𝑦 ∧ 𝑏) = 𝑐 ∨ 𝑏

=𝑏

=𝑏

=𝑐∧𝑎

=𝑐

=𝑎∧𝑐

= 𝑐 ∧ (𝑏 ∨ 𝑎)

=𝑦∧𝑐

= 𝑎 ∧ (𝑏 ∨ 𝑐)

= 𝑦 ∧ (𝑐 ∨ 𝑏)

✏ The Birkhoff distributivity criterion (next) demonstrates that a lattice is distributive if and only if it does not contain either the N5 or M3 lattices. If a lattice does contain either of these, it is not distributive. If a lattice is distributive, it does not contain either the N5 or M3 lattices. There was a similar theorem for modular lattices and the N5 lattice (Theorem 5.2 page 82). Theorem 6.2 (Birkhoff distributivity criterion). 9 Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. r r r ⎧ r t ⎪ 𝙇 does not contain 𝑁5 as a sublattice and r h 𝙇 is distributive ⟺ r ⎨ r r r m ⎪ r ⎩ 𝙇 does not contain 𝑀3 as a sublattice ✎PROOF: 1. Proof that 𝙇 is distributive ⟹ 𝙇 does not contain 𝑁5: This follows directly from Lemma 6.1 (page 92). 2. Proof that 𝙇 is distributive ⟹ 𝙇 does not contain 𝑀3: This follows directly from Lemma 6.2 (page 93). 9

📘 Burris and Sankappanavar (1981) page 12, 📘 Birkhoff (1948) page 134, 📃 Birkhoff and Hall (1934)

version 0.30


Daniel J. Greenhoe


page 95


3. Proof that 𝙇 is distributive ⟸ 𝑁5 ∉ 𝐿 and 𝑀3 ∉ 𝐿: (a) Proof that this statement is equivalent to 10 (𝙇 is nondistributive) ∧ (𝑁5 ∉ 𝐿) ⟹ (𝑀3 ∈ 𝐿) ∶ Let 𝑃 ≡ 𝑄 denote that statement 𝑃 is equivalent to statement 𝑄. Then … (𝙇 is distributive) ⟸ (𝑁5 ∉ 𝐿) ∧ (𝑀3 ∉ 𝐿) ≡ (𝙇 is nondistributive) ⟹ (𝑁5 ∈ 𝐿) ∨ (𝑀3 ∈ 𝐿)

contrapositive

≡ ¬(𝙇 is nondistributive) ∨ [(𝑁5 ∈ 𝐿) ∨ (𝑀3 ∈ 𝐿)]

by definition of ⟹ page 185

≡ [¬(𝙇 is nondistributive) ∨ (𝑁5 ∈ 𝐿)] ∨ (𝑀3 ∈ 𝐿)

by associative property page 185

≡ ¬¬[¬(𝙇 is nondistributive) ∨ ¬(𝑁5 ∉ 𝐿)] ∨ (𝑀3 ∈ 𝐿)

by involutary property page 185

≡ ¬[(𝙇 is nondistributive) ∧ (𝑁5 ∉ 𝐿)] ∨ (𝑀3 ∈ 𝐿)

by de Morgan's law page 185

≡ (𝙇 is nondistributive) ∧ (𝑁5 ∉ 𝐿) ⟹ (𝑀3 ∈ 𝐿)

by definition of ⟹ page 185

(b) Proof that 𝙇 is not distributive and 𝑁5 ∉ 𝐿 ⟹ 𝑀3 ∈ 𝐿: i. Because 𝑁5 ∉ 𝐿 and by Theorem 5.2 (page 82), 𝙇 is modular (so we can use the modularity property of Definition 5.3 page 80). ii. We will show that the five values defined below form an 𝑀3 lattice: 1 𝑏 ≜ (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) ∧ (𝑦 ∨ 𝑧) 𝑎 ≜ (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ∨ (𝑦 ∧ 𝑧)

𝑦̇

𝑥̇

𝑥̇ ≜ (𝑥 ∧ 𝑏) ∨ 𝑎

𝑧̇

𝑦 ̇ ≜ (𝑦 ∧ 𝑏) ∨ 𝑎 𝑧 ̇ ≜ (𝑧 ∧ 𝑏) ∨ 𝑎 iii. Proof that 𝑎 ≤ 𝑏:

0

𝑎 = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ∨ (𝑦 ∧ 𝑧)

by definition of 𝑎 (item (3(b)ii))

= (𝑥 ∧ 𝑦 ∧ 𝑥) ∨ (𝑥 ∧ 𝑧 ∧ 𝑧) ∨ (𝑦 ∧ 𝑧 ∧ 𝑧)

by idempotent property of lattices (page 62)

≤ (𝑥 ∨ 𝑥 ∨ 𝑦) ∧ (𝑦 ∨ 𝑧 ∨ 𝑧) ∧ (𝑥 ∨ 𝑧 ∨ 𝑧)

by minimax inequality Theorem 4.5 page 64

= (𝑥 ∨ 𝑦) ∧ (𝑦 ∨ 𝑧) ∧ (𝑥 ∨ 𝑧)

by idempotent property of lattices (page 62)

= (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) ∧ (𝑦 ∨ 𝑧)


=𝑏

by definition of 𝑏 (item (3(b)ii)) ⋀{ 𝑥 𝑦 𝑥 } ⎫ ⎧ ⎪ ⎪ ⎪ ⎪ ⋀{ 𝑥 𝑧 𝑧 } ⎬ ⋁⎨ ⎪ ⎪ ⎪ ⎩ ⋀{ 𝑦 𝑧 𝑧 } ⎪ ⎭

≤

⎧ ⎪ ⋀⎨ ⎪ ⎩

⋁ 𝑥 𝑥 𝑦

⋁ 𝑦 𝑧 𝑧

⋁ 𝑥 𝑧 𝑧

⎫ ⎪ ⎬ ⎪ ⎭

iv. Proof that 𝑎 ≤ 𝑥̇ ≤ 𝑦 ̇ ≤ 𝑧 ̇ ≤ 𝑏: A. By item (3(b)iii), 𝑎 ≤ 𝑏. B. By definition of ∧, (𝑥 ∧ 𝑏) must be less than or equal to 𝑏. C. By definition of ∨, (𝑥 ∧ 𝑏) ∨ 𝑎 must be greater than or equal to 𝑎. 10

email Consider the 3 statements: A: L is nondistributive B: M3 can be embedded in L C: N5 can be embedded in L Theorem 3.6 says: A (B or C) is true of lattices. This is equivalent to: (A --> (B or C)) and ((B or C) --> A) which is equivalent to ((A --> (B or C)) and ((B --> A) and (C --> A)) The last two implications are your (1) and (2). So what remains after these two is to show A --> (B or C) is true. But this is equivalent to not A or B or C which is equivalent to not A or not (not C) or B which is equivalent (by de Morgan’s Laws) to not (A and not C)) or B which is equivalent to (A and not C) --> B which is your (3).

Many many thanks to University of Waterloo Professor Emeritus Stanley Burris for his brilliant help with the logical structure of this proof. (If you are viewing this text as a pdf file, zoom in on the figure to the left to see text from Professor Burris' 2007 October 9 email.)

Note: one could have used (A and not B) --> C, which is your (4), instead of (3). But no need to prove both. The above is just an application of propositional logic. You can also use truth tables to show that A (B or C) and ((A and not C) --> B) and ((B --> A) and (C --> A)) have the same the same truth values, no matter how you assign TRUE/FALSE to each of A,B,C. So they are equivalent. Stanley Burris Professor Emeritus Dept. of Pure Mathematics University of Waterloo Waterloo, Ontario Canada N2L 3G1


Daniel J. Greenhoe


version 0.30

page 96


D. By definition of 𝑥̇ (item (3(b)ii)), 𝑎 ≤ 𝑥̇ ≤ 𝑏. E. The proofs for 𝑎 ≤ 𝑦 ̇ ≤ 𝑏 and 𝑎 ≤ 𝑧̇ ≤ 𝑏 are essentially identical to that of 𝑎 ≤ 𝑥̇ ≤ 𝑏. v. Proof that 𝑥̇ ∧ 𝑦 ̇ = 𝑥̇ ∧ 𝑧 ̇ = 𝑦 ̇ ∧ 𝑧 ̇ = 𝑎: 𝑥̇ ∧ 𝑦 ̇ = [(𝑥 ∧ 𝑏) ∨ 𝑎] ∧ 𝑦 ̇ ⏟⏟⏟⏟⏟⏟⏟

by definition of 𝑥̇ item (3(b)ii)

𝑥̇

= [(𝑥 ∧ 𝑏)∧𝑦]∨𝑎 ̇

by modularity page 80

= [(𝑥 ∧ 𝑏) ∧ ((𝑦 ∧ 𝑏) ∨ 𝑎)] ∨ 𝑎 ⏟⏟⏟⏟⏟⏟⏟

by definition of 𝑦 ̇ item (3(b)ii)

𝑦̇

= [(𝑥 ∧ 𝑏) ∧ (𝑦∨𝑎)∧𝑏] ∨ 𝑎


= [(𝑥 ∧ 𝑏) ∧ (𝑦 ∨ 𝑎)] ∨ 𝑎


⎞ ⎡⎛ ⎜ ⎟ ⎢ = ⎢⎜𝑥 ∧ [(𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) ∧ (𝑦 ∨ 𝑧)]⎟ ∧ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⎢⎜ ⎟ 𝑏 ⎣⎝ ⎠ ⎛ ⎞⎤ ⎜𝑦 ∨ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ [(𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ∨ (𝑦 ∧ 𝑧)]⎟⎥ ∨ 𝑎 ⎜ ⎟⎥ ⎝ ⎠⎦ 𝑎 = [(𝑥 ∧ (𝑦 ∨ 𝑧)) ∧ (𝑦 ∨ (𝑥 ∧ 𝑧))] ∨ 𝑎

by definitions of 𝑎 and 𝑏 item (3(b)ii) by absorption property page 62

= [𝑥 ∧ (𝑦∨((𝑦 ∨ 𝑧)∧(𝑥 ∧ 𝑧)))] ∨ 𝑎


= [𝑥 ∧ (𝑦 ∨ (𝑥 ∧ 𝑧))] ∨ 𝑎

because (𝑥 ∧ 𝑧) ≤ (𝑦 ∨ 𝑧)

= [(𝑥 ∧ 𝑧)∨(𝑥∧𝑦)] ∨ 𝑎


= [(𝑥 ∧ 𝑧) ∨ (𝑥 ∧ 𝑦)] ∨ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ [(𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ∨ (𝑦 ∧ 𝑧)] by definition of 𝑎 item (3(b)ii) 𝑎

= (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ∨ (𝑦 ∧ 𝑧)


=𝑎

by definition of 𝑎 item (3(b)ii)

vi. To prove that 𝑥̇ ∧ 𝑧̇ = 𝑎, simply replace 𝑦 ̇ with 𝑧̇ and 𝑦 with 𝑧 in item (3(b)v). vii. To prove that 𝑦 ̇ ∧ 𝑧 ̇ = 𝑎, simply replace 𝑥̇ with 𝑧̇ and 𝑥 with 𝑧 in item (3(b)v). viii. Proof that 𝑥̇ ∨ 𝑦 ̇ = 𝑏: 𝑥̇ ∨ 𝑦 ̇ = [(𝑥 ∧ 𝑏) ∨ 𝑎] ∨ 𝑦 ̇ ⏟⏟⏟⏟⏟⏟⏟

by definition of 𝑥̇ item (3(b)ii)

𝑥̇

= [(𝑥∨𝑎)∧𝑏] ∨ 𝑦 ̇


= [(𝑥 ∨ 𝑎)∨𝑦]∧𝑏 ̇


= [(𝑥 ∨ 𝑎) ∨ ((𝑦 ∧ 𝑏) ∨ 𝑎)] ∧ 𝑏 ⏟⏟⏟⏟⏟⏟⏟⏟⏟

by definition of 𝑦 ̇ item (3(b)ii)

𝑦̇

= [(𝑥 ∨ 𝑎) ∨ (𝑦 ∧ 𝑏)] ∧ 𝑏


⎡⎛ ⎞ = ⎢⎜𝑥 ∨ [(𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ∨ (𝑦 ∧ 𝑧)] ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⎟ ∨ ⎢⎜ ⎟ ⎣⎝ ⎠ 𝑎 ⎛ ⎞⎤ ⎜ ⎟ ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) ∧ (𝑦 ∨ 𝑧)]⎟⎥⎥ ∧ 𝑏 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⎜𝑦 ∧ [(𝑥 ⎜ ⎟⎥ 𝑏 ⎝ ⎠⎦ = [(𝑥 ∨ (𝑦 ∧ 𝑧)) ∨ (𝑦 ∧ (𝑥 ∨ 𝑧))] ∧ 𝑏 version 0.30


by definitions of 𝑎 and 𝑏 item (3(b)ii) by absorption property page 62 Daniel J. Greenhoe


page 97


= [𝑥 ∨ (𝑦 ∧ 𝑧) ∨ (𝑦 ∧ (𝑥 ∨ 𝑧))] ∧ 𝑏

by associative property page 62

= [𝑥 ∨ (𝑦∧[(𝑦 ∧ 𝑧)∨(𝑥 ∨ 𝑧)])] ∧ 𝑏


= [𝑥 ∨ (𝑦 ∧ (𝑥 ∨ 𝑧))] ∧ 𝑏

by Definition 3.21 and Definition 3.22

= [(𝑥 ∨ 𝑧)∧(𝑥∨𝑦)] ∧ 𝑏


= [(𝑥 ∨ 𝑧) ∧ (𝑥 ∨ 𝑦)] ∧ [(𝑥 ∨ 𝑧) ∧ (𝑥 ∨ 𝑦) ∧ (𝑦 ∨ 𝑧)] by definition of 𝑏 item (3(b)ii) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑏

= (𝑥 ∨ 𝑧) ∧ (𝑥 ∨ 𝑦) ∧ (𝑦 ∨ 𝑧)


=𝑏

by definition of 𝑏 item (3(b)ii)

ix. To prove that 𝑥̇ ∨ 𝑧 ̇ = 𝑏, simply replace 𝑦 ̇ with 𝑧̇ and 𝑦 with 𝑧 in item (3(b)viii). x. To prove that 𝑦 ̇ ∨ 𝑧 ̇ = 𝑏, simply replace 𝑥̇ with 𝑧̇ and 𝑥 with 𝑧 in item (3(b)viii).

✏ Theorem 6.3 (cancellation criterion). 11 Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. t 𝑥 ∨ 𝑧 = 𝑦 ∨ 𝑧 ∀𝑥,𝑦,𝑧∈𝑋 and (1) h 𝙇 is distributive ⟺ ⟹ 𝑥=𝑦 {{ } 𝑥 ∧ 𝑧 = 𝑦 ∧ 𝑧 ∀𝑥,𝑦,𝑧∈𝑋 (2) } m ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ cancellation property

✎PROOF: 1. Proof that distributive property ⟹ cancellation property: 𝑥 = 𝑥(𝑥 + 𝑧)

by absorbtive property (Theorem 4.3 page 62)

= 𝑥(𝑦 + 𝑧)

by (1)

= 𝑥𝑦 + 𝑥𝑧

by distributive hypothesis

= 𝑥𝑦 + 𝑦𝑧

by (2)

= 𝑦𝑥 + 𝑦𝑧


= 𝑦(𝑥 + 𝑧)


= 𝑦(𝑦 + 𝑧)

by (1)

=𝑦

by absorbtive property (Theorem 4.3 page 62)

2. Proof that distributive property ⟸ cancellation property: (a) Define 𝑎 ≜ 𝑥(𝑦 + 𝑧) 𝑏 ≜ 𝑦(𝑥 + 𝑧) 𝑐 ≜ 𝑧(𝑥 + 𝑦) 𝑑 ≜ (𝑥 + 𝑦)(𝑥 + 𝑧)(𝑦 + 𝑧) 11

📘 Blyth (2005) pages 67–68, 📃 Birkhoff and Hall (1934)


Daniel J. Greenhoe


version 0.30

page 98


(b) Proof that 𝑎𝑏 = 𝑥𝑦, 𝑎𝑐 = 𝑥𝑧, and 𝑏𝑐 = 𝑦𝑧: 𝑎𝑏 = [𝑥(𝑦 + 𝑧)][𝑦(𝑥 + 𝑧)]

by item (2a)

= [𝑥(𝑥 + 𝑧)][𝑦(𝑦 + 𝑧)]


= 𝑥𝑦


𝑎𝑐 = [𝑥(𝑦 + 𝑧)][𝑧(𝑥 + 𝑦)]

by item (2a)

= [𝑥(𝑥 + 𝑦)][𝑧(𝑧 + 𝑦)]


= 𝑥𝑧


𝑏𝑐 = [𝑦(𝑥 + 𝑧)][𝑧(𝑥 + 𝑦)]

by item (2a)

= [𝑦(𝑦 + 𝑥)][𝑧(𝑧 + 𝑥)]


= 𝑦𝑧


(c) Proof of some inequalities: 𝑎 = 𝑥(𝑦 + 𝑧)

by item (2a)

≤ (𝑥 + 𝑦)(𝑦 + 𝑧)

by definition of ∨

≤ (𝑥 + 𝑦)[(𝑥 + 𝑦) + 𝑧]


=𝑥+𝑦


𝑎 = 𝑥(𝑦 + 𝑧)

by item (2a)

= 𝑥(𝑧 + 𝑦)


≤ (𝑥 + 𝑧)(𝑧 + 𝑦)


≤ (𝑥 + 𝑧)[(𝑥 + 𝑧) + 𝑦]


=𝑥+𝑧


𝑏 = 𝑦(𝑥 + 𝑧)

by item (2a)

≤ (𝑥 + 𝑦)(𝑥 + 𝑧)


≤ (𝑥 + 𝑦)[(𝑥 + 𝑦) + 𝑧]


=𝑥+𝑦


𝑐 = 𝑧(𝑥 + 𝑦)

by item (2a)

≤ (𝑥 + 𝑧)(𝑥 + 𝑦)


≤ (𝑥 + 𝑧)[(𝑥 + 𝑧) + 𝑦]


=𝑥+𝑧


(d) Proof that 𝙇 is modular: 1

i. Consider the following 𝑁5 lattice:

𝑦 𝑥

𝑧 0

ii. For the 𝑁5 lattice, the cancellation property does not hold because 1 = 𝑥 + 𝑧 = 𝑦 + 𝑧 = 1 and 0 = 𝑥𝑧 = 𝑦𝑧 = 0, but yet 𝑥 ≠ 𝑦. iii. Because 𝑁5 does not support the cancellation property and by the hypothesis that 𝙇 does support the cancellation property, 𝙇 therefore does not contain 𝑁5. iv. Because 𝙇 does not contain 𝑁5 and by Theorem 5.2 (page 82), 𝙇 is modular. version 0.30


Daniel J. Greenhoe


page 99


(e) Proof that 𝑎 + 𝑏 = 𝑎 + 𝑐 = 𝑏 + 𝑐 = 𝑑: 𝑎 + 𝑏 = 𝑎 + 𝑦(𝑥 + 𝑧)

by definition of 𝑐 (item (2a) page 97)

= (𝑎 + 𝑦)(𝑥 + 𝑧)

by modularity: item (2c) and item (2d)

= [𝑥(𝑦 + 𝑧) + 𝑦](𝑥 + 𝑧)

by definition of 𝑎 (item (2a) page 97)

= [𝑦 + 𝑥(𝑦 + 𝑧)](𝑥 + 𝑧)


= (𝑦 + 𝑥)(𝑦 + 𝑧)(𝑥 + 𝑧)


= (𝑥 + 𝑦)(𝑥 + 𝑧)(𝑦 + 𝑧)


=𝑑

by definition of 𝑑 (item (2a) page 97)

𝑎 + 𝑐 = 𝑎 + 𝑧(𝑥 + 𝑦)


= (𝑎 + 𝑧)(𝑥 + 𝑦)


= [𝑥(𝑦 + 𝑧) + 𝑧](𝑥 + 𝑦)


= [𝑧 + 𝑥(𝑦 + 𝑧)](𝑥 + 𝑦)


= (𝑧 + 𝑥)(𝑦 + 𝑧)(𝑥 + 𝑦)


= (𝑥 + 𝑦)(𝑥 + 𝑧)(𝑦 + 𝑧)


=𝑑


𝑏 + 𝑐 = 𝑏 + 𝑧(𝑥 + 𝑦)


= (𝑏 + 𝑧)(𝑥 + 𝑦)


= [𝑦(𝑥 + 𝑧) + 𝑧](𝑥 + 𝑦)


= [𝑧 + 𝑦(𝑥 + 𝑧)](𝑥 + 𝑦)


= (𝑧 + 𝑦)(𝑥 + 𝑧)(𝑥 + 𝑦)


= (𝑥 + 𝑦)(𝑥 + 𝑧)(𝑦 + 𝑧)


=𝑑


(f) Proof that (𝑎 + 𝑦𝑧) + 𝑐 = (𝑏 + 𝑥𝑧) + 𝑐 and (𝑎 + 𝑦𝑧)𝑐 = (𝑏 + 𝑥𝑧)𝑐: (𝑎 + 𝑦𝑧) + 𝑐 = (𝑎 + 𝑏𝑐) + 𝑐

by item (2b)

= 𝑎 + (𝑐 + 𝑐𝑏)


=𝑎+𝑐


=𝑑

by item (2e)

=𝑏+𝑐

by item (2e)

= 𝑏 + (𝑐 + 𝑐𝑎)


= (𝑏 + 𝑎𝑐) + 𝑐


= (𝑏 + 𝑥𝑧) + 𝑐

by item (2b)

(𝑎 + 𝑦𝑧)𝑐 = 𝑐(𝑎 + 𝑦𝑧)


= 𝑐(𝑎 + 𝑏𝑐)

by item (2b)

= (𝑏𝑐 + 𝑎)𝑐


= 𝑏𝑐 + 𝑎𝑐


= 𝑎𝑐 + 𝑏𝑐


= (𝑎𝑐 + 𝑏)𝑐


= (𝑏 + 𝑎𝑐)𝑐


= (𝑏 + 𝑥𝑧)𝑐

by item (2b)

(g) Proof that 𝑎 + 𝑦𝑧 = 𝑏 + 𝑥𝑧: by item (2f) and cancellation hypothesis. 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 100


(h) Proof that 𝑎 + 𝑦𝑧 = 𝑑: 𝑎 + 𝑦𝑧 = (𝑎 + 𝑦𝑧) + (𝑎 + 𝑦𝑧)


= (𝑎 + 𝑦𝑧) + (𝑏 + 𝑥𝑧)

by item (2g)

= (𝑎 + 𝑏𝑐) + (𝑏 + 𝑎𝑐)

by item (2b)

= (𝑎 + 𝑎𝑐) + (𝑏 + 𝑏𝑐)


=𝑎+𝑏


=𝑑

by item (2e)

(i) Proof that 𝑧(𝑥 + 𝑦) = 𝑧𝑥 + 𝑧𝑦 (distributivity): 𝑧(𝑥 + 𝑦) = 𝑐

by item (2a)

= 𝑐(𝑐 + 𝑎)


= 𝑐(𝑎 + 𝑐)


= 𝑐𝑑

by item (2e)

= 𝑐(𝑎 + 𝑦𝑧)

by item (2h)

= 𝑐(𝑎 + 𝑏𝑐)

by item (2b)

= (𝑏𝑐 + 𝑎)𝑐


= 𝑏𝑐 + 𝑎𝑐


= 𝑦𝑧 + 𝑥𝑧

by item (2b)

= 𝑧𝑥 + 𝑧𝑦


✏

Algebraic characterizations 12

Let 𝘼 ≜ ( 𝑋, ∨, ∧ ; ≤) be an algebraic structure. = 𝑥 ⎧ 1. 𝑥 ∧ 𝑥 ⎪ 2. 𝑥 ∨ 1 = 1∨𝑥=1 p ⎪ 𝘼 is a r = 1∧𝑥=𝑥 ⟺ ⎨ 3. 𝑥 ∧ 1 p { distributive lattice } ⎪ 4. 𝑥 ∧ (𝑦 ∨ 𝑧) = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ⎪ 5. (𝑦 ∨ 𝑧) ∧ 𝑥 = (𝑦 ∧ 𝑥) ∨ (𝑧 ∧ 𝑥) ⎩

Proposition 6.1.

Theorem 6.4. 14 Let 𝘼 ≜ ( 𝑋, ∨, ∧ ; ≤) be an algebraic structure. t 1. 𝑥 ∧ (𝑥 ∨ 𝑦) = 𝑥 𝘼 is a h ⟺ { 2. 𝑥 ∧ (𝑦 ∨ 𝑧) = (𝑧 ∧ 𝑥) ∨ (𝑦 ∧ 𝑥) m { distributive lattice } 12 13 14

and

∀𝑥∈𝑋

and

∀𝑥∈𝑋

and


and


⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

13

Let 𝘼 ≜ ( 𝑋, ∨, ∧ ; ≤) be an algebraic structure. 1. 𝑥 ∧ 𝑥 = 𝑥 ⎧ ⎪ 2. 𝑥 ∨ 𝑦 = 𝑦∨𝑥 ⎪ p 3. 𝑥 ∧ 𝑦 = 𝑦∧𝑥 ⎪ 𝘼 is a r ⟺ ⎨ p { distributive lattice } ⎪ 4. 𝑥 ∧ (𝑦 ∧ 𝑧) = (𝑥 ∧ 𝑦) ∧ 𝑧 ⎪ 5. 𝑥 ∧ (𝑥 ∨ 𝑦) = 𝑥 ⎪ ⎩ 6. 𝑥 ∧ (𝑦 ∨ 𝑧) = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)

Proposition 6.2.

∀𝑥∈𝑋

∀𝑥∈𝑋

and

∀𝑥,𝑦∈𝑋

and

∀𝑥,𝑦∈𝑋

and


and

∀𝑥,𝑦∈𝑋

and

∀𝑥,𝑦,𝑧∈𝑋.

∀𝑥,𝑦∈𝑋 ∀𝑥,𝑦,𝑧∈𝑋

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

and

}

📘 Birkhoff (1948) pages 135–136, 📘 Birkhoff and Birkhoff (1946) ⟨???⟩ 📘 Padmanabhan and Rudeanu (2008) page 58, 📘 Birkhoff (1948) pages 134–135 ⟨Ex.6⟩ 📘 Padmanabhan and Rudeanu (2008) page 59, 📘 Sholander (1951) page 28 ⟨P1, P2⟩

version 0.30


Daniel J. Greenhoe


page 101


✎PROOF: 1. Proof that 𝑥𝑥 = 𝑥 (meet idempotent property): 𝑥𝑥 = 𝑥[𝑥(𝑥 + 𝑥)]

by 1

= 𝑥(𝑥𝑥 + 𝑥𝑥)

by 2

= 𝑥𝑥𝑥 + 𝑥𝑥𝑥

by 2

= 𝑥𝑥𝑥(𝑥 + 𝑥) + 𝑥𝑥𝑥(𝑥 + 𝑥)

by 1

= 𝑥𝑥(𝑥𝑥 + 𝑥𝑥) + 𝑥𝑥(𝑥𝑥 + 𝑥𝑥)

by 2

= 𝑥𝑥 + 𝑥𝑥

by 1

= 𝑥(𝑥 + 𝑥)

by 2

=𝑥

by 1

2. Proof that 𝑥 + 𝑥 = 𝑥 (join idempotent property): 𝑥 + 𝑥 = 𝑥𝑥 + 𝑥𝑥

by meet idempotent property (item (1) page 101)

= 𝑥(𝑥 + 𝑥)

by 2

=𝑥

by 1

3. Proof that 𝑥𝑦 = 𝑦𝑥 (meet commutative property): 𝑥𝑦 = 𝑥𝑦 + 𝑥𝑦

by join idempotent property (item (2) page 101)

= 𝑦(𝑥 + 𝑥)

by 2

= 𝑦𝑥

by join idempotent property (item (2) page 101)

4. Proof that 𝑥(𝑦 + 𝑧) = 𝑥𝑦 + 𝑥𝑧 (conjunctive distributive property): 𝑥(𝑦 + 𝑧) = 𝑦𝑥 + 𝑧𝑥 = 𝑥𝑦 + 𝑥𝑧

by 2 by meet commutative property (item (3) page 101)

5. Proof that 𝑥 + 𝑥𝑦 = 𝑥 (join absorptive property): 𝑥 = 𝑥(𝑥 + 𝑦)

by 1

= 𝑦𝑥 + 𝑥𝑥

by 2

= 𝑦𝑥 + 𝑥


= (𝑦𝑥 + 𝑥)(𝑦𝑥 + 𝑥)


= 𝑥(𝑦𝑥 + 𝑥) + 𝑦𝑥(𝑦𝑥 + 𝑥)

by 2

= 𝑥(𝑦𝑥 + 𝑥) + 𝑦𝑥

by 1

= [𝑥𝑥 + (𝑦𝑥)𝑥] + 𝑦𝑥

by 2

= 𝑥(𝑦𝑥 + 𝑥) + 𝑦𝑥

by 2

= 𝑥(𝑦𝑥 + 𝑥𝑥) + 𝑦𝑥


= 𝑥[𝑥(𝑥 + 𝑦)] + 𝑦𝑥

by 2

= 𝑥𝑥 + 𝑦𝑥

by 1

= 𝑥 + 𝑦𝑥


= 𝑥 + 𝑥𝑦

by meet commutative property (item (3) page 101)


Daniel J. Greenhoe


version 0.30

page 102


6. Proof that 𝑥 + 𝑦 = 𝑦 + 𝑥 (join commutative property): 𝑥 + 𝑦 = (𝑥 + 𝑦)(𝑥 + 𝑦)


= 𝑦(𝑥 + 𝑦) + 𝑥(𝑥 + 𝑦)

by 2

= 𝑦(𝑥 + 𝑦) + 𝑥

by 1

= (𝑦𝑦 + 𝑥𝑦) + 𝑥

by 2

= (𝑦 + 𝑥𝑦) + 𝑥


= (𝑦 + 𝑦𝑥) + 𝑥


=𝑦+𝑥

by join absorptive property (item (5) page 101)

7. Proof that (𝑥 + 𝑦) + 𝑧 = 𝑥 + (𝑦 + 𝑧) (join associative property): (a) Let 𝑃 ≜ (𝑥 + 𝑦) + 𝑧 and 𝑄 ≜ 𝑥 + (𝑦 + 𝑧) (b) Proof that 𝑃 𝑥 = 𝑥, 𝑃 𝑦 = 𝑦, and 𝑃 𝑧 = 𝑧: 𝑃 𝑥 = [(𝑥 + 𝑦) + 𝑧]𝑥

by definition of 𝑃

(item (7a) page 102)

= 𝑥[(𝑥 + 𝑦) + 𝑧]


= 𝑥(𝑥 + 𝑦) + 𝑥𝑧

by conjunctive distributive property (item (4) page 101)

= 𝑥 + 𝑥𝑧

by 1

=𝑥


𝑃 𝑦 = [(𝑥 + 𝑦) + 𝑧]𝑦



= 𝑦[(𝑥 + 𝑦) + 𝑧]


= 𝑦(𝑥 + 𝑦) + 𝑦𝑧


= 𝑦(𝑦 + 𝑥) + 𝑦𝑧

by join commutative property (item (6) page 102)

= 𝑦 + 𝑦𝑧

by 1

=𝑦


𝑃 𝑧 = [(𝑥 + 𝑦) + 𝑧]𝑧



= 𝑧[(𝑥 + 𝑦) + 𝑧]


= 𝑧[𝑧 + (𝑥 + 𝑦)]


=𝑧

by 1

(c) Proof that 𝑄𝑥 = 𝑥, 𝑄𝑦 = 𝑦, and 𝑄𝑧 = 𝑧: 𝑄𝑥 = [𝑥 + (𝑦 + 𝑧)]𝑥 = 𝑥[𝑥 + (𝑦 + 𝑧)]


=𝑥

by 1

𝑄𝑦 = [𝑥 + (𝑦 + 𝑧)]𝑦

by definition of 𝑄 (item (7a) page 102)

= 𝑦[𝑥 + (𝑦 + 𝑧)]


= 𝑦𝑥 + 𝑦(𝑦 + 𝑧)


= 𝑦𝑥 + 𝑦

by 2

= 𝑦 + 𝑦𝑥


=𝑦


𝑄𝑧 = [𝑥 + (𝑦 + 𝑧)]𝑧

version 0.30



= 𝑧[𝑥 + (𝑦 + 𝑧)]


= 𝑧𝑥 + 𝑧(𝑦 + 𝑧)


= 𝑧(𝑧 + 𝑦) + 𝑧𝑥


= 𝑧 + 𝑧𝑥

by 1

= 𝑧 + 𝑧𝑥

by 1

=𝑧



Daniel J. Greenhoe


page 103


(d) Proof that (𝑥 + 𝑦) + 𝑧 = 𝑥 + (𝑦 + 𝑧): (𝑥 + 𝑦) + 𝑧 = 𝑄𝑥 + (𝑄𝑦 + 𝑄𝑧)

by item (7c)

= 𝑄𝑥 + 𝑄(𝑦 + 𝑧)


= 𝑄[𝑥 + (𝑦 + 𝑧)]


= 𝑄𝑃


= 𝑃𝑄


= 𝑃𝑄


= 𝑃 [𝑥 + (𝑦 + 𝑧)]


= 𝑃 𝑥 + 𝑃 (𝑦 + 𝑧)


= 𝑃 𝑥 + (𝑃 𝑦 + 𝑃 𝑧)


= 𝑥 + (𝑦 + 𝑧)

by item (7b)

8. Proof that 𝑥 + 𝑦𝑧 = (𝑥 + 𝑦)(𝑥 + 𝑧) (disjunctive distributive property): (𝑥 + 𝑦)(𝑥 + 𝑧) = (𝑥 + 𝑦)𝑥 + (𝑥 + 𝑦)𝑧


= 𝑥(𝑥 + 𝑦) + 𝑧(𝑥 + 𝑦)


= 𝑥 + 𝑧(𝑥 + 𝑦)

by 1

= 𝑥 + (𝑧𝑥 + 𝑧𝑦)


= 𝑥 + (𝑥𝑧 + 𝑦𝑧)


= (𝑥 + 𝑥𝑧) + 𝑦𝑧

by join associatiave property (item (7) page 102)

= 𝑥 + 𝑦𝑧


9. Proof that (𝑥𝑦)𝑧 = 𝑥(𝑦𝑧) (meet associative property): (a) Let 𝑃 ≜ (𝑥𝑦)𝑧 and 𝑄 ≜ 𝑥(𝑦𝑧) (b) Proof that 𝑃 + 𝑥 = 𝑥, 𝑃 + 𝑦 = 𝑦, and 𝑃 + 𝑧 = 𝑧: 𝑃 + 𝑥 = (𝑥𝑦)𝑧 + 𝑥



= 𝑥 + (𝑥𝑦)𝑧


= [𝑥 + (𝑥𝑦)][𝑥 + 𝑧]

by disjunctive distributive property (item (8) page 103)

= 𝑥[𝑥 + 𝑧]

by 1

=𝑥

by 1

𝑃 + 𝑦 = (𝑥𝑦)𝑧 + 𝑦



= 𝑦 + (𝑥𝑦)𝑧


= 𝑦 + (𝑦𝑥)𝑧


= [𝑦 + (𝑦𝑥)][𝑦 + 𝑧]


= 𝑦[𝑦 + 𝑧]

by 1

=𝑦

by 1

𝑃 + 𝑧 = (𝑥𝑦)𝑧 + 𝑧



= 𝑧 + (𝑥𝑦)𝑧


= 𝑧 + 𝑧(𝑦𝑥)


=𝑧

by 1


Daniel J. Greenhoe


version 0.30

page 104


(c) Proof that 𝑄 + 𝑥 = 𝑥, 𝑄 + 𝑦 = 𝑦, and 𝑄 + 𝑧 = 𝑧: 𝑄 + 𝑥 = 𝑥(𝑦𝑧) + 𝑥


= 𝑥 + 𝑥(𝑦𝑧)


=𝑥

by 1

𝑄 + 𝑦 = 𝑥(𝑦𝑧) + 𝑦


= 𝑦 + 𝑥(𝑦𝑧)


= (𝑦 + 𝑥)(𝑦 + 𝑦𝑧)


= (𝑦 + 𝑥)𝑦

by 1

= 𝑦(𝑦 + 𝑥)


=𝑦

by 1

𝑄 + 𝑧 = 𝑥(𝑦𝑧) + 𝑧


= 𝑧 + 𝑥(𝑦𝑧)


= (𝑧 + 𝑥)(𝑧 + 𝑦𝑧)


= (𝑧 + 𝑥)(𝑧 + 𝑧𝑦)


= (𝑧 + 𝑥)𝑧

by 1

= 𝑧(𝑧 + 𝑥)


=𝑧

by 1

(d) Proof that (𝑥𝑦)𝑧 = 𝑥(𝑦𝑧): (𝑥𝑦)𝑧 = [(𝑄 + 𝑥)(𝑄 + 𝑦)](𝑄 + 𝑧)

by item (9c)

= (𝑄 + 𝑥𝑦)(𝑄 + 𝑧)


= 𝑄 + (𝑥𝑦)𝑧


=𝑄+𝑃


=𝑃 +𝑄


= 𝑃 + 𝑥(𝑦𝑧)


= (𝑃 + 𝑥)(𝑃 + 𝑦𝑧)


= (𝑃 + 𝑥)[(𝑃 + 𝑦)(𝑃 + 𝑧)]


= 𝑥(𝑦𝑧)

by item (9b)


10. Proof that 𝘼 is a distributive lattice: (a) Proof that 𝘼 is a lattice: i. ii. iii. iv. v.

𝘼 is idempotent by item (1) and item (2). 𝘼 is commutative by item (3) and item (6). 𝘼 is associative by item (9) and item (7). 𝘼 is absorptive by 1 and item (5). Because 𝘼 is idempotent, commutative, associative, and absorptive, then by Theorem 4.3 (page 62), 𝘼 is a lattice.

(b) Proof that 𝘼 is distributive: by item (4) and Definition 6.2 (page 90).

✏

6.2.3

Properties

Distributive lattices are a special case of modular lattices. That is, all distributive lattices are modular, but not all modular lattices are distributive (next theorem). An example is the M3 lattice— it version 0.30


Daniel J. Greenhoe


page 105


is modular, but yet it is not distributive (Lemma 6.2 page 93). Theorem 6.5. 15 Let ( 𝑋, ∨, ∧ ; ≤) be a lattice. t ⟹ h ( 𝑋, ∨, ∧ ; ≤) is distributive ⟸ / m

( 𝑋, ∨, ∧ ; ≤) is modular.

✎PROOF: 1. Proof that distributivity ⟹ modularity: 𝑥 ∨ (𝑦 ∧ 𝑧) = (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧)


= 𝑦 ∧ (𝑥 ∨ 𝑧)

by 𝑥 ≤ 𝑦 hypothesis

2. Proof that distributivity ⟸ / modularity: By Lemma 6.2 page 93, the 𝑀3 lattice is modular, but yet it is non-distributive.

✏ Theorem 6.6 (Birkhoff's Theorem). 16 Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. Let 𝟚𝑋 be the power set of some set 𝑋. t 𝙇 is 𝙇 is isomorphic to a sublattice of ( 𝟚𝑋 , ∪, ∩ ; ⊆) h ⟹ { for some set 𝑋. } m { distributive } Theorem 6.7. Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. tautology ⎧ t ⎪ 𝑁 ⎞ 𝑁 𝙇 is h ⟹ ⎨ ⎛⎜ ⎟ (𝑥 ∨ 𝑦) 𝑥 ∨𝑦= m { distributive } ⋀ 𝑛 ⎪ ⎜⋀ 𝑛 ⎟ 𝑛=1 𝑛=1 ⎩ ⎝ ⎠

dual

⎫ ⎪ ⎛ ⎞ ⎜ 𝑥𝑛 ⎟ ∧ 𝑦 = (𝑥 ∧ 𝑦) ⎬ ⋁ 𝑛 ⎪ ⎜⋁ ⎟ 𝑛=1 ⎭ ⎝𝑛=1 ⎠ 𝑁

𝑁

✎PROOF: 𝑁 1. Proof that (⋀𝑁 𝑛=1 𝑥𝑛 ) ∨ 𝑦 = ⋁𝑛=1 (𝑥𝑛 ∨ 𝑦) (by induction):

Proof for 𝑁 = 1 case: ⎛𝑁=1 ⎞ ⎜ 𝑥𝑛 ⎟ ∨ 𝑦 = 𝑥1 ∨ 𝑦 ⎟ ⎜⋀ ⎝ 𝑛=1 ⎠

by definition of ∧

𝑁=1

=

⋀

(𝑥𝑛 ∨ 𝑦)


𝑛=1

Proof for 𝑁 = 2 case: ⎛𝑁=2 ⎞ ⎜ 𝑥𝑛 ⎟ ∨ 𝑦 = (𝑥1 ∨ 𝑦) ∧ (𝑥2 ∨ 𝑦) ⎟ ⎜⋀ ⎝ 𝑛=1 ⎠


𝑁=2

=

⋀

(𝑥𝑛 ∨ 𝑦)


𝑛=1

15 16

📘 Birkhoff (1948) page 134, 📘 Burris and Sankappanavar (1981) page 11 📘 Saliǐ (1988) page 24


Daniel J. Greenhoe


version 0.30

page 106


Proof that (𝑁 case) ⟹ (𝑁 + 1 case): 𝑁 ⎛𝑁+1 ⎞ ⎜ 𝑥𝑛 ⎟ ∨ 𝑦 = 𝑥𝑛 ∧ 𝑥𝑁+1 ∨ 𝑦 ⎜⋀ ⎟ [(⋀ ) ] 𝑛=1 𝑛=1 ⎝ ⎠


𝑁

=

[(⋀ 𝑛=1

𝑥𝑛

)


∨ 𝑦 ∧ (𝑥𝑁+1 ∨ 𝑦) ]

𝑁

=

(𝑥𝑛 ∨ 𝑦) ∧ (𝑥𝑁+1 ∨ 𝑦) ] [⋀ 𝑛=1

by left hypothesis

𝑁+1

=

⋀

(𝑥𝑛 ∨ 𝑦)


𝑛=1

𝑁 2. Proof that (⋁𝑁 𝑛=1 𝑥𝑛 ) ∧ 𝑦 = ⋀𝑛=1 (𝑥𝑛 ∧ 𝑦): by principle of duality (Theorem 4.4 page 63).

✏ Theorem 6.8. 17 Let ( 𝑋, ∨, ∧ ; ≤) be a lattice. t h (𝑋, ≤) is linearly ordered ⟹ (⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑋, ∨, ∧ ; ≤) is distributive ⏟ m ordered set

lattice

✎PROOF: 𝑥 ≤ 𝑦 ≤ 𝑧 ⟹ 𝑥 ∧ (𝑦 ∨ 𝑧)

=𝑥∧𝑧

=𝑥

=𝑥∨𝑥

= (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)

𝑥 ≤ 𝑧 ≤ 𝑦 ⟹ 𝑥 ∧ (𝑦 ∨ 𝑧)

=𝑥∧𝑦

=𝑥

=𝑥∨𝑥

= (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)

𝑧 ≤ 𝑥 ≤ 𝑦 ⟹ 𝑥 ∧ (𝑦 ∨ 𝑧)

=𝑥∧𝑦

=𝑥

=𝑥∨𝑧

= (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)

𝑦 ≤ 𝑧 ≤ 𝑥 ⟹ 𝑥 ∧ (𝑦 ∨ 𝑧)

=𝑥∧𝑧

=𝑧

=𝑦∨𝑧

= (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)

𝑦 ≤ 𝑥 ≤ 𝑧 ⟹ 𝑥 ∧ (𝑦 ∨ 𝑧)

=𝑥∧𝑧

=𝑥

=𝑦∨𝑥

= (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)

𝑧 ≤ 𝑦 ≤ 𝑥 ⟹ 𝑥 ∧ (𝑦 ∨ 𝑧)

=𝑥∧𝑦

=𝑦

=𝑦∨𝑧

= (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)

✏ Theorem 6.9. 18 Let 𝑌 𝑋 ≜ {𝖿 ∶ 𝑋 → 𝑌 } (the set of all functions from the set 𝑋 to the set 𝑌 ). t ( 𝑌 , >, ? ; ⋜) is a distributive lattice ⟹ ( 𝑌 𝑋 , ∨, ∧ ; ≤) is a distributive lattice h where 𝖿 ≤ 𝗀 ⟺ 𝖿(𝑥) ⋜ 𝗀(𝑥) ∀𝑥 ∈ 𝑋 m ✎PROOF: [𝖿 ∧ (𝗀 ∨ 𝗁)](𝑥) = 𝖿 (𝑥) ? (𝗀(𝑥) > 𝗁(𝑥)) = (𝖿(𝑥) ? 𝗀(𝑥)) > (𝖿 (𝑥) ? 𝗁(𝑥)) = [𝖿 ∧ 𝗀](𝑥) ∨ [𝖿 ∧ 𝗁](𝑥)

because ( 𝑌 , >, ? ; ⋜) is distributive because ( 𝑌 , >, ? ; ⋜) is distributive

✏ 17 18

📘 MacLane and Birkhoff (1999) page 484 📘 MacLane and Birkhoff (1999) page 484

version 0.30


Daniel J. Greenhoe


page 107


6.2.4 Examples Example 6.1. 19 For any pair of natural numbers 𝑛, 𝑚 ∈ ℕ, let 𝑛|𝑚 represent the relation “𝑚 divides 𝑛”, lcm(𝑛, 𝑚) the least common multiple of 𝑛 and 𝑚, and gcd(𝑛, 𝑚) the greatest common divisor of 𝑛 and 𝑚. e x ( ℕ, gcd, lcm ; |) is a distributive lattice.

30 6

10

2

3 1

15 5

✎PROOF: 1. For all 𝑚 ∈ ℕ, 𝑚 can be analyzed as a product of prime factors such that 𝖾(𝑘)

𝑚 = 2𝖾(1) 3𝖾(2) 5𝖾(3) 7𝖾(4) ⋯ 𝑝𝑘

where 𝖾(𝑛) is a function 𝖾 ∶ ℕ → 𝕎 expressing the number of prime factors 𝑝𝑛 in 𝑚. For example, 84 = 22 31 71

⟹

𝖾(1) = 2, 𝖾(2) = 1, 𝖾(3) = 0, 𝖾(4) = 1, 𝖾(5) = 0, 𝖾(6) = 0, …

2. Because 𝕎 is a chain and by Theorem 6.8 page 106, ( 𝕎, ∨, ∧ ; ≤) is a distributive lattice where ≤ is the standard ordering on 𝕎 and ∨ and ∧ are defined in terms of ≤. 3. Let 𝕎 ℕ represent the set of all functions 𝖾 ∶ ℕ → 𝕎. By Theorem 6.9 page 106, ( 𝕎 ℕ , >, ? ; ⋜) is also a distributive lattice where ⋜ is defined in terms of ≤ as 𝖾⋜𝖿

⟺

𝖾(𝑛) ≤ 𝖿 (𝑛)

∀𝑛 ∈ ℕ.

4. Again by Theorem 6.9 page 106, ( ℕ, gcd, lcm ; |) is a distributive lattice because 𝑚|𝑘 if 𝑒𝑚 (𝑛) ⋜ 𝑒𝑘 (𝑛).

✏ Proposition 6.3. 20 Let 𝑋𝑛 be a finite set with order 𝑛 = | 𝑋𝑛 |. Let 𝑙𝑛 be the number of unlabeled lattices on 𝑋𝑛 , 𝑚𝑛 the number of unlabeled modular lattices on 𝑋𝑛 , and 𝑑𝑛 the number of unlabeled distributive lattices on 𝑋𝑛 . 𝑛 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 t 𝑙 1 1 1 1 2 5 15 53 222 1078 5994 37622 𝑛 h 𝑚 1 1 1 1 2 4 8 16 34 72 157 343 m 𝑛 𝑑𝑛 1 1 1 1 2 3 5 8 15 26 47 82 151 269 494 Example 6.2. 21 There are a total of five unlabeled lattices on a five element set; and of these five, three are distributive (Proposition 6.3 page 107). Example 4.11 (page 67) illustrated all five of the unlabeled lattices, Example 5.5 (page 86) illustrated the 4 modular lattices, and the following table illustrates the 3 distributive lattices. Note that none of these lattices are complemented (none are Boolean Definition 8.1 page 115). non-distributive distributive e x

19

📘 MacLane and Birkhoff (1999) page 484, 📘 Sheffer (1920) page 310 ⟨footnote 1⟩ 𝑙𝑛 : 💻 Sloane (2014) ⟨http://oeis.org/A006966⟩ | 𝑚𝑛 : 💻 Sloane (2014) ⟨http://oeis.org/A006981⟩ | 𝑑𝑛 : 💻 Sloane (2014) ⟨http://oeis.org/A006982⟩ | 𝑙𝑛 : 📘 Heitzig and Reinhold (2002) | 𝑚𝑛 : 📘 Thakare et al. (2002)? | 𝑑𝑛 : 📘 Erné et al. (2002), page 17 21 📘 Erné et al. (2002), pages 4–5 20


Daniel J. Greenhoe


version 0.30

page 108


Example 6.3. 22 There are a total of 15 unlabeled lattices on a six element set; and of these 15, five are distributive (Proposition 6.3 page 107). Example 4.12 (page 67) illustrated all 15 of the unlabeled lattices, Example 5.6 (page 86) illustrated the 8 modular lattices, and the following illustrates the 5 distributive lattices. Note that none of these lattices are complemented (none are Boolean Definition 8.1 page 115). distributive lattices on 6 element sets e x

Example 6.4. 23 There are a total of 53 unlabeled lattices on a seven element set; and of these, 8 are distributive (Proposition 6.3 page 107). Example 4.13 (page 68) illustrated all 53 of the unlabeled lattices, Example 5.8 (page 87) illustrated the 16 modular lattices, and the following illustrates the 8 distributive lattices. Note that none of these lattices are complemented (none are Boolean Definition 8.1 page 115). distributive lattices on 7 element sets e x

22 23

📘 Erné et al. (2002), pages 4–5 📘 Erné et al. (2002), pages 4–5

version 0.30


Daniel J. Greenhoe


CHAPTER 7 COMPLEMENTED LATTICES

7.1 Definitions Definition 7.1. 1 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice (Definition 4.4 page 71). An element 𝑥′ ∈ 𝑋 is a complement of an element 𝑥 in 𝙇 if ′ 1. 𝑥 ∧ 𝑥 = 0 (non-contradiction) and ′ 2. 𝑥 ∨ 𝑥 = 1 (excluded middle). d An element 𝑥′ in 𝙇 is the unique complement of 𝑥 in 𝙇 if 𝑥′ is a complement of 𝑥 and 𝑦′ is a e ′ ′ f complement of 𝑥 ⟹ 𝑥 = 𝑦 . 𝙇 is complemented if every element in 𝑋 has a complement in 𝑋. 𝙇 is uniquely complemented if every element in 𝑋 has a unique complement in 𝑋. A complemented lattice that is not uniquely complemented is multiply complemented. A complemented lattice is optionally denoted ( 𝑋, ∨, ∧, 0, 1 ; ≤). Definition 7.1 (previous) introduced the concept of a complement of a lattice. Definition 7.2 (next) introduces the concept of a relative complement in an interval (Definition A.2 page 265). Definition 7.2. 2 Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. An element 𝑦 ∈ 𝑋 is a relative complement of 𝑥 in [𝑎, 𝑏] with respect to 𝙇 if 1. 𝑥 ∨ 𝑦 = 𝑏 and d e 2. 𝑥 ∧ 𝑦 = 𝑎. f A lattice 𝙇 is relatively complemented if every element in every closed interval [𝑎, 𝑏] in 𝙇 has a complement in [𝑎, 𝑏].

7.2 Examples Example 7.1. 3 The lattice ( 𝟚{𝑥,𝑦,𝑧} , ∪, ∩ ; ⊆) of Example 4.2 page 66 is a complemented lattice. The “lattice complement” of each element 𝐴 is simply the “set complement” 𝐴𝖼 ≜ 𝟚{𝑥,𝑦,𝑧} ⧵𝐴: 1 2 3

📘 Stern (1999) page 9, 📘 Birkhoff (1948) page 23 📘 Birkhoff (1948) page 23 📘 Svozil (1994) page 72

page 110

7.3. PROPERTIES

𝐴𝖼 𝖼∅ = {𝑥, 𝑦, 𝑧} 𝖼{𝑥} = {𝑦, 𝑧} 𝖼{𝑦} = {𝑥, 𝑧} 𝖼{𝑥, 𝑦} = {𝑧} 𝖼{𝑧} = {𝑥, 𝑦} 𝖼{𝑥, 𝑧} = {𝑦} 𝖼{𝑦, 𝑧} = {𝑥} 𝖼{𝑥, 𝑦, 𝑧} = ∅

e x

𝐴 ∪ 𝐴𝖼 ∅ ∪ {𝑥, 𝑦, 𝑧} = {𝑥, 𝑦, 𝑧} {𝑥} ∪ {𝑦, 𝑧} = {𝑥, 𝑦, 𝑧} {𝑦} ∪ {𝑥, 𝑧} = {𝑥, 𝑦, 𝑧} {𝑥, 𝑦} ∪ {𝑧} = {𝑥, 𝑦, 𝑧} {𝑧} ∪ {𝑥, 𝑦} = {𝑥, 𝑦, 𝑧} {𝑥, 𝑧} ∪ {𝑦} = {𝑥, 𝑦, 𝑧} {𝑦, 𝑧} ∪ {𝑥} = {𝑥, 𝑦, 𝑧} {𝑥, 𝑦, 𝑧} ∪ ∅ = {𝑥, 𝑦, 𝑧}

𝐴 ∩ 𝐴𝖼 ∅ ∩ {𝑥, 𝑦, 𝑧} = ∅ {𝑥} ∩ {𝑦, 𝑧} = ∅ {𝑦} ∩ {𝑥, 𝑧} = ∅ {𝑥, 𝑦} ∩ {𝑧} =∅ {𝑧} ∩ {𝑥, 𝑦} = ∅ {𝑥, 𝑧} ∩ {𝑦} =∅ {𝑦, 𝑧} ∩ {𝑥} =∅ {𝑥, 𝑦, 𝑧} ∩ ∅ =∅

Example 7.2 (factors of 12). 4 The lattice 𝙇 ≜ ( {1, 2, 3, 4, 6, 12}, lcm, gcd ; |) (illustrated to the right) is non-complemented. In particular, the elements 2 and 6 have no complements in 𝙇: 12 lcm(2, 3) = 6 ≠ 12 gcd(2, 3) = 1 lcm(2, 4) = 4 ≠ 12 gcd(2, 4) = 2 ≠ 1 6 4 lcm(2, 6) = 6 ≠ 12 gcd(2, 2) = 2 ≠ 1 3 2 lcm(6, 3) = 6 ≠ 12 gcd(6, 3) = 3 ≠ 1 lcm(6, 4) = 12 gcd(6, 4) = 2 ≠ 1 1 Example 7.3. 5 The lattice illustrated in the figure to the right is complemented. In this complemented lattice, complements are not unique. For example, the 𝑥 complement of 𝑥 is both 𝑦 and 𝑧, the complement of 𝑦 is both 𝑥 and 𝑧, and the complement of 𝑧 is both 𝑥 and 𝑦. Example 7.4. Here are some more examples: non-complemented lattices

1 𝑦 0

𝑧

uniquely complemented lattices

multiply complemented lattices

Example 7.5. Of the 53 unlabeled lattices on a 7 element set (Example 4.13 page 68), 0 are complemented with unique complements, e x 17 are complemented with multiple complements, and 36 are non-complemented.

7.3

Properties

Theorem 7.1 (next) is a landmark theorem in mathematics. Theorem 7.1. 4 5 6

6

📘 Durbin (2000) page 271, 📘 Saliǐ (1988) pages 26–27 📘 Durbin (2000) page 271 📃 Dilworth (1945) page 123, 📘 Saliǐ (1988) page 51, 📘 Grätzer (2003) page 378 ⟨Corollary 3.8⟩

version 0.30


Daniel J. Greenhoe


page 111

CHAPTER 7. COMPLEMENTED LATTICES

For every lattice 𝙇, there exists a lattice 𝙐 such that 1. 𝙇 ⊆ 𝙐 (𝙇 is a sublattice of 𝙐 ) and 2. 𝙐 is uniquely complemented.

t h m

“I therefore propose the following problem…”. With these words, Edward Huntington in a 1904 paper introduced one of the most famous problems in mathematical history;7 a question that took some 40 years to answer, and that in the end had a very surprising solution. Huntington's problem was essentially this: Are all uniquely complemented lattices also distributive?(Huntington, 1904)305 This question is significant because if a lattice is both complemented and distributive, then it is uniquely complemented (Corollary 7.1—next) and, more importantly, is a Boolean algebra (Definition 8.1 page 115). Being a Boolean algebra is very significant in that it implies the lattice has several powerful properties including that it satisfies de Morgan's laws (Theorem 4.3 page 62) and that it is isomorphic to an algebra of sets (Theorem 14.4 page 214). A uniquely complemented lattice that satifies any one of a number of other conditions is distributive (Theorem 7.2 page 111, Literature item 3 page 112). So there was ample evidence that the answer to Huntington's question is “yes”. But the final answer to Huntington's problem is actually “no”— an answer that took the mathematical community 40 years to find. The resulting effort had a profound impact on lattice theory in general. In fact, George Grätzer, in a 2007 paper, identified uniquely complemented lattices as one of the “two problems that shaped a century of lattice theory”.(Grätzer, 2007)696 This final solution to Huntington's problem was found by Robert Dilworth and published in a 1945 paper.8 And the answer is this: Every lattice is a sublattice of a uniquely complemented lattice (Theorem 7.1 page 110). To understand why this answers the question, consider either the M3 lattice (Definition 6.3 page 93) or the N5 lattice (Definition 5.4 page 82). Neither of these lattices are distributive (Theorem 6.2 page 94), but yet either of them can be a sublattice in a uniquely complemented lattice (by Dilworth's theorem). That is, it is therefore possible to have a lattice that is both uniquely complemented and non-distributive. Corollary 7.1. 9 Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. c 1. 𝙇 is distributive and o ⟹ { } 2. 𝙇 is complemented r

{𝙇 is uniquely complemented}

✎PROOF: 𝙇 is complemented ⟺ ∀𝑥 ∈ 𝐿∃𝑎, 𝑏

such that

𝑎, 𝑏 are complements of 𝑥 in 𝙇

by definition of complement page 109

⟺ 𝑥 ∨ 𝑎 = 1, 𝑥 ∨ 𝑏 = 1, 𝑥 ∧ 𝑎 = 0, 𝑥 ∧ 𝑏 = 0

by definition of complement page 109

⟹ 𝑎=𝑏


⟹ 𝙇 is uniquely complemented

✏ Theorem 7.2 (Huntington properties).

10

Let 𝙇 be a lattice.

7

For more discussion, see Literature item 7 page 113 📃 Dilworth (1945) page 123 9 📘 MacLane and Birkhoff (1999) page 488, 📘 Saliǐ (1988) page 30 ⟨Theorem 10⟩ 10 📘 Roman (2008) page 103, 📘 Adams (1990) page 79, 📘 Saliǐ (1988) page 40, 📃 Dilworth (1945) page 123, 📘 Grätzer (2007), page 698 8


Daniel J. Greenhoe


version 0.30

page 112

⎧ ⎪ ⎨ ⎪ ⎩

t h m

7.4. LITERATURE

𝙇 is uniquely complemented

or ⎧ 𝙇 is modular ⎫ ⎪ 𝙇 is atomic or ⎪ ⎫ 𝙇 is ⎪ ⎪ ⎪ or ⎬ ⟹ ⎬ and ⎨ 𝙇 is ortho-complemented { distributive } ⎪ ⎪ 𝙇 has finite width or ⎪ ⎭ ⎪ 𝙇 has de Morgan properties ⎪ ⎩ ⎭ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ Huntington properties

Theorem 7.3 (Peirce's Theorem). 11 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice. Let ∁𝑦 ≜ {𝑦′ ∈ 𝑋|𝑦′ is a complement of 𝑦}. t 1. 𝙇 is uniquely complemented and h {∀𝑦′ ∈ ∁𝑦, 𝑥 ≰ 𝑦′ ⟹ 𝑥 ∧ 𝑦 ≠ 0} ⟹ { } 2. 𝙇 is distributive m

7.4

Literature

📚 Literature survey: 1. General treatment of lattice varieties: 📘 Jipsen and Rose (1992) 2. Distributive lattices: 📘 Grätzer (1971) 📘 Balbes and Dwinger (1975) 📘 Dilworth (1984) 3. Uniquely complemented lattices: 📃 Dilworth (1945) ⟨“Every lattice is a sublattice of a lattice with unique complements.”⟩ 📘 Saliǐ (1988) 📘 Adams (1990) pages 79–84 📘 Grätzer (2007) 📘 Roman (2008) page 103 📘 Bergman (1929) ⟨uniquely complemented + modular = distributive⟩ 📘 Birkhoff (1940) ⟨uniquely complemented + ortho-complemented = distributive⟩ 📘 Birkhoff and Ward (1939a) ⟨uniquely complemented + atomic = distr.⟩ 📘 Birkhoff and Ward (1939b) ⟨uniquely complemented + atomic = distributive⟩ 4. Projective distributive lattices: 📘 Balbes (1967) 📘 Balbes and Horn (1970) 5. Median property: 📘 Birkhoff and Kiss (1947a) 📘 Birkhoff and Kiss (1947b) 📘 Grau (1947) 📘 Evans (1977) 📘 Isbell (1980) 📘 Bandelt and Hedlíková (1983) 📘 Birkhoff and Ward (1987) pages 1–8 📘 Artamonov (2000) page 554 ⟨median algebras⟩ 📘 Grätzer (2008) page 356 6. Properties of lattices (a) The fact that lattices are not in general distributive was not always universally accepted. In a famous 1880 paper, Charles S. Peirce(Peirce, 1880b)33 presents distributivity as a property of all lattices but says that “the proof is too tedius to give”. 11

📘 Saliǐ (1988) pages 38–39 ⟨“Peirce's Theorem”⟩, 📃 Peirce ⟨1902 January 31 entry⟩, 📃 Peirce (1903) ⟨letter to Huntington⟩, 📃 Peirce (1904) ⟨letter to Huntington⟩, 📃 Huntington (1904) version 0.30


Daniel J. Greenhoe


page 113

CHAPTER 7. COMPLEMENTED LATTICES 7. Note about Huntington's problem concerning uniquely complemented lattices:

(a) Salii12 suggests that Huntington's problem is actually motivated by and a simple extension of Peirce's Theorem (Theorem 7.3 page 112). That is, Huntington's problem is equivalent to asking if the uniquely complemented property is equivalent to the left hypothesis in Peirce's Theorem. (b) George Grätzer in a 2007 paper seems to indicate that Huntington's 1904 paper13 is not the original source of “Huntington's problem”. In particular, Grätzer says ”…Neither gives any references as to the origin of the problem. G. Birkhoff and M. Ward, 1933, reference E. V. Huntington, 1904, for the lattice axioms, which Huntington stated as being due to E. Schröder, but not for the problem. If the reader is surprised, I suggest he try to read the original paper of E. V. Huntington, and there he may find the clue. In my earlier papers on the subject, I reference only R. P. Dilworth, 1945, but in my lattice books (e.g., [7]) I give the correct reference. But I have no recollection of reading E. V. Huntington, 1904, until the preparation for this article.” (📘 Grätzer (2007), page 699) The reference [7] is 📘 Grätzer (2003). In this reference, Dilworth's 1945 theorem is presented on page 378, and its historical background is discussed on page 392. However, this discussion does not seem to give credit for Huntington's problem to anyone other than Huntington (1904). Perhaps it is Peirce that Grätzer has in mind with these comments—but so far the person referred to by Grätzer is unclear (to me). See also http://groups.google.com/group/sci.math/browse_thread/ thread/b7790be1efe8946e# 8. General treatment of lattice varieties: 📘 Jipsen and Rose (1992) 9. Atomic lattices: 📘 Birkhoff (1938), page 800 ⟨see footnote ‡⟩

📚

12 13

📘 Saliǐ (1988) pages 38–39 ⟨“Peirce's Theorem”⟩ 📃 Huntington (1904) page 305


Daniel J. Greenhoe


version 0.30

page 114

version 0.30

7.4. LITERATURE


Daniel J. Greenhoe


CHAPTER 8 BOOLEAN LATTICES

❝That

the symbolic processes of algebra, invented as tools of numerical calculation, should be competent to express every act of thought, and to furnish the grammar and dictionary of an all-containing system of logic, would not have been believed until it was proved.…by Mr. Boole. The unity of the forms of thought in all the applications of reason, however remotely separated, will one day be matter of notoriety and common wonder: and Boole's name will be remembered in connection with one of the most important steps towards the attainment of knowledge.❞ Augustus de Morgan (1806–1871), British mathematician and logician,

1

8.1 Definition and properties A Boolean algebra (next definition) is a bounded (Definition 4.4 page 71), distributive and complemented (Definition 7.1 page 109), lattice (Definition 4.3 page 61).

(Definition 6.2 page 90),

Definition 8.1. 2 The bounded lattice (Definition 4.4 page 71) 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) is Boolean if 1. 𝙇 is complemented (Definition 7.1 page 109) and d . 2. 𝙇 is distributive (Definition 6.2 page 90) e f A bounded lattice 𝙇 that is Boolean is a Boolean algebra or a Boolean lattice. A Boolean lattice with 2𝘕 elements is denoted 𝙇𝘕 2. Several examples of Boolean lattices are illustrated in Example 9.2 (page 140). Proposition 8.1. 1 2

quote: 📘 DeMorgan (1872) page 80 image: http://en.wikipedia.org/wiki/Augustus_De_Morgan 📘 MacLane and Birkhoff (1999) page 488, 📘 Jevons (1864)

page 116

8.2. ORDER PROPERTIES

The algebraic structure 𝘼 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) is a Boolean algebra (Definition 8.1 page 115) if 1. ( 𝑋, ∨, ∧, 0, 1 ; ≤) is a bounded lattice (Definition 4.4 page 71) and 2. 𝑥 ∧ (𝑦 ∨ 𝑧) = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ∀𝑥,𝑦,𝑧∈𝑋 (distributive) and . ′ 3. 𝑥 ∧ 𝑥 = 0 ∀𝑥∈𝑋 (non-contradiction) and ′ 4. 𝑥 ∨ 𝑥 = 1 ∀𝑥∈𝑋 (excluded middle).

p r p

✎PROOF:

✏

This follows directly from Definition 8.1 (page 115).

Boolean algebras support the principle of duality (next theorem). Theorem 8.1 (Principle of duality). 3 Let 𝘽 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a Boolean algebra. 𝜙 is an identity on 𝘽 in terms ⟹ 𝐓𝜙 is also an identity on 𝘽 t {of the operations } h ∨, ∧, ′, 0, and 1 m where the operator 𝐓 performs the following mapping on the operations in 𝑋 𝑋 : 0 → 1, 1 → 0, ∨ → ∧, ∧→∨

✎PROOF:

For each of the identities in the definition of Boolean algebras (Proposition 8.5 operator 𝐓 produces another identity that is also in the definition: 𝐓(1𝑎) 𝐓(1𝑏) 𝐓(2𝑎) 𝐓(2𝑏) 𝐓(3𝑎) 𝐓(3𝑏) 𝐓(4𝑎) 𝐓(4𝑏)

= = = = = = = =

𝐓[𝑥 ∨ 𝑦 𝐓[𝑥 ∧ 𝑦 𝐓[𝑥 ∨ (𝑦 ∧ 𝑧) 𝐓[𝑥 ∧ (𝑦 ∨ 𝑧) 𝐓[𝑥 ∨ 0 𝐓[𝑥 ∧ 1 𝐓[𝑥 ∨ 𝑥′ 𝐓[𝑥 ∧ 𝑥′

= = = = = = = =

𝑦 ∨ 𝑥] 𝑦 ∧ 𝑥] (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧)] (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)] 𝑥] 𝑥] 1] 0]

= = = = = = = =

[𝑥 ∧ 𝑦 [𝑥 ∨ 𝑦 [𝑥 ∧ (𝑦 ∨ 𝑧) [𝑥 ∨ (𝑦 ∧ 𝑧) [𝑥 ∧ 1 [𝑥 ∨ 0 [𝑥 ∧ 𝑥′ [𝑥 ∨ 𝑥′

= = = = = = = =

𝑦 ∧ 𝑥] 𝑦 ∨ 𝑥] (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)] (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧)] 𝑥] 𝑥] 0] 1]

= = = = = = = =

page

131), the

(1𝑏) (1𝑎) (2𝑏) (2𝑎) (3𝑏) (3𝑎) (4𝑏) (4𝑎)

Therefore, if the statement 𝜙 is consistent with regards to the Boolean algebra 𝘽, then 𝐓𝜙 is also consistent with regards to the Boolean algebra 𝘽. ✏

8.2

Order properties

The definition of Boolean algebras given by Definition 8.1 is a set of postulates known as Huntington's FIRST SET. Lemma 8.1 (next) gives a link between Huntington's FIRST SET of Boolean algebra postulates and the classic 10 set of Boolean algebra postulates (Theorem 8.2 page 120). Lemma 8.1. 3 4

4

Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice.

📘 Givant and Halmos (2009) pages 20–22 ⟨Chapter 4⟩, 📘 Sikorski (1969), page 8 📃 Huntington (1904) pages 292–296 ⟨“1st set”⟩, 📘 Joshi (1989) pages 224–227

version 0.30


Daniel J. Greenhoe


page 117

CHAPTER 8. BOOLEAN LATTICES

If ⎧ ⎪ ⎨ ⎪ ⎩

l e m

∀𝑥, 𝑦, 𝑧 ∈ 𝑋 ➀ ➁ ➂ ➃

then ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

1. 2. 3. 4. 5.

𝑥∨𝑦 𝑥 ∨ (𝑦 ∧ 𝑧) 𝑥∨0 𝑥 ∨ 𝑥′

= 𝑦∨𝑥 = (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) = 𝑥 = 1

∀𝑥, 𝑦, 𝑧 ∈ 𝑋 𝑥∨𝑥 = 𝑥 ∨ (𝑦 ∨ 𝑧) = 𝑥 ∨ (𝑥 ∧ 𝑦) = 𝑥∨1 = (𝑥 ∨ 𝑦)′ =

𝑥 (𝑥 ∨ 𝑦) ∨ 𝑧 𝑥 1 𝑥′ ∧ 𝑦′

𝑥∧𝑦 𝑥 ∧ (𝑦 ∨ 𝑧) 𝑥∧1 𝑥 ∧ 𝑥′

= = = =

𝑥∧𝑥 𝑥 ∧ (𝑦 ∧ 𝑧) 𝑥 ∧ (𝑥 ∨ 𝑦) 𝑥∧0 (𝑥 ∧ 𝑦)′

= = = = =

𝑦∧𝑥 (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) 𝑥 0

𝑥 (𝑥 ∧ 𝑦) ∧ 𝑧 𝑥 0 𝑥′ ∨ 𝑦′

(commutative)

and

(distributive) (identity) (complemented)

and and

(idempotent)

and

(associative)5

and

(absorptive)

and

(bounded)

and

(de Morgan's laws).

⎫ ⎪ ⎬ ⎪ ⎭

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

✎PROOF:

For each pair of properties, it is only necessary to prove one of them, as the other follows by the principle of duality (Theorem 8.1 page 116). Let the join ∨ be represented by +, the operation meet ∧ represented by ⋅ or juxtaposition, and let ∧ have algebraic precedence over ∨.

1. Proof that 𝑥 + 𝑥 = 𝑥 and 𝑥𝑥 = 𝑥 (idempotent properties): by identity property,

➂b

by complemented property,

➃a

= 𝑥 + (𝑥𝑥 )

by distributive property,

➁a

=𝑥+0


➃b

=𝑥

by identity property,

➂a

𝑥 + 𝑥 = (𝑥 + 𝑥) ⋅ 1 ′

= (𝑥 + 𝑥)(𝑥 + 𝑥 ) ′

2. Proof that 𝑥 + 1 = 1 and 𝑥 ⋅ 0 = 0 (bounded properties): 𝑥 + 1 = (𝑥 + 1) ⋅ 1

➂b

by commutative property,

➀b

′


➃a

′


➁a


➂b


➃a

= 1 ⋅ (𝑥 + 1) = (𝑥 + 𝑥 )(𝑥 + 1) = 𝑥 + (𝑥 ⋅ 1) =𝑥+𝑥


′

=1

3. Proof that 𝑥 + (𝑥𝑦) = 𝑥 and 𝑥(𝑥 + 𝑦) = 𝑥: (absorptive properties) by identity property,

➂b

= 𝑥 ⋅ (1 + 𝑦)


➁b

= 𝑥 ⋅ (𝑦 + 1)


➀a

=𝑥⋅1

by item (2)

=𝑥


𝑥 + (𝑥 ⋅ 𝑦) = (𝑥 ⋅ 1) + (𝑥𝑦)

➂b

4. Proof that (𝑥 + 𝑦) + 𝑧 = 𝑥 + (𝑦 + 𝑧) and (𝑥𝑦)𝑧 = 𝑥(𝑦𝑧) (associative properties): Let 𝑎 ≜ 𝑥(𝑦𝑧) and 𝑏 ≜ (𝑥𝑦)𝑧. 5

K.D. Joshi comments that having the associative property as a result of an axiom rather than as an axiom, is a very unusual and “remarkable property” in the world of algebras. 📘 Joshi (1989) pages 225–226 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 118


(a) Proof that 𝑎 + 𝑥 = 𝑏 + 𝑥: 𝑎 + 𝑥 = 𝑥(𝑦𝑧) + 𝑥


= 𝑥(𝑦𝑧) + 𝑥1


➂b

= 𝑥(𝑦𝑧 + 1)


➁a

= 𝑥(1)

by bounded property,

item (2)

=𝑥


➂b

= 𝑥(𝑥 + 𝑧)

by absorptive property,

item (3)

= (𝑥 + 𝑥𝑦)(𝑥 + 𝑧)


item (3)

= 𝑥 + (𝑥𝑦)𝑧


➁b

= (𝑥𝑦)𝑧 + 𝑥


➀a,b

=𝑏+𝑥

by definition of 𝑏

(b) Proof that 𝑎 + 𝑥′ = 𝑏 + 𝑥′ : 𝑎 + 𝑥′ = 𝑥(𝑦𝑧) + 𝑥′


′


➀a,b


➁b


➃a


➂b


➁b


➂b


➀b


➃a

= (𝑥 + 𝑥𝑦)(𝑥 + 𝑧)


➁b

= 𝑥′ + (𝑥𝑦)𝑧


➁b


➀a

= 𝑥 + 𝑥(𝑦𝑧) ′

′

= (𝑥 + 𝑥)(𝑥 + 𝑦𝑧) ′

= 1 ⋅ (𝑥 + 𝑦𝑧) ′

= 𝑥 + 𝑦𝑧 ′

′

= (𝑥 + 𝑦)(𝑥 + 𝑧) ′

′

= [(𝑥 + 𝑦) ⋅ 1](𝑥 + 𝑧) ′

′

= [1 ⋅ (𝑥 + 𝑦)](𝑥 + 𝑧) ′

′

′

= [(𝑥 + 𝑥 )(𝑥 + 𝑦)](𝑥 + 𝑧) ′

′

= (𝑥𝑦)𝑧 + 𝑥 =𝑏+𝑥

′

′


(c) Proof that 𝑥(𝑦𝑧) = (𝑥𝑦)𝑧: 𝑥(𝑦𝑧) ≜ 𝑎


=𝑎+𝑎

item (1)


➂a,b

′

′


➃a,b

′

′


➁a

= 𝑎 + 𝑎1 + 0 = 𝑎 + 𝑎(𝑥 + 𝑥 ) + 𝑥𝑥 = 𝑎 + 𝑎𝑥 + 𝑎𝑥 + 𝑥𝑥 ′

′


➀a,b

′

′

by idempotent property,

item (1)

′

′

= 𝑎 + 𝑎𝑥 + 𝑥𝑎 + 𝑥𝑥

= 𝑎𝑎 + 𝑎𝑥 + 𝑥𝑎 + 𝑥𝑥


➁a

′


➁a

′

by item (4a)

= 𝑎(𝑎 + 𝑥 ) + 𝑥(𝑎 + 𝑥 ) = (𝑎 + 𝑥)(𝑎 + 𝑥 ) = (𝑏 + 𝑥)(𝑎 + 𝑥 ) ′

= (𝑏 + 𝑥)(𝑏 + 𝑥 )

by item (4b)

= (𝑏 + 𝑥)𝑏 + (𝑏 + 𝑥)𝑥

′

′


➁a


➀b

′


➁a

′

= 𝑏 + 𝑏𝑥 + 𝑥 𝑏 + 𝑥 𝑥


item (1)

′


➀b

= 𝑏(𝑏 + 𝑥) + 𝑥 (𝑏 + 𝑥) ′

= 𝑏𝑏 + 𝑏𝑥 + 𝑥 𝑏 + 𝑥 𝑥 ′

′

= 𝑏 + 𝑏𝑥 + 𝑏𝑥 + 𝑥 𝑥 version 0.30



Daniel J. Greenhoe


page 119


= 𝑏 + 𝑏(𝑥 + 𝑥′ ) + 𝑥′ 𝑥


➁a

=𝑏+𝑏⋅1+0


➃a,b

=𝑏+𝑏


➂a,b

=𝑏


item (1)

≜ (𝑥𝑦)𝑧


5. Proof that (𝑥 + 𝑦)′ = 𝑥′ 𝑦′ and (𝑥𝑦)′ = 𝑥′ + 𝑦′ : (de Morgan properties) (a) Proof that (𝑥 + 𝑦) + (𝑥′ 𝑦′ ) = 1: (𝑥 + 𝑦) + (𝑥′ 𝑦′ ) = [(𝑥 + 𝑦) + 𝑥′ ][(𝑥 + 𝑦) + 𝑦′ ]


➁a


➀a


➂b


➁b


➃a


➁a

= [𝑥 + 𝑥][𝑦 + 𝑦]


item (3)

= [1][1]


➃a

=1


item (2)


➁b


➂a


➃b

′

′

= [𝑥 + (𝑥 + 𝑦)][𝑦 + (𝑥 + 𝑦)] ′

′

= [(𝑥 + (𝑥 + 𝑦))1][(𝑦 + (𝑥 + 𝑦))1] ′

′

= [1(𝑥 + (𝑥 + 𝑦))][1(𝑦 + (𝑦 + 𝑥))] ′

′

′

′

= [(𝑥 + 𝑥)(𝑥 + (𝑥 + 𝑦))][(𝑦 + 𝑦)(𝑦 + (𝑦 + 𝑥))] ′

′

= [𝑥 + (𝑥(𝑥 + 𝑦))][𝑦 + (𝑦(𝑦 + 𝑥))] ′

′

(b) Proof that (𝑥 + 𝑦)(𝑥′ 𝑦′ ) = 0: (𝑥 + 𝑦)(𝑥′ 𝑦′ ) = [𝑥(𝑥′ 𝑦′ )] + [𝑦(𝑥′ 𝑦′ )] ′ ′

′ ′

= [0 + 𝑥(𝑥 𝑦 )] + [0 + 𝑦(𝑥 𝑦 )] = [(𝑥𝑥′ ) + 𝑥(𝑥′ 𝑦′ )] + [(𝑦𝑦′ ) + 𝑦(𝑥′ 𝑦′ )] ′

′ ′

′

′ ′

= [𝑥(𝑥 + 𝑥 𝑦 )] + [𝑦(𝑦 + 𝑥 𝑦 )] = [𝑥𝑥′ ] + [𝑦𝑦′ ]


➁b


item (3)

= [0] + [0]


➃b

=0


item (2)

(c) Proof that (𝑥 + 𝑦)′ = 𝑥′ 𝑦′ : The quanities (𝑥 + 𝑦) and 𝑥′ 𝑦′ are complements of each other as demonstrated by item (5a) ((𝑥 + 𝑦) + (𝑥′ 𝑦′ ) = 1) and item (5b) ((𝑥 + 𝑦)(𝑥′ 𝑦′ ) = 0). Therefore, (𝑥 + 𝑦)′ = 𝑥′ 𝑦′ .

✏ Proposition 8.2. 6 Let 𝘽 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a Boolean algebra. The pair (𝑋, ≤) is an ordered set. In particular, p 1. 𝑥 ≤ 𝑥 ∀𝑥∈𝑋 (reflexive) r 2. 𝑥 ≤ 𝑦 and 𝑦 ≤ 𝑧 ⟹ 𝑥 ≤ 𝑧 ∀𝑥,𝑦,𝑧∈𝑋 (transitive) p 3. 𝑥 ≤ 𝑦 and 𝑦 ≤ 𝑥 ⟹ 𝑥 = 𝑦 ∀𝑥,𝑦∈𝑋 (anti-symmetric).

and and

✎PROOF: 1. Proof that ≤ is reflexive in (𝑋, ≤): 𝑥≤𝑥 ⟺ 𝑥∨𝑥=𝑥 ⟺ true 6

by definition of ≤ (Definition 8.1 page 115) by Lemma 8.1 page 116

📘 Sikorski (1969), page 7


Daniel J. Greenhoe


version 0.30

page 120


2. Proof that ≤ is transitive in (𝑋, ≤): {(𝑥 ≤ 𝑦) and (𝑦 ≤ 𝑧)} ⟺ {(𝑥 ∨ 𝑦 = 𝑦) and (𝑦 ∨ 𝑧 = 𝑧)} by definition of ≤ (Definition 8.1 page 115) ⟹ (𝑥 ∨ 𝑧) = 𝑥 ∨ (𝑦 ∨ 𝑧) = (𝑥 ∨ 𝑦) ∨ 𝑧

by associative property of Lemma 8.1 page 116

=𝑦∨𝑧 =𝑧 3. Proof that ≤ is anti-symmetric in (𝑋, ≤): {(𝑥 ≤ 𝑦) and (𝑦 ≤ 𝑥)} ⟺ {(𝑥 ∨ 𝑦 = 𝑦) and (𝑦 ∨ 𝑥 = 𝑥)} by definition of ≤ (Definition 8.1 page 115) ⟺ {(𝑥 ∨ 𝑦 = 𝑦) and (𝑥 ∨ 𝑦 = 𝑥)} by commutative property of Definition 8.1 page 115 ⟺ 𝑥=𝑥∨𝑦=𝑦 ⟹ 𝑥=𝑦

✏ Proposition 8.3. Let ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a Boolean algebra. p 𝑥 ∨ 𝑦 is the least upper bound of 𝑥 and 𝑦 in (𝑋, ≤). r 𝑥 ∧ 𝑦 is the greatest lower bound of 𝑥 and 𝑦 in (𝑋, ≤). p Theorem 8.2 (classic 10 Boolean properties). 7 𝘼 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) is a Boolean algebra 𝑥∨𝑥 = 𝑥 𝑥∧𝑥 𝑥∧𝑦 𝑥∨𝑦 = 𝑦∨𝑥 𝑥 ∨ (𝑦 ∨ 𝑧) = (𝑥 ∨ 𝑦) ∨ 𝑧 𝑥 ∧ (𝑦 ∧ 𝑧) 𝑥 ∨ (𝑥 ∧ 𝑦) = 𝑥 𝑥 ∧ (𝑥 ∨ 𝑦) t 𝑥 ∨ (𝑦 ∧ 𝑧) = (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) 𝑥 ∧ (𝑦 ∨ 𝑧) h 𝑥∨0 = 𝑥 𝑥∧1 m 𝑥∧0 𝑥∨1 = 1 ′ 𝑥∨𝑥 = 1 𝑥 ∧ 𝑥′ ′ ′ ′ (𝑥 ∨ 𝑦) = 𝑥 ∧𝑦 (𝑥 ∧ 𝑦)′ ′ ′ = 𝑥 (𝑥 ) property with emphasis on ∨

⟺ ∀𝑥, 𝑦, 𝑧 ∈ 𝑋 = 𝑥 = 𝑦∧𝑥 = (𝑥 ∧ 𝑦) ∧ 𝑧 = 𝑥 = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) = 𝑥 = 0 = 0 = 𝑥′ ∨ 𝑦′

(idempotent)

and

(commutative)

and

(associative)

and

(absorptive)

and

(distributive)

and

(identity)

and

(bounded)

and

(complemented)

and

(de Morgan)

and

(involutory) .

dual property with emphasis on ∧

property name

✎PROOF: 1. Proof that Proposition 8.5 (page 131) ⟹ Theorem 8.2 (page 120): 1. 2. 3. 4. 5. 6.

Proof that 𝘼 is idempotent: Proof that 𝘼 is commutative: Proof that 𝘼 is associative: Proof that 𝘼 is absorptive: Proof that 𝘼 is distributive: Proof that 𝘼 is identity:

by 1 by 1 by 2 by 3 by 2 by 3

of Lemma 8.1 of Proposition 8.5 of Lemma 8.1 of Lemma 8.1 of Proposition 8.5 of Proposition 8.5

page 116 page 131 page 116 page 116 page 131 page 131

7

📃 Huntington (1904) pages 292–293 ⟨“1st set”⟩, 📘 Huntington (1933) page 280 ⟨“4th set”⟩, 📘 MacLane and Birkhoff (1999) page 488, 📘 Givant and Halmos (2009) page 10, 📘 Müller (1909) pages 20–21, 📘 Schröder (1890), 📘 Whitehead (1898) pages 35–37 version 0.30


Daniel J. Greenhoe


page 121


7. 8. 9. 10.

Proof that 𝘼 is bounded: Proof that 𝘼 is complemented: Proof that 𝘼 is involutory: Proof that 𝘼 is de Morgan:

2. Proof that Proposition 8.5 (page 131) 1. Proof that 𝘼 is commutative: 2. Proof that 𝘼 is distributive: 3. Proof that 𝘼 is identity: 4. Proof that 𝘼 is complemented:

by 4 by 4 by by 5

of Lemma 8.1 of Proposition 8.5 Corollary 7.1 of Lemma 8.1

page 116 page 131 page 111 page 116

⟸ Theorem 8.2 (page 120): by 2 of Theorem 8.2 page 120 by 5 of Theorem 8.2 page 120 by 6 of Theorem 8.2 page 120 by 8 of Theorem 8.2 page 120

✏ Lemma 8.2. l ( 𝑋, ∨, ∧, 0, 1 ; ≤) e ⟹ { m is a Boolean algebra }

1. 2.

𝑥′ ∨ (𝑥 ∧ 𝑦) = 𝑥′ ∨ 𝑦 𝑥 ∨ (𝑥′ ∧ 𝑦) = 𝑥 ∨ 𝑦

∀𝑥,𝑦∈𝑋

(Sasaki hook)

and

∀𝑥,𝑦∈𝑋

✎PROOF: ′ 𝑥′ ∨ (𝑥 ∧ 𝑦) = 𝑥 ∨ (𝑥′ ∧ 𝑦) ∨ (𝑥 ∧ 𝑦) ⏟⏟⏟⏟⏟⏟⏟⏟⏟

by absorption property (Theorem 8.2 page 120)

𝑥′

= 𝑥′ ∨ [(𝑥′ ∨ 𝑥) ∧ 𝑦]

by associative and distributive properties (Theorem 8.2 page 120)

′

by excluded middle property (Theorem 8.2 page 120)

′


= 𝑥 ∨ [1 ∧ 𝑦] =𝑥 ∨𝑦 ′

𝑥 ∨ (𝑥 ∧ 𝑦) = 𝑥 ∨ (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦) ⏟⏟⏟⏟⏟⏟⏟


𝑥

= 𝑥 ∨ [(𝑥 ∨ 𝑥′ ) ∧ 𝑦]

by associative and distributive properties (Theorem 8.2 page 120)

= 𝑥 ∨ [1 ∧ 𝑦]


=𝑥∨𝑦


✏ Theorem 8.3. 8 Let | 𝑋 | be the number of elements in a finite set 𝑋. t h 𝘼 is a Boolean algebra ⟹ | 𝘼 | = 2𝑛 for some 𝑛 ∈ ℕ. m Example 8.1. The table below illustrates some lattices that are Boolean algebras. Boolean algebras as algebras of sets {𝑥, 𝑧} {𝑥, 𝑦, 𝑧} e x

{𝑥, 𝑦} {𝑥} ∅

{𝑥}

{𝑦} ∅

{𝑥, 𝑦}

{𝑦, 𝑧}

{𝑥}

{𝑧} ∅

{𝑦}

Theorem 8.4. 8

📘 Joshi (1989) page 237


Daniel J. Greenhoe


version 0.30

page 122

t h m


If ( 𝑋, ∨, ∧, 0, 1 ; ≤) is a Boolean algebra then tautology dual ⎧ ⎪ ⎛𝑁 ⎞ 𝑁 𝑁 ⎛𝑁 ⎞ ⎪ ¬⎜ 𝑥 ⎟ ⎜ ⎟ = = (¬𝑥 ) ¬ 𝑥𝑛 (¬𝑥 ) ⎪ ⎜⋀ 𝑛 ⎟ ⋁ 𝑛 ⋀ 𝑛 ⎜⋁ ⎟ 𝑛=1 𝑛=1 𝑛=1 𝑛=1 ⎨ ⎝ ⎠ ⎝ ⎠ 𝑁 𝑁 ⎪ ⎛𝑁 ⎞ ⎛𝑁 ⎞ ⎪ ⎜ 𝑥 ⎟∨𝑦 = ⎜ ⎟ (𝑥 ∨ 𝑦) 𝑥 ∧ 𝑦 = (𝑥 ∧ 𝑦) ⋁ 𝑛 ⋁ 𝑛⎟ ⋀ 𝑛 ⎪ ⎜⋀ 𝑛 ⎟ ⎜ 𝑛=1 𝑛=1 ⎩ ⎝𝑛=1 ⎠ ⎝𝑛=1 ⎠

∀𝑥𝑛 ∈𝑋,𝑁∈ℕ

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

✎PROOF: 𝑁 1. Proof that ¬ (⋀𝑁 𝑛=1 𝑥𝑛 ) = ⋁𝑛=1 (¬𝑥𝑛 ) (by induction):

Proof for 𝑁 = 1 case: ⎛𝑁=1 ⎞ ¬⎜ 𝑥𝑛 ⎟ = ¬𝑥𝑛 ⎜⋀ ⎟ ⎝ 𝑛=1 ⎠


𝑁=1

=

⋁

(¬𝑥𝑛 )


𝑛=1

Proof for 𝑁 = 2 case: ⎛𝑁=2 ⎞ ¬⎜ 𝑥𝑛 ⎟ = (¬𝑥1 ) ∨ (¬𝑥2 ) ⎜⋀ ⎟ ⎝ 𝑛=1 ⎠


𝑁=2

=

⋁

(¬𝑥𝑛 )


𝑛=1

Proof that (𝑁 case) ⟹ (𝑁 + 1 case): 𝑁 ⎛𝑁+1 ⎞ ⎜ ⎟ ¬ 𝑥𝑛 = ¬ 𝑥𝑛 ∧ 𝑥𝑁 ⎜⋀ ⎟ [(⋀ ) ] 𝑛=1 𝑛=1 ⎝ ⎠


𝑁

=

¬

( ⋀ 𝑛=1

𝑥𝑛

)

∨ (¬𝑥𝑁+1 )


𝑁

=

(¬𝑥𝑛 ) ∨ (¬𝑥𝑁+1 ) [⋁ ] 𝑛=1

by left hypothesis

𝑁+1

=

⋁

(¬𝑥𝑛 )


𝑛=1

𝑁 2. Proof that ¬ (⋁𝑁 𝑛=1 𝑥𝑛 ) = ⋀𝑛=1 (¬𝑥𝑛 ): 𝑁

¬

(⋁ 𝑛=1

𝑁

𝑥𝑛

)

=¬

(¬¬𝑥𝑛 ) (⋁ ) 𝑛=1


𝑁

= ¬¬

(¬𝑥𝑛 ) (⋀ ) 𝑛=1

by previous result 1.

𝑁

=

(¬𝑥 ) ⋀ 𝑛


𝑛=1

version 0.30


Daniel J. Greenhoe


page 123

CHAPTER 8. BOOLEAN LATTICES 𝑁 3. Proof that (⋀𝑁 𝑛=1 𝑥𝑛 ) ∨ 𝑦 = ⋁𝑛=1 (𝑥𝑛 ∨ 𝑦) (by induction):

Proof for 𝑁 = 1 case: ⎛𝑁=1 ⎞ ⎜ 𝑥𝑛 ⎟ ∨ 𝑦 = 𝑥1 ∨ 𝑦 ⎜⋀ ⎟ 𝑛=1 ⎝ ⎠


𝑁=1

=

⋀

(𝑥𝑛 ∨ 𝑦)


𝑛=1

Proof for 𝑁 = 2 case: ⎛𝑁=2 ⎞ ⎜ 𝑥𝑛 ⎟ ∨ 𝑦 = (𝑥1 ∨ 𝑦) ∧ (𝑥2 ∨ 𝑦) ⎜⋀ ⎟ 𝑛=1 ⎝ ⎠


𝑁=2

=

⋀

(𝑥𝑛 ∨ 𝑦)


𝑛=1

Proof that (𝑁 case) ⟹ (𝑁 + 1 case): 𝑁 ⎛𝑁+1 ⎞ ⎜ ⎟ 𝑥𝑛 ∨ 𝑦 = 𝑥𝑛 ∧ 𝑥𝑁+1 ∨ 𝑦 ⎜⋀ ⎟ [(⋀ ) ] 𝑛=1 𝑛=1 ⎝ ⎠


𝑁

=

[(⋀ 𝑛=1

𝑥𝑛

)

∨ 𝑦 ∧ (𝑥𝑁+1 ∨ 𝑦) ]


𝑁

=

(𝑥𝑛 ∨ 𝑦) ∧ (𝑥𝑁+1 ∨ 𝑦) [⋀ ] 𝑛=1

by left hypothesis

𝑁+1

=

⋀

(𝑥𝑛 ∨ 𝑦)


𝑛=1

𝑁 4. Proof that (⋁𝑁 𝑛=1 𝑥𝑛 ) ∧ 𝑦 = ⋀𝑛=1 (𝑥𝑛 ∧ 𝑦): 𝑁

(⋁ 𝑛=1

𝑁

𝑥𝑛

)

∧ 𝑦 = ¬¬

[(⋁ 𝑛=1

𝑥𝑛

)

∧𝑦


]

𝑁

𝑥𝑛 ∨ (¬𝑦) =¬ ¬ ) ] [ (⋁ 𝑛=1


𝑁

=¬

(¬𝑥𝑛 ) ∨ (¬𝑦) [(⋀ ) ] 𝑛=1


𝑁

=¬

[(¬𝑥𝑛 ) ∨ (¬𝑦)] (⋀ ) 𝑛=1


𝑁

=

(⋁ 𝑛=1

¬[(¬𝑥𝑛 ) ∨ (¬𝑦)]

)


𝑁

=

(𝑥 ∧ 𝑦) ⋁ 𝑛


𝑛=1


Daniel J. Greenhoe


version 0.30

page 124

8.3

8.3. ADDITIONAL OPERATIONS

Additional operations 2

Propositional logic has a total of 24 = 16 operations in the class of functions {0, 1}{0, 1} (see page 187). The 16 logic operations of propositional logic can all be represented using the logic operations of disjunction ∨, conjunction ∧, and negation ¬. Using these representations, all 16 operations can be generalized to Boolean algebras using the equivalent Boolean algebra/lattice operations of join, meet, and complement.9 Several of these additional operations for Boolean algebras are defined in Definition 8.2 (next). Definition 8.2 (additional Boolean algebra operations). 10 Let ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a Boolean algebra. The following table defines additional operations in 𝑋 𝑋×𝑋 in terms of ∨, ∧, and ′. Let 𝑥′ ≜ ′𝑥 and 𝑦′ ≜ ′𝑦. name symbol definition ′ rejection ↓ 𝑥↓𝑦 ≜ 𝑥 ∧ 𝑦′ ∀𝑥,𝑦∈𝑋 exception − 𝑥 − 𝑦 ≜ 𝑥 ∧ 𝑦′ ∀𝑥,𝑦∈𝑋 ′ adjunction ÷ 𝑥÷𝑦 ≜ 𝑥∨𝑦 ∀𝑥,𝑦∈𝑋 Sheffer stroke | 𝑥|𝑦 ≜ 𝑥 ′ ∨ 𝑦′ ∀𝑥,𝑦∈𝑋 ′ ′ Boolean addition △ 𝑥 △ 𝑦 ≜ (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦 ) ∀𝑥,𝑦∈𝑋 inhibit 𝑥 ⊖ 𝑥 ⊖ 𝑦 ≜ 𝑥′ ∧ 𝑦 ∀𝑥,𝑦∈𝑋 implication ⇒ 𝑥 ⇒ 𝑦 ≜ 𝑥′ ∨ 𝑦 ∀𝑥,𝑦∈𝑋 ′ ′ biconditional ⇔ 𝑥 ⇔ 𝑦 ≜ (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦 ) ∀𝑥,𝑦∈𝑋 Theorem 8.5. 11 ∨ (join) ∧ (meet) t h △ (Boolean addition) m − (exception) ÷ (adjunction)

is the dual of ↓ is the dual of | is the dual of ⇔ is the dual of ⇒ is the dual of ⊖

(rejection) (Sheffer stroke) (biconditional) (implication) (inhibit 𝑥)

✎PROOF: (join)

(𝑥 ∨ 𝑦)′ = 𝑥′ ∧ 𝑦′ =𝑥↓𝑦

(meet)

′

(rejection)

′

(𝑥 ∧ 𝑦) = 𝑥 ∨ 𝑦 = 𝑥|𝑦 ′

by de Morgan's law property (Theorem 8.2 page 120) by definition of rejection ↓ (Definition 8.2 page 124)

′

by de Morgan's law property (Theorem 8.2 page 120)

(Sheffer stroke) ′ ′

′

(Boolean addition) (𝑥 △ 𝑦) = (𝑥 𝑦 ∨ 𝑥𝑦 ) = (𝑥 ∨ 𝑦′ )(𝑥′ ∨ 𝑦) ′

′ ′

by definition of Sheffer stroke | (Definition 8.2 page 124) by def. of Boolean addition △ (Definition 8.2 page 124) by de Morgan's law property (Theorem 8.2 page 120)

′

= 𝑥𝑥 ∨ 𝑥𝑦 ∨ 𝑦 𝑥 ∨ 𝑦 𝑦

by distributive property (Theorem 8.2 page 120)

′ ′

= 𝑥𝑦 ∨ 𝑥 𝑦 =𝑥⇔𝑦 (exception)

′

(biconditional) by def. of biconditional ⇔ (Definition 8.2 page 124)

′ ′

(𝑥 − 𝑦) = (𝑥𝑦 ) = 𝑥′ ∨ 𝑦

by de Morgan's law property (Theorem 8.2 page 120)

=𝑥⇒𝑦 (adjunction)

′

by definition of exception − (Definition 8.2 page 124) (implication)

′ ′

(𝑥 ÷ 𝑦) = (𝑥 ∨ 𝑦 )

by definition of implication ⇒ (Definition 8.2 page 124) by definition of adjunction ÷ (Definition 8.2 page 124)

9

📘 Givant and Halmos (2009), page 32 📘 Givant and Halmos (2009) pages 32–33, 📘 Bernstein (1934) page 876 ⟨implication ⊃⟩, 📘 Huntington (1933) page 276, 📘 Taylor (1920) page 243, 📘 Bernstein (1914) page 93, 📘 Sheffer (1913) pages 487–488, 📘 Peirce (1902) page 216, 📘 Peirce (1880a) pages 218–221 11 📘 Givant and Halmos (2009) page 33 10

version 0.30


Daniel J. Greenhoe


page 125


= 𝑥′ 𝑦

by de Morgan's law property (Theorem 8.2 page 120) (inhibit 𝑥)

′ ′

(𝑥 𝑦) = (𝑥 ) =𝑥 ⦹

(complement 𝑥)

by definition of complement 𝑥

by involutory property (Theorem 8.2 page 120)

= 𝑥◨𝑦

(transfer 𝑥)

(𝑥 𝑦) = (𝑦 ) =𝑦 ⦹

(complement 𝑦)

by definition of transfer 𝑥 ◨ by definition of complement 𝑦

⦹

′ ′

′

by definition of inhibit 𝑥 ⊖ (Definition 8.2 page 124) ⦹

=𝑥⊖𝑦 ′


= 𝑥◧𝑦

(transfer 𝑦)

by definition of transfer 𝑦 ◧

✏ Theorem 8.6. 12 Let ( 𝑋, ∨, ∧, ⟺ 𝑦′ ≤ t 𝑥≤𝑦 h 𝑥≤𝑦 ⟺ 𝑥 − m 𝑥≤𝑦 ⟺ 𝑥 ⇒

0, 1 ; ≤) be a Boolean algebra. 𝑥′ ∀𝑥,𝑦∈𝑋 𝑦=0 ∀𝑥,𝑦∈𝑋 𝑦=1 ∀𝑥,𝑦∈𝑋

✎PROOF: 1. Proof that 𝑥 ≤ 𝑦 ⟺ 𝑦′ ≤ 𝑥′ : 𝑥≤𝑦 ⟺ 𝑥∧𝑦=𝑥 ′

⟺ (𝑥 ∧ 𝑦) = 𝑥 ′

′

′

′

⟺ 𝑥 ∨𝑦 =𝑥 ⟺ 𝑦 ≤𝑥

′

′

by definition of meet ∧,

Definition 3.22 page 58

by de Morgan's law property,

Theorem 8.2 page 120

by de Morgan's law property,

Theorem 8.2 page 120

by definition of join ∨,


2. Proof that 𝑥 ≤ 𝑦 ⟹ 𝑥 − 𝑦 = 0: 𝑥 − 𝑦 = 𝑥 ∧ 𝑦′ ≤𝑦∧𝑦

by definition of exception −,

′


by left hypothesis

=0

by definition of complement,


3. Proof that 𝑥 ≤ 𝑦 ⟸ 𝑥 − 𝑦 = 0: 𝑥 − 𝑦 = 0 ⟺ 𝑥 ∧ 𝑦′ = 0

by definition of exception −,


⟺

✏

8.4 Representation A Boolean algebra ( 𝑋, ∨, ∧, 0, 1 ; ≤) can be represented in terms of five operators (see Theorem 8.2 page 120): the binary operators join ∨ and meet ∧, the unary operator complement ′, and the nullary opeartors 0 and 1. 12

📘 Givant and Halmos (2009) page 39


Daniel J. Greenhoe


version 0.30

page 126

8.4. REPRESENTATION

However, it is also possible to represent a Boolean algebra with fewer operators— in fact, as few as one operator. When a set of operators can completely represent all the operators of a Boolean algebra, then that set is called functionally complete (next definition). Definition 8.3. 13 Let 𝘽 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a Boolean algebra. d A set of operators 𝛷 is functionally complete in 𝘽 if e ∨, ∧, ′, 0, and 1 f can all be expressed in terms of 𝛷. Here are some examples of functionally complete sets: {↓} (rejection) {|} (Sheffer stroke) {÷, 0} (adjunction and 0) {−, 1} (exception and 1) (join and complement) {∨, ′} ∧, ′ (meet and complement) { } {△, ∧, 1} (Boolean addition, meet, and 1) {△, ∨, 1} (Boolean addition, join, and 1) {△, −, ′} (Boolean addition, exception, and complement)

Theorem 8.9 Theorem 8.10 Theorem 8.12 Theorem 8.13 Theorem 8.7 Theorem 8.8 Theorem 8.14 Theorem 8.15 Theorem 8.16

page 127 page 127 page 128 page 129 page 126 page 126 page 129 page 130 page 130

Theorem 8.7. Let 𝘽 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a Boolean algebra. The set {∨, ′} is functionally complete with respect to 𝘽. In particular, t 𝑥 ∧ 𝑦 = (𝑥′ ∨ 𝑦)′ ∀𝑥,𝑦∈𝑋 h 0 = (𝑥 ∨ 𝑥′ )′ ∀𝑥∈𝑋 m ′ 1 = 𝑥∨𝑥 ∀𝑥∈𝑋 ✎PROOF: 𝑥 ∧ 𝑦 = (𝑥 ∧ 𝑦)″ ′

′ ′

= (𝑥 ∨ 𝑦 ) 1 = 𝑥 ∨ 𝑥′ 0=1

by involutory property Theorem 8.2 page 120 by de Morgan's Law property Theorem 8.2 page 120 by complement property Theorem 8.2 page 120

′

= (𝑥 ∨ 𝑥′ )′

by complement property Theorem 8.2 page 120

✏ Theorem 8.8. Let 𝘽 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a Boolean algebra. The set {∧, ′} is functionally complete with respect to 𝘽. In particular, t 𝑥 ∨ 𝑦 = (𝑥′ ∧ 𝑦)′ ∀𝑥,𝑦∈𝑋 h ′ 0 = 𝑥∧𝑥 ∀𝑥∈𝑋 m ′ ′ ∀𝑥∈𝑋 1 = (𝑥 ∧ 𝑥 ) ✎PROOF: 𝑥 ∨ 𝑦 = (𝑥 ∨ 𝑦)″ ′

′ ′

= (𝑥 ∧ 𝑦 ) 0 = 𝑥 ∧ 𝑥′

by involutory property Theorem 8.2 page 120 by de Morgan's Law property Theorem 8.2 page 120 by complement property Theorem 8.2 page 120

1 = 0′ = (𝑥 ∧ 𝑥′ )′


✏ 13

📘 Whitesitt (1995) page 69

version 0.30


Daniel J. Greenhoe


page 127


Theorem 8.9. 14 Let 𝘽 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a Boolean algebra. Let ↓ represent the rejection operator (Definition 8.2 page 124). The set {↓} is functionally complete with respect to 𝘽. In particular, 𝑥 ∨ 𝑦 = (𝑥 ↓ 𝑦) ↓ (𝑥 ↓ 𝑦) ∀𝑥,𝑦∈𝑋 t 𝑥 ∧ 𝑦 = (𝑥 ↓ 𝑥) ↓ (𝑦 ↓ 𝑦) ∀𝑥,𝑦∈𝑋 h ′ 𝑥 = 𝑥↓𝑥 ∀𝑥∈𝑋 m 0 = 𝑥 ↓ (𝑥 ↓ 𝑥) ∀𝑥∈𝑋 1 = [𝑥 ↓ (𝑥 ↓ 𝑥)] ↓ [𝑥 ↓ (𝑥 ↓ 𝑥)] ∀𝑥∈𝑋 ✎PROOF: 𝑥′ = (𝑥 ∨ 𝑥)′


=𝑥↓𝑥

by definition of ↓ page 124

𝑥 ∨ 𝑦 = (𝑥 ∨ 𝑦)

″


′


= (𝑥 ↓ 𝑦)

= (𝑥 ↓ 𝑦) ↓ (𝑥 ↓ 𝑦) 𝑥 ∧ 𝑦 = (𝑥 ∧ 𝑦) ′

by previous result

″


′ ′

by de Morgan's Law page 120

= (𝑥 ∨ 𝑦 ) = 𝑥′ ↓ 𝑦′


= (𝑥 ↓ 𝑥) ↓ (𝑦 ↓ 𝑦)

by previous result

0=1

′

= (𝑥 ∨ 𝑥′ )′


′

= 𝑥 ↓ (𝑥 ) = 𝑥 ↓ (𝑥 ↓ 𝑥)


1 = (𝑥 ∨ 𝑥 ′ )


′ ″

= (𝑥 ∨ 𝑥 ) = (𝑥 ∨ 𝑥′ )′ ↓ (𝑥 ∨ 𝑥′ )′ = [𝑥 ↓ (𝑥′ )] ↓ [𝑥 ↓ (𝑥′ )]

by Theorem 8.2 page 120 by definition of ↓ page 124

= [𝑥 ↓ (𝑥 ↓ 𝑥)] ↓ [𝑥 ↓ (𝑥 ↓ 𝑥)]

✏ Theorem 8.10. Let 𝘽 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a Boolean algebra. Let | represent the Sheffer stroke operator (Definition 8.2 page 124). The set {|} is functionally complete with respect to 𝘽. In particular, 𝑥 ∨ 𝑦 = (𝑥|𝑥)|(𝑦|𝑦) ∀𝑥,𝑦∈𝑋 t 𝑥 ∧ 𝑦 = (𝑥|𝑦)|(𝑥|𝑦) ∀𝑥,𝑦∈𝑋 h ′ 𝑥 = 𝑥|𝑥 ∀𝑥∈𝑋 m 0 = [𝑥|(𝑥|𝑥)]|[𝑥|(𝑥|𝑥)] ∀𝑥∈𝑋 1 = 𝑥|(𝑥|𝑥) ∀𝑥∈𝑋 ✎PROOF: 𝑥′ = (𝑥 ∧ 𝑥)′


= 𝑥|𝑥

by definition of | page 124 ″


′ ′

by de Morgan's Law page 120

𝑥 ∨ 𝑦 = (𝑥 ∨ 𝑦) ′

= (𝑥 ∧ 𝑦 ) 14

📘 Givant and Halmos (2009) page 33


Daniel J. Greenhoe


version 0.30

page 128

8.4. REPRESENTATION

= 𝑥′ |𝑦′

by definition of | page 124

= (𝑥|𝑥)|(𝑦|𝑦)

by first result

″

𝑥 ∧ 𝑦 = (𝑥 ∧ 𝑦) = (𝑥|𝑦)


′


= (𝑥|𝑦)|(𝑥|𝑦) 1=0

by first result

′

= (𝑥 ∧ 𝑥′ )′


′

= 𝑥|(𝑥 ) = 𝑥|(𝑥|𝑥)


0 = (𝑥 ∧ 𝑥′ )


′ ″

= (𝑥 ∧ 𝑥 ) = (𝑥 ∧ 𝑥′ )′ |(𝑥 ∧ 𝑥′ )′ ′

by Theorem 8.2 page 120 by definition of | page 124

′

= [𝑥|(𝑥 )]|[𝑥|(𝑥 )] = [𝑥|(𝑥|𝑥)]|[𝑥|(𝑥|𝑥)]

✏ Besides the rejection singleton {↓} and the Sheffer stroke singleton {|}, there are no single opertor sets that are functionally complete (next theorem). Theorem 8.11. 15 Let 𝘽 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a Boolean algebra. Let ↓ be the rejection operator and | be the Sheffer stroke operator. t h {+} is functionally complete in 𝘽 ⟹ + =↓ or + = | m Theorem 8.12. Let 𝘽 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a Boolean algebra. Let ÷ represent the adjunction operator (Definition 8.2 page 124). The set {÷, 0} is functionally complete with respect to 𝘽. In particular, 𝑥 ∨ 𝑦 = 𝑥 ÷ (0 ÷ 𝑦) ∀𝑥,𝑦∈𝑋 t h 𝑥 ∧ 𝑦 = 0 ÷ [(0 ÷ 𝑥) ÷ 𝑦] ∀𝑥,𝑦∈𝑋 m 𝑥′ = 0 ÷ 𝑥 ∀𝑥∈𝑋 1 = 𝑥÷𝑥 ∀𝑥∈𝑋 ✎PROOF: 𝑥′ = 0 ∨ 𝑥′


=0÷𝑥 𝑥∨𝑦=𝑥∨𝑦

by definition of ÷ (Definition 8.2 page 124) ″


′

= 𝑥 ÷ (𝑦 ) = 𝑥 ÷ (0 ÷ 𝑦)

by definition of ÷ (Definition 8.2 page 124) by previous result

′ ′

′

𝑥 ∧ 𝑦 = (𝑥 ∨ 𝑦 ) = (𝑥′ ÷ 𝑦)′

= [(0 ÷ 𝑥) ÷ 𝑦]

by de Morgan's law property Theorem 8.2 page 120 by definition of ÷ (Definition 8.2 page 124) ′

= 0 ÷ [(0 ÷ 𝑥) ÷ 𝑦] 1=𝑥∨𝑥 =𝑥÷𝑥

′

by previous result by previous result by complement property Theorem 8.2 page 120 by definition of ÷ (Definition 8.2 page 124)

✏ 15

📘 Quine (1979) page 49, 📘 Żyliński (1925) page 208 ⟨↓= 𝜙15 , | = 𝜙2 ⟩

version 0.30


Daniel J. Greenhoe


page 129


Theorem 8.13. 16 Let 𝘽 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a Boolean algebra. Let − represent the exception operator (Definition 8.2 page 124). The set {−, 1} is functionally complete with respect to 𝘽. In particular, 𝑥 ∨ 𝑦 = 1 − [(1 − 𝑥) − 𝑦] ∀𝑥,𝑦∈𝑋 t h 𝑥 ∧ 𝑦 = 𝑥 − (1 − 𝑦) ∀𝑥,𝑦∈𝑋 m ′ 𝑥 = 1−𝑥 ∀𝑥∈𝑋 0 = 𝑥−𝑥 ∀𝑥∈𝑋 ✎PROOF: 𝑥′ = 1 ∧ 𝑥′


=1−𝑥 𝑥∧𝑦=𝑥∧𝑦

by definition of − (Definition 8.2 page 124)

″

by Theorem 8.2 page 120 ′

= 𝑥 − (𝑦 ) = 𝑥 − (1 − 𝑦)

by definition of − (Definition 8.2 page 124) by previous result

′ ′

′

𝑥 ∨ 𝑦 = (𝑥 ∧ 𝑦 ) = (𝑥′ − 𝑦)′

by de Morgan's law property Theorem 8.2 page 120 by definition of − (Definition 8.2 page 124) ′

= [(1 − 𝑥) − 𝑦]

by previous result

= 1 − [(1 − 𝑥) − 𝑦]

by previous result

0=𝑥∧𝑥

′


=𝑥−𝑥


✏ Theorem 8.14. 17 Let 𝘽 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a Boolean algebra. The set {△, ∧, 1} is functionally complete with respect to 𝘽. In particular, t 𝑥 ∨ 𝑦 = 𝑥𝑦 △ 𝑥 △ 𝑦 ∀𝑥,𝑦∈𝑋 h ′ 𝑥 = 𝑥 △ 1 ∀𝑥∈𝑋 m 0 = 𝑥△𝑥 ∀𝑥∈𝑋 ✎PROOF: 𝑥′ = 𝑥′ ∨ 0


′

= (𝑥 ∧ 1) ∨ (𝑥 ∧ 0) = (𝑥′ ∧ 1) ∨ (𝑥 ∧ 1′ )


=𝑥△1

by definition of △ (Definition 8.2 page 124) by Theorem 8.2 page 120

0=0∨0 ′

′

= (𝑥 ∧ 𝑥) ∨ (𝑥 ∧ 𝑥 ) =𝑥△𝑥

by Theorem 8.2 page 120 by definition of △ (Definition 8.2 page 124)

𝑥𝑦 ⊕ 𝑥 ⊕ 𝑦 = (𝑥𝑦) △ (𝑥 △ 𝑦) ′

by associative property Theorem 8.2 page 120 ′

= (𝑥𝑦) ⊕ (𝑥 𝑦 ∨ 𝑥𝑦 ) = (𝑥𝑦)′ (𝑥′ 𝑦 ∨ 𝑥𝑦′ ) ∨ (𝑥𝑦)(𝑥′ 𝑦 ∨ 𝑥𝑦′ )′ ′

′

′

′

′

′

by definition of △ (Definition 8.2 page 124) by definition of △ (Definition 8.2 page 124) ′ ′

= (𝑥 ∨ 𝑦 )(𝑥 𝑦 ∨ 𝑥𝑦 ) ∨ (𝑥𝑦)[(𝑥 𝑦) (𝑥𝑦 ) ] by de Morgan's law Theorem 8.2 page 120 ′ ′ ′ ′ ″ ′ ′ ″ = (𝑥 ∨ 𝑦 )(𝑥 𝑦 ∨ 𝑥𝑦 ) ∨ (𝑥𝑦)[(𝑥 ∨ 𝑦 )(𝑥 ∨ 𝑦 )] by de Morgan's law Theorem 8.2 page 120 = (𝑥′ ∨ 𝑦′ )(𝑥′ 𝑦 ∨ 𝑥𝑦′ ) ∨ (𝑥𝑦)[(𝑥 ∨ 𝑦′ )(𝑥′ ∨ 𝑦)] = (𝑥′ 𝑦 ∨ 𝑥𝑦′ ) ∨ (𝑥𝑦)[𝑥𝑦 ∨ 𝑥′ 𝑦′ ] 16 17

📘 Bernstein (1914) pages 89–91 📘 Roth (2006) page 42


Daniel J. Greenhoe


version 0.30

page 130

8.4. REPRESENTATION

= (𝑥′ 𝑦 ∨ 𝑥𝑦′ ) ∨ 𝑥𝑦 = (𝑥′ 𝑦 ∨ 𝑥𝑦′ ) ∨ (𝑥𝑦 ∨ 𝑥𝑦) ′

by idempotent property Theorem 8.2

′

= (𝑥𝑦 ∨ 𝑥 𝑦) ∨ (𝑥𝑦 ∨ 𝑥𝑦 ) = (𝑥 ∨ 𝑥′ )𝑦 ∨ 𝑥(𝑦 ∨ 𝑦′ )

by Theorem 8.2 page 120 by distributive property Theorem 8.2

= (1)𝑦 ∨ 𝑥(1) =𝑥∨𝑦

✏ Theorem 8.15. Let 𝘽 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a Boolean algebra. The set {△, ∨, 1} is functionally complete with respect to 𝘽. In particular, t 𝑥 ∧ 𝑦 = [(𝑥 △ 1) ∨ (𝑦 △ 1)] △ 1 ∀𝑥,𝑦∈𝑋 h ′ 𝑥 = 𝑥△1 ∀𝑥∈𝑋 m 0 = 𝑥△𝑥 ∀𝑥∈𝑋 ✎PROOF: 0=𝑥△𝑥 𝑥′ = 𝑥 △ 1 𝑥 ∧ 𝑦 = (𝑥′ ∨ 𝑦′ )′ = [(𝑥 △ 1) ∨ (𝑦 △ 1)] △ 1

✏ Theorem 8.16. Let 𝘽 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a Boolean algebra. The set {△, −, ′} is functionally complete with respect to 𝘽. In particular, t 𝑥 ∨ 𝑦 = (𝑥 − 𝑦) △ 𝑦 ∀𝑥,𝑦∈𝑋 h 𝑥 ∧ 𝑦 = 𝑥 − (𝑥 − 𝑦) ∀𝑥,𝑦∈𝑋 m 0 = 𝑥△𝑥 ∀𝑥∈𝑋 ✎PROOF: 𝑥 ∨ 𝑦 = 𝑥(𝑦 ∨ 𝑦′ ) ∨ 𝑦 = 𝑥𝑦 ∨ 𝑥𝑦′ ∨ 𝑦


= (𝑦 ∨ 𝑥𝑦) ∨ 𝑥𝑦 = 𝑦 ∨ 𝑥𝑦

′

by associative property (Theorem 8.2 page 120)

′


′

= (𝑦 ∨ 𝑥 𝑦) ∨ 𝑥𝑦 ′

′

by absorptive property (Theorem 8.2 page 120) ′

′

= (𝑦 ∨ 𝑥 )𝑦 ∨ (𝑥𝑦 )𝑦 ′ ′

′

= (𝑥𝑦 ) 𝑦 ∨ (𝑥𝑦 )𝑦 ′

′

by distributive and idempotent properties (Theorem 8.2 page 120) by de Morgan's law property (Theorem 8.2 page 120)

= (𝑥𝑦 ) △ 𝑦

by definition of △ (Definition 8.2 page 124)

= (𝑥 − 𝑦) △ 𝑦


𝑥 ∧ 𝑦 = 𝑥𝑥′ ∨ 𝑥𝑦 = 𝑥(𝑥′ ∨ 𝑦) ″ ′ ′

= 𝑥(𝑥 𝑦 ) ′ ′

= 𝑥(𝑥𝑦 )

= 𝑥(𝑥 − 𝑦)

by de Morgan's law property (Theorem 8.2 page 120) by involutory property (Theorem 8.2 page 120)

′

= 𝑥 − (𝑥 − 𝑦) 0 = 𝑥𝑥

by distributive and idempotent properties (Theorem 8.2 page 120)

by definition of − (Definition 8.2 page 124) by definition of − (Definition 8.2 page 124)

′

= 𝑥 − (𝑥 − 𝑥′ )

by previous result

✏ version 0.30


Daniel J. Greenhoe


page 131


8.5 Characterizations ❝The

algebra of symbolic logic…has recently assumed some importance as an independent calculus; it may therefore be not without interest to consider it from a purely mathematical or abstract point of view…❞ Edward V. Huntington (1874–1952), American mathematician18

Order characterizations An order characterization of Boolean algebras has already been given by Definition 8.1 (page 115): A lattice is a Boolean algebra if and only if it is distributive and complemented. Proposition 8.4. 19 Let 𝑨 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded and complemented lattice. ′ p 𝑨 is a 1. 1 = 0 and r ⟺ ′ ′ ′ ′ { 2. (𝑥 ∧ 𝑦 ) = 𝑦 ∨ (𝑥 ∧ 𝑦 ) ∀𝑥,𝑦∈𝑋 (Elkan's law) } p { Boolean algebra }

Algebraic characterizations This section presents several algebraic characterizations. One such characterization has already been provided by Theorem 8.2 (page 120)— the standard properties of Boolean algebras characterized by 19 identities. If a system satisfies these 19 identities, then that system is a Boolean algebra. However, the set of 19 identities is very much an over-specification. It is possible to characterize Boolean algebras using much fewer relationships, from which all of the 19 identities of Theorem 8.2 can be derived. Here are some of these reduced characterizations: Huntington's first set: (1904) 8 relationships, Proposition 8.5 page 131 Huntington's fourth set: (1933) 4 relationships, Proposition 8.6 page 133 Huntington's fifth set: (1933) 3 relationships, Proposition 8.7 page 133 Stone: (1935) 7 relationships, Proposition 8.8 page 133 Byrne's Formulation A: (1946) 3 relationships, Proposition 8.9 page 134 Byrne's Formulation B: (1946) 2 relationships, Proposition 8.10 page 136 All of these characterizations use 3 variables. It might be reasonable to ask if there exists a characterization that uses only two variables. The answer is “No”, as demonstrated by the next theorem. Theorem 8.17. 20 t h There does not exist a characterization of Boolean algebras consisting of only 2 variables. m Proposition 8.5 (Huntington's first set). 21 Let 𝑋 be a set, ≤ a relation in 𝟚𝑋𝑋 , ∨ and ∧ binary operations in 𝑋 𝑋×𝑋 , ′ an unary operation in 𝑋 𝑋 , and 0 and 1 nullary operations on 𝑋. 18

quote: 📃 Huntington (1904) page 288 image: http://en.wikipedia.org/wiki/Edward_V._Huntington 19 📘 Kondo and Dudek (2008) page 1035, 📘 Elkan et al. (1994), page 3 ⟨Elkan's law⟩ 20 📘 Sikorski (1969), page 3, 📘 Diamond and McKinsey (1947) page 961, 📘 Gerrish (1978), page 36 21 📘 Gerrish (1978), page 35, 📘 Saliǐ (1988) page 33 ⟨“Huntington's Theorem”⟩, 📘 Joshi (1989) page 222 ⟨(B1)–(B4)⟩, 📃 Huntington (1904) pages 292–293 ⟨“1st set”⟩, 📘 Huntington (1933) page 277 ⟨“1st set”⟩, 📘 Givant and Halmos (2009) page 10 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 132


( 𝑋, ∨, ∧, 0, 1 ; ≤) is a Boolean algebra if for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 1. 𝑥 ∨ 𝑦 = 𝑦∨𝑥 𝑥∧𝑦 = 𝑦∧𝑥 (commutative) 2. 𝑥 ∨ (𝑦 ∧ 𝑧) = (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) 𝑥 ∧ (𝑦 ∨ 𝑧) = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) (distributive) 3. 𝑥 ∨ 0 = 𝑥 𝑥∧1 = 𝑥 (identity) ′ ′ 4. 𝑥 ∨ 𝑥 = 1 𝑥∧𝑥 = 0 (complemented) and where the relation ≤ is defined as 𝑥 ≤ 𝑦 ⟺ 𝑥 ∨ 𝑦 = 𝑦 ∀𝑥,𝑦∈𝑋.

p r p

The property 𝑥 ∨ 𝑥′ = 1 is referred to as “the law of the excluded middle”. The property 𝑥 ∧ 𝑥′ = 0 is referred to as “the law of non-contradiction”. ✎PROOF: 1. Proof that 𝘼 is a Boolean algebra ⟹ 𝘼 is a distributive complemented lattice: (a) Proof that 𝘼 is distributive: by Definition 8.1 page 115 (b) Proof that 𝘼 is complemented: by Definition 8.1 page 115 (c) Proof that 𝘼 is bounded: by Lemma 8.1 page 116 (d) Proof that 𝘼 is a lattice: i. Proof that 𝘼 is idempotent: by Lemma 8.1 page 116 ii. Proof that 𝘼 is commutative: by Definition 8.1 page 115 iii. Proof that 𝘼 is associative: by Lemma 8.1 page 116 iv. Proof that 𝘼 is absorptive: by Lemma 8.1 page 116 v. Therefore, by Theorem 4.3 (page 62), 𝘼 is a lattice 2. Proof that 𝘼 is a Boolean algebra ⟸ 𝘼 is a distributive complemented lattice: (a) Proof that 𝘼 is commutative: by property of lattices, Theorem 4.3 page 62 (b) Proof that 𝘼 is distributive: by right hypothesis (c) Proof that 𝘼 has identity: 𝑥 ∨ 0 = 𝑥 ∨ (𝑥 ∧ 𝑥′ ) =𝑥

by complemented property in right hypothesis by absorptive property of lattices Theorem 4.3 page 62

′

𝑥 ∧ 1 = 𝑥 ∧ (𝑥 ∨ 𝑥 ) =𝑥

by complemented property in right hypothesis by absorptive property of lattices Theorem 4.3 page 62

(d) Proof that 𝘼 is complemented: by right hypothesis

✏ Huntington's fourth set (next) characterizes Boolean algebras in terms of the standard properties of idempotent, commutative, and associative (see Theorem 8.2 page 120), and also in terms of an additional property called Huntington's axiom,22 or (in terms of 𝑥 and 𝑦), 𝑥 commutes 𝑦. Huntington's axiom is significant in the context of orthomodular lattices in that an orthomodular lattice that satisfies Huntington's axiom is a Boolean algebra.23 22 23

📘 Givant and Halmos (2009) page 13 ⟨problem 7⟩ 📘 Renedo et al. (2003), page 72 ⟨Definition 3⟩, 📘 Beran (1985) page 52, 📘 Beran (1982)

version 0.30


Daniel J. Greenhoe


page 133


Proposition 8.6 (Huntington's fourth set). 24 Let 𝑨 ≜ ( 𝑋, ∨, ∧ ; ≤) be an algebraic structure. 𝑨 is a Boolean algebra ⟺ 1. 𝑥 ∨ 𝑥 = 𝑥 ∀𝑥∈𝑋 (idempotent) and ⎧ ⎫ p ⎪ 2. 𝑥 ∨ 𝑦 r = 𝑦∨𝑥 ∀𝑥,𝑦∈𝑋 (commutative) and ⎪ p ⎨ 3. (𝑥 ∨ 𝑦) ∨ 𝑧 = 𝑥 ∨ (𝑦 ∨ 𝑧) ∀𝑥,𝑦,𝑧∈𝑋 (associative) and ⎬ ⎪ ⎪ ′ ′ ′ ′ ′ ∀𝑥,𝑦∈𝑋. (Huntington's axiom) ⎩ 4. (𝑥 ∨ 𝑦 ) ∨ (𝑥 ∨ 𝑦) = 𝑥 ⎭ ✎PROOF: 1. Proof that [𝘼 is a Boolean algebra] ⟹ [𝘼 satisfies the 4 pairs of properties]: (a) Proof that 𝑥 ∨ 𝑥 = 𝑥 (idempotent property with respect to ∨): by 1a of Lemma 8.1 (page 116). (b) Proof that 𝑥 ∨ 𝑦 = 𝑦 ∨ 𝑥 (commutative property with respect to ∨): by 1a of this proposition. (c) Proof that (𝑥 ∨ 𝑦) ∨ 𝑧 = 𝑥 ∨ (𝑦 ∨ 𝑧) (associative property with respect to ∨): by 2a of Lemma 8.1 (page 116). (d) Proof that (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦′ ) = 𝑥 (Huntington's axiom): (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦′ ) = 𝑥 ∧ (𝑦 ∨ 𝑦′ )

by 2a

(distributive property wrt ∨)

=𝑥∧1

by 3a

(complemented property wrt ∨)

=𝑥

by 4b

(identity property wrt ∧)

2. Proof that [𝘼 is a Boolean algebra] ⟸ [𝘼 satisfies the 4 pairs of properties]: (a) Proof that 𝑥 ∨ 𝑦 = 𝑦 ∨ 𝑥: by 2 of Definition 8.1 page 115. (b) Proof that 𝑥 ∧ 𝑦 = 𝑦 ∧ 𝑥: (c) Proof that 𝑥 ∨ (𝑦 ∧ 𝑧) = (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧): (d) Proof that 𝑥 ∧ (𝑦 ∨ 𝑧) = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧): (e) Proof that 𝑥 ∨ 𝑥′ = 1: (f) Proof that 𝑥 ∧ 𝑥′ = 0: (g) Proof that 𝑥 ∨ 0 = 𝑥: (h) Proof that 𝑥 ∧ 1 = 𝑥:

✏ Proposition 8.7 (Huntington's fifth set). 25 Let 𝑨 ≜ ( 𝑋, ∨, ∧ ; ≤) be an algebraic structure. 𝑨 is a Boolean algebra ⟺ ″ p 1. 𝑥 = 𝑥 ∀𝑥,𝑦,𝑧∈𝑋 and ⎫ ⎧ r ⎪ ⎪ ′ ′ ∀𝑥,𝑦∈𝑋 and ⎬ p ⎨ 2. 𝑥 ∨ (𝑦 ∨ 𝑦 ) = 𝑥 ′ ⎪ ⎪ = [(𝑦′ ∨ 𝑥)′ ∨ (𝑧′ ∨ 𝑥)′ ]′ ∀𝑥,𝑦,𝑧∈𝑋. ⎩ 3. 𝑥 ∨ (𝑦 ∨ 𝑧) ⎭ 24 25

📘 Huntington (1933) page 280 ⟨“4th set”⟩ 📘 Givant and Halmos (2009) page 13, 📘 Huntington (1933) page 286 ⟨“5th set”⟩


Daniel J. Greenhoe


version 0.30

page 134


Proposition 8.8 (Stone). 26 Let 𝑨 ≜ ( 𝑋, ∨, ∧ ; ≤) be an algebraic structure. 𝑨 is a Boolean algebra ⟺ 1. 𝑥 ∨ 𝑦 = 𝑦∨𝑥 (join commutative) ∀𝑥,𝑦∈𝑋 ⎧ ⎪ 2. 𝑥 ∧ (𝑦 ∨ 𝑧) = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ∀𝑥,𝑦,𝑧∈𝑋 (left distributive) p ⎪ 3. (𝑥 ∨ 𝑦) ∧ 𝑧 = (𝑥 ∧ 𝑧) ∨ (𝑦 ∧ 𝑧) ∀𝑥,𝑦,𝑧∈𝑋 (right distributive) r ⎪ = 𝑥 ∀𝑥∈𝑋 (join identity) p ⎨ 4. 𝑥 ∨ 0 ′ ′ ′ ⎪ 5. ∃𝑥 such that 𝑥 ∨ 𝑥 = 1 and 𝑥 ∧ 𝑥 = 0 ∀𝑥∈𝑋 (complemented) ⎪ 6. 𝑥 ∨ 𝑥 = 𝑥 ∀𝑥∈𝑋 (idempotent) ⎪ = 𝑥 ∀𝑥∈𝑋 ⎩ 7. 𝑥 ∧ 𝑥

and and and and and and

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

Proposition 8.9 (Byrne's FORMULATION A). 27 Let 𝑨 ≜ ( 𝑋, ∨, ∧ ; ≤) be an algebraic structure. 𝑨 is a Boolean algebra ⟺ p 1. 𝑥 ∨ 𝑦 = 𝑦∨𝑥 ∀𝑥,𝑦∈𝑋 (commutative) and ⎫ ⎧ r ⎪ ⎪ (𝑥 (𝑦 2. ∨ 𝑦) ∨ 𝑧 = 𝑥 ∨ ∨ 𝑧) ∀𝑥,𝑦,𝑧∈𝑋 (associative) and ⎬ p ⎨ ′ ′ ⎪ ⎪ ∀𝑥,𝑦,𝑧∈𝑋. ⎩ 3. 𝑥 ∨ 𝑦 = 𝑧 ∨ 𝑧 ⟺ 𝑥 ∨ 𝑦 = 𝑥 ⎭ ✎PROOF: 1. Proof that 𝑨 is a Boolean algebra ⟹ 3 identities: (a) commutative property: By Theorem 8.2 (page 120), all Boolean algebras are commutative. (b) associative property: By Theorem 8.2 (page 120), all Boolean algebras are associative. (c) Proof that 𝑥 ∨ 𝑦′ = 𝑦 ∨ 𝑦′ ⟹ 𝑥 ∨ 𝑦 = 𝑥: 𝑥∨𝑦=𝑦∨𝑥

by Boolean hypothesis and Theorem 8.2 page 120 ′ ′

= 𝑦 ∨ (𝑥 ) = 𝑦 ∨ (𝑥′ )′

by Boolean hypothesis and Theorem 8.2 page 120 by Boolean hypothesis and Theorem 8.2 page 120

′ ′

′

= 𝑥 ∨ (𝑥 ) = 𝑥′ ∨ 𝑥

by 𝑥 ∨ 𝑦′ = 𝑦 ∨ 𝑦′ hypothesis

=𝑥

by Boolean hypothesis and Theorem 8.2 page 120


(d) Proof that 𝑥 ∨ 𝑦′ = 𝑦 ∨ 𝑦′ ⟸ 𝑥 ∨ 𝑦 = 𝑥: 𝑥 ∨ 𝑦′ = (𝑥 ∨ 𝑦) ∨ 𝑦′ ′

by 𝑥 ∨ 𝑦 = 𝑥 hypothesis

= 𝑥 ∨ (𝑦 ∨ 𝑦 ) =𝑥∨1


=𝑥



2. Proof that 𝑨 is a Boolean algebra ⟸ 3 identities: (a) Proof that 𝑥 ∨ 𝑥 = 𝑥 (idempotent property): because 𝑥 ∨ 𝑥′ = 𝑥 ∨ 𝑥′ and by identity 3 (b) Proof that 𝑥 ∨ 𝑥′ = 𝑦 ∨ 𝑦′ : by item (2a) and identity 3 (c) Proof that 𝑥 ∨ 𝑦 = 𝑥 and 𝑦 ∨ 𝑧 = 𝑦 ⟹ 𝑥 ∨ 𝑧 = 𝑥: 𝑥 ∨ 𝑧 = (𝑥 ∨ 𝑦) ∨ 𝑧

26 27


= 𝑥 ∨ (𝑦 ∨ 𝑧)

by identity 2 (associative property)

=𝑥∨𝑦

by 𝑦 ∨ 𝑧 = 𝑦 hypothesis

=𝑥


📘 Stone (1935) page 705 📘 Givant and Halmos (2009) page 13, 📘 Byrne (1946) page 270 ⟨“FORMULATION A”⟩

version 0.30


Daniel J. Greenhoe


page 135


(d) Proof that 𝑥″ = 𝑥 (involutory property): 𝑥″ ∨ 𝑥′ = 𝑥′ ∨ 𝑥″ =𝑧∨𝑧 ″

by identity 1 (commutative property)

′

(8.1)

by item (2b)

𝑥 ∨𝑥=𝑥

″

by equation (8.1) and identity 3

(8.2)

‴

′

𝑥 ∨𝑥 =𝑥

‴

by equation (8.2)

(8.3)

⁗

″

⁗

by equation (8.2)

(8.4)

by equation (8.4), equation (8.5), and item (2c)

(8.5)


(8.6)


(8.7)

by equation (8.3)

(8.8)

𝑥

∨𝑥 =𝑥

⁗

𝑥

⁗

𝑥

∨𝑥=𝑥

⁗

′

∨𝑥 =𝑧∨𝑧

′

‴

′

‴

‴

𝑥 ∨𝑥 =𝑥

′

𝑥 =𝑥 ∨𝑥

′

= 𝑥 ′ ∨ 𝑥‴ =𝑥


′

by equation (8.7)

‴

′

by equation (8.8)

′

by item (2b)

𝑥∨𝑥 =𝑥∨𝑥 =𝑧∨𝑧 ″

𝑥∨𝑥 =𝑥 ″

(8.9)

by equation (8.9) and identity 3 ″

𝑥 =𝑥 ∨𝑥 =𝑥∨𝑥

(8.10)

by equation (8.2)

″


=𝑥

by equation (8.10)

(e) Proof that 𝑥 ∨ (𝑥′ ∨ 𝑦)″ = 𝑧 ∨ 𝑧′ : 𝑥 ∨ (𝑥′ ∨ 𝑦)″ = 𝑥 ∨ (𝑥′ ∨ 𝑦)

by item (2d) (involutory property)

′

= (𝑥 ∨ 𝑥 ) ∨ 𝑦


′


′

by item (2b)

= 𝑦 ∨ (𝑥 ∨ 𝑥 ) = 𝑦 ∨ (𝑦 ∨ 𝑦 ) = (𝑦 ∨ 𝑦) ∨ 𝑦 =𝑦∨𝑦

′


′

by item (2a)

′

by item (2b)

=𝑧∨𝑧

(f) Proof that 𝑥 ∨ (𝑥′ ∨ 𝑦)′ = 𝑥: by item (2e) and identity 3 (g) Proof that 𝑥 ∨ 𝑦″ ∨ (𝑥 ∨ 𝑦)′ = 𝑧 ∨ 𝑧′ : 𝑥 ∨ 𝑦″ ∨ (𝑥 ∨ 𝑦)′ = 𝑥 ∨ 𝑦 ∨ (𝑥 ∨ 𝑦)′ =𝑧∨𝑧

by item (2d)

′

by item (2b)

(h) Proof that 𝑥 ∨ (𝑥 ∨ 𝑦)′ = 𝑥 ∨ 𝑦′ : 𝑥 ∨ (𝑥 ∨ 𝑦)′ = 𝑥 ∨ (𝑥 ∨ 𝑦)′ ∨ 𝑦′

by item (2g) and identity 3

′

′


′

′ ′

by item (2f)

= 𝑥 ∨ 𝑦 ∨ (𝑥 ∨ 𝑦)

= 𝑥 ∨ 𝑦 ∨ [(𝑥 ∨ 𝑦 ) 𝑧] = 𝑥 ∨ 𝑦′

by item (2f)

(i) Proof that [(𝑥′ ∨ 𝑦′ )′ ∨ (𝑥′ ∨ 𝑦)′ ] ∨ 𝑥′ = 𝑧 ∨ 𝑧′ : ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ [(𝑥 ∨ 𝑦 ) ∨ (𝑥 ∨ 𝑦) ] ∨ 𝑥 = 𝑥 ∨ [(𝑥 ∨ 𝑦 ) ∨ (𝑥 ∨ 𝑦) ] by identity 1 (commutative property) = [𝑥′ ∨ (𝑥′ ∨ 𝑦′ )′ ] ∨ (𝑥′ ∨ 𝑦)′ by identity 2 (associative property)

= (𝑥′ ∨ 𝑦″ ) ∨ (𝑥′ ∨ 𝑦)′ ′

′

′

= (𝑥 ∨ 𝑦) ∨ (𝑥 ∨ 𝑦) =𝑧∨𝑧 2016 December 27 Tuesday UTC

′

Daniel J. Greenhoe

by item (2h) by item (2d) (involutory) by item (2b)


version 0.30

page 136


(j) Proof that (𝑥′ ∨ 𝑦′ )′ ∨ (𝑥′ ∨ 𝑦)′ = 𝑥 (Huntington's axiom): ′

′ ′

′

′

′

′ ′

′

′

∨ 𝑦 ) ∨ (𝑥 ∨ 𝑦) ∨ ⏟ 𝑥 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (𝑥 ∨ 𝑦 ) ∨ (𝑥 ∨ 𝑦) = (𝑥 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ “𝑥” in identity 3

“𝑥” in identity 3

=𝑥 ∨ (𝑥′ ∨ 𝑦)′ ∨ (𝑥′ ∨ 𝑦′ )′ ⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑥 by item (2f) ′ ′ ′

=𝑥 ∨ (𝑥 ∨ 𝑦 ) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

by item (2i) and identity 3

“𝑦”

by identity 1 (commutative property) by item (2f)

𝑥 by item (2f)

=𝑥

by item (2f)

(k) The three identities therefore imply that 𝑨 i. ii. iii. iv.

is idempotent (item (2a)), is commutative (identity 1), is associative (identity 2), and satisfies Huntington's axiom (item (2j)).

Therefore, by Proposition 8.6 page 133 (Huntington's Fourth Set), 𝑨 is a Boolean algebra.

✏ Proposition 8.10 (Byrne's FORMULATION B). 28 Let 𝑨 ≜ ( 𝑋, ∨, ∧ ; ≤) be an algebraic structure. ⟺ p 𝑨 is a Boolean algebra ′ ′ r 1. 𝑥 ∨ 𝑦 = 𝑧 ∨ 𝑧 ⟺ 𝑥∨𝑦=𝑥 ∀𝑥,𝑦,𝑧∈𝑋 and p { 2. (𝑥 ∨ 𝑦) ∨ 𝑧 } = (𝑦 ∨ 𝑧) ∨ 𝑥 ∀𝑥,𝑦,𝑧∈𝑋. Theorem 8.18. 29 Let 𝘼 ≜ ( 𝑋, ∨, ∧ ; ≤) be an algebraic structure. 𝘼 is a Boolean algebra ⟺ t 1. 𝑥 ∧ (𝑥 ∨ 𝑦) = 𝑥 ∀𝑥,𝑦∈𝑋 ⎧ h ⎪ = (𝑧 ∧ 𝑥) ∨ (𝑦 ∧ 𝑥) ∀𝑥,𝑦,𝑧∈𝑋 m ⎨ 2. 𝑥 ∧ (𝑦 ∨ 𝑧) ′ ′ ⎪ 3. ∃𝑦 such that 𝑥 ∧ (𝑦 ∨ 𝑦 ) = 𝑥 ∨ (𝑦 ∧ 𝑦′ ) ∀𝑥,𝑦∈𝑋. ⎩

and and

⎫ ⎪ ⎬ ⎪ ⎭

✎PROOF: 1. Proof that 𝘼 is a distributive lattice: by 1 and 2 and by Theorem 6.4 (page 100). 2. Define 0 ≜ 𝑥 ∧ 𝑥′ and 1 ≜ 𝑥 ∨ 𝑥′ . 3. Proof that 0 is the join-identity element and that 1 is the meet-identity element: 𝑥 ∨ 0 = 𝑥 ∨ (𝑦 ∧ 𝑦′ )

by definition of 0 (item (2) page 136) ′

= (𝑥 ∨ 𝑥) ∨ (𝑦 ∧ 𝑦 ) = 𝑥 ∨ [𝑥 ∨ (𝑦 ∧ 𝑦′ )] ′

= 𝑥 ∨ [𝑥 ∧ (𝑦 ∨ 𝑦 )] =𝑥 𝑥 ∧ 1 = 𝑥 ∧ (𝑦 ∨ 𝑦′ ) = (𝑥 ∧ 𝑥) ∧ (𝑦 ∨ 𝑦 ) = 𝑥 ∧ [𝑥 ∧ (𝑦 ∨ 𝑦′ )] ′

= 𝑥 ∧ [𝑥 ∨ (𝑦 ∧ 𝑦 )] =𝑥 29

by associative property of lattices (Theorem 4.3 page 62) by 3 by absorptive property of lattices (Theorem 4.3 page 62) by definition of 1 (item (2) page 136)

′

28

by idempotent property of lattices (Theorem 4.3 page 62)

by idempotent property of lattices (Theorem 4.3 page 62) by associative property of lattices (Theorem 4.3 page 62) by 3 by absorptive property of lattices (Theorem 4.3 page 62)

📘 Byrne (1946) page 271 ⟨“FORMULATION B”⟩ 📘 Sholander (1951) pages 28–29, 𝑃 1, 𝑃 2, 𝑃 3∗

version 0.30


Daniel J. Greenhoe


page 137


4. Proof that 𝘼 is bounded with 0 being the greatest lower bound and 1 being the least upper bound: 𝑥 ∧ 0 = (𝑥 ∨ 0) ∧ 0

by identity property (item (3) page 136)

= 0 ∧ (0 ∨ 𝑥)

by commutative property of lattices (Theorem 4.3 page 62)

=0


𝑥 ∨ 1 = (𝑥 ∧ 1) ∨ 1

by identity property (item (3) page 136)

= 1 ∨ (1 ∧ 𝑥)


=1


5. Proof that 𝘼 is complemented: Because 𝘼 is bounded with greatest lower bound 0 and least upper bound 1 (item (4)) and because 𝑥 ∧ 𝑥′ = 0 and 𝑥 ∨ 𝑥′ = 1 (definition of 0 and 1 (item (2) page 136)). 6. Proof that 𝘼 is a Boolean algebra: Because 𝘼 is distributive (item (1)) and complemented (item (5)), and by Definition 8.1 (page 115).

✏

8.6 Literature 📚 Literature survey: 1. General information about Boolean algebras: 📘 Sikorski (1969) 📘 Dwinger (1971) 📘 Dwinger (1961) 📘 Halmos (1972) 📘 Monk (1989) 📘 Givant and Halmos (2009) 2. Characterizations: (a) Survey of characterizations: 📘 Padmanabhan and Rudeanu (2008) (b) Characterizations in terms of traditional binary operations join ∨, meet ∧, and complement ′: 📃 Huntington (1904) ⟨ ⟩📘 Huntington (1933) ⟨ ⟩📘 Diamond (1933) 📘 Diamond (1934) 📘 Stone (1935) 📘 Hoberman and McKinsey (1937) 📘 Frink (1941) ⟨4 identities involving ∨, ∧, ′⟩ 📘 Newman (1941) 📘 Braithwaite (1942) 📘 Byrne (1946) ⟨Form. A and B⟩ 📘 Gerrish (1978) ⟨independence of Huntington's characterizations⟩ (c) Characterizations in terms of non-traditional binary operations: 📘 Sheffer (1913) ⟨rejection ↓⟩ 📘 Bernstein (1914) ⟨exception −⟩ 📘 Bernstein (1916) ⟨rejection ↓⟩ 📘 Bernstein (1933) ⟨rejection ↓⟩ 📘 Bernstein (1934) ⟨implication ⇒⟩ 📘 Bernstein (1936) ⟨complete disjuction △⟩ 📘 Byrne (1948) ⟨inclusion⟩ 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 138

8.6. LITERATURE 📘 Byrne (1951) ⟨ring operations⟩ 📘 Miller (1952) ⟨ring operations⟩

(d) Characterizations in terms of ternary operations: 📘 Whiteman (1937)ternary rejection (e) Characterizations involving Elkan's law: 📘 Kondo and Dudek (2008) ⟨for bounded lattices⟩ 📘 Renedo et al. (2003) ⟨for orthomodular lattices⟩ 📘 Trillas et al. (2004) ⟨for orthocomplemented lattices⟩ 3. Analytic properties: 📘 Vladimirov (2002) 4. Miscellaneous: 📘 Montague and Tarski (1954) 📘 Rudeanu (1961) ⟨referenced by 📘 Sikorski (1969)⟩ 5. Actually, “Boolean algebras” are not really “algebras”. Rather, they are “a commutative ring with unit, without nilpotents, and having idempotents which stood for classes” 📘 Hailperin (1981), page 184 6. Pioneering works related to Boolean algebras: 📘 Boole (1847) 📘 Boole (1854) 📘 Jevons (1864) ⟨join and meet operations⟩ 📘 Peirce (1870a) ⟨order concepts⟩ 📃 Huntington (1904) ⟨axiomization⟩ 7. History of development of Boolean algebra: 📘 Burris (2000)

📚

version 0.30


Daniel J. Greenhoe


CHAPTER 9 ORTHOCOMPLEMENTED LATTICES

Orthocomplemented lattices (Definition 9.1 page 140) are a kind of generalization of Boolean algebras. The relationship between lattices of several types, including orthocomplemented and Boolean lattices, is stated in Theorem 9.7 (page 150) and illustrated in Figure 9.1 (page 139).

bounded (Definition 4.4 page 71)

complemented (Definition 7.1 page 109)

modular

orthocomplemented



distributive

orthomodular



modular orthocomplemented (Definition 9.5 page 150)

boolean (Definition 8.1 page 115)

Figure 9.1: lattice of orthocomplemented lattices

page 140

9.1

9.1. ORTHOCOMPLEMENTED LATTICES

Orthocomplemented Lattices

9.1.1

Definition

Definition 9.1. 1 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice (Definition 4.4 page 71). An element 𝑥⟂ ∈ 𝑋 is an orthocomplement of an element 𝑥 ∈ 𝑋 if ⟂⟂ 1. 𝑥 = 𝑥 (involutory) and ⟂ d 2. 𝑥 ∧ 𝑥 = 0 (non-contradiction) and e ⟂ ⟂ 3. 𝑥 ≤ 𝑦 ⟹ 𝑦 ≤ 𝑥 ∀𝑦∈𝑋 (antitone). f The lattice 𝙇 is orthocomplemented (𝙇 is an orthocomplemented lattice) if every element 𝑥 in 𝑋 has an orthocomplement. Definition 9.2.

2

1

𝑞⟂

d The O6 lattice is the ordered set ({0, 𝑝, 𝑞, 𝑝⟂ , 𝑞 ⟂ , 1}, ≤) with cover relation e ≺= {(0, 𝑝) , (0, 𝑞) , (𝑝, 𝑞 ⟂ ) , (𝑞, 𝑝⟂ ) , (𝑝⟂ , 1) , (𝑞 ⟂ , 1)}. f The 𝑂6 lattice is illustrated by the Hasse diagram to the right.

𝑝

𝑝⟂ 𝑞

0

Example 9.1. 3 e The 𝑂6 lattice (Definition 9.2 page 140) is an x orthocomplemented lattice (Definition 9.1 page 140). Example 9.2. 4 There are a total of 10 orthocomplemented lattices with 8 elements or less. These 10, along with 3 other orthocomplemented lattices with 10 elements, are illustrated next: Lattices that are orthocomplemented but non-orthomodular and hence also not modular orthocomplemented and non-Boolean: 1

𝑦⟂

1

𝑦⟂

𝑥⟂ 𝑦

𝑥

0

𝑂6 lattice

1.

1 𝑥⟂

⟂ 𝑧⟂ 𝑦 𝑝

𝑥

𝑦 0 5.

𝑝⟂ 𝑧

2.

⟂

𝑝⟂

𝑝 𝑥

0

𝑝

𝑥

𝑦

3.

𝑝 𝑧

𝑞

𝑧

𝑥⟂ 𝑞⟂

𝑝⟂ 𝑦

𝑥 0

0 6.

𝑦 0

𝑥⟂

4.

1

𝑦⟂

𝑤 𝑦

𝑥⟂

𝑝

0 𝑥⟂ ⟂

𝑧⟂ 𝑦⟂

1 ⟂ 𝑧⟂ 𝑦

⟂

𝑥

𝑂8 lattice

𝑥

𝑦

𝑦

1

𝑤

1

𝑥⟂

7.

Lattices that are orthocomplemented and orthomodular but not modular orthocomplemented and hence also non-Boolean:

1

📘 Stern (1999) page 11, 📘 Beran (1985) page 28, 📘 Kalmbach (1983) page 16, 📘 Gudder (1988) page 76, 📘 Loomis (1955) page 3, 📃 Birkhoff and Neumann (1936) page 830 ⟨L71–L73⟩ 2 📘 Kalmbach (1983) page 22, 📓 Holland (1970), page 50, 📘 Beran (1985) page 33, 📘 Stern (1999) page 12, The O6 lattice is also called the Benzene ring or the hexagon. 3 📘 Holland (1963), page 50 4 📘 Beran (1985) pages 33–42, 📃 Maeda (1966) page 250, 📘 Kalmbach (1983) page 24 ⟨Figure 3.2⟩, 📘 Stern (1999) page 12, 📘 Holland (1970), page 50 version 0.30


Daniel J. Greenhoe


page 141

CHAPTER 9. ORTHOCOMPLEMENTED LATTICES 1

1 𝑑⟂

𝑦⟂ 𝑧⟂ 𝑝

𝑥

𝑥

𝑝⟂

𝑧

𝑦

𝑐⟂

⟂

𝑧⟂

𝑦⟂

𝑥⟂

𝑏⟂

𝑥

𝑦

𝑧

𝑐

𝑏

𝑎

0

𝑎⟂

𝑑

0

8.

9.

Lattices that are orthocomplemented, orthomodular, and modular orthocomplemented but non-Boolean: 1 𝑤

1 𝑦

𝑥

𝑥

𝑧

𝑦

𝑧

𝑦⟂

𝑧⟂

0

𝑥⟂

0

10. 𝑀4 lattice 11. 𝑀6 lattice Lattices that are orthocomplemented, orthomodular, modular orthocomplemented and Boolean:

1

1

1 𝑟⟂

𝑝

𝑝⟂

𝑞⟂

𝑝⟂ 𝑞

𝑝

12.

0

0

0 1

𝐿1 lattice

𝐿2 lattice

13.

𝐿22

14.

𝑟

0

lattice

15.

𝐿32 lattice

1 𝑠⟂ 𝑑

𝑏

𝑎

𝑏⟂

𝑑⟂ 𝑝

𝑞

𝑝⟂

𝑞⟂

𝑟⟂

𝑟

𝑎⟂

𝑠

0 4 2

16.𝐿

lattice 1

𝑗⟂ 𝑎

𝑟⟂

𝑠⟂

𝑡⟂ 𝑔⟂

ℎ⟂

⟂

𝑖

𝑑⟂

𝑓⟂ 𝑒⟂

𝑏 𝑝

𝑐

𝑑 𝑞

𝑞⟂

𝑒

𝑓

𝑟

𝑠

𝑔

𝑝⟂

𝑐⟂

𝑏⟂

ℎ

𝑖

𝑎⟂ 𝑗

𝑡

0 5 2

17.𝐿

lattice

Example 9.3. 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 142

9.1. ORTHOCOMPLEMENTED LATTICES 𝙔⟂

𝘕

The structure ( 𝟚ℝ , +, ∩, ∅, 𝙃 ; ⊆) is an orthocomplemented lattice where ℝ𝘕 is an Euclidean space with dimension 𝘕 𝘕 𝟚ℝ is the set of all subspaces of ℝ𝘕 𝙑 + 𝙒 is the Minkowski sum of subspaces 𝙑 and 𝙒 𝙑 ∩ 𝙒 is the intersection of subspaces 𝙑 and 𝙒

e x

𝙕 𝙔 𝙕⟂

𝙓

Example 9.4. The structure ( 𝟚𝙃 , ⊕, ∩, ∅, 𝙃 ; ⊆) is an orthocomplemented lattice where 𝙃 is a Hilbert space 𝟚𝙃 is the set of all closed subspaces of 𝙃 e x 𝙓 + 𝙔 is the Minkowski sum of subspaces 𝙓 and 𝙔 𝙓 ⊕ 𝙔 ≜ (𝙓 + 𝙔 )− is the closure of 𝙓 + 𝙔 𝙓 ∩ 𝙔 is the intersection of subspaces 𝙓 and 𝙔

9.1.2

Properties

⎧ t ⎪ h m ⎨ ⎪ ⎩

5

Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an algebraic structure. 0⟂ = 1 (boundary condition) ⎧ (1). 𝙇 is an ⟂ ⎪ ⎫ (2). 1 = 0 (boundary condition) ⎪ ⎪ ortho⟂ ⟂ ⟂ ⟹ ⎨ (3). (𝑥 ∨ 𝑦) = 𝑥 ∧ 𝑦 ∀𝑥,𝑦∈𝑋 (disjunctive de morgan) complemented ⎬ ⎪ (4). (𝑥 ∧ 𝑦)⟂ = 𝑥⟂ ∨ 𝑦⟂ ∀𝑥,𝑦∈𝑋 (conjunctive de morgan) ⎪ lattice ⎭ ⎪ ⟂ ∀𝑥∈𝑋 (excluded middle). ⎩ (5). 𝑥 ∨ 𝑥 = 1

Theorem 9.1.

and and and and

Let 𝑥⟂ ≜ ¬𝑥, where ¬ is an ortho negation function (Definition 11.3 page 158). Then, this theorem follows ✏ directly from Theorem 11.5 (page 162).

✎PROOF:

Corollary 9.1. Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a lattice (Definition 4.3 page 61). c 𝙇 is orthocomplemented 𝙇 is complemented o ⟹ } { (Definition 7.1 page 109) } r { (Definition 9.1 page 140) ✎PROOF:

This follows directly from the definition of orthocomplemented lattices (Definition 9.1 page 140) and complemented lattices (Definition 7.1 page 109). ✏

Example 9.5. 1

e x

1

𝑎

𝑏

𝑝

𝑞 0

The 𝑂6 lattice (Definition 9.2 page 140) illustrated to the left is both orthocomplemented (Definition 9.1 page 140) and multiply complemented (Definition 7.1 page 109). The lattice illustrated to the right is multiply complemented, but is non-orthocomplemented.

𝑎 𝑝

𝑞

𝑟

0

✎PROOF: 1. Proof that 𝑂6 lattice is multiply complemented: 𝑏 and 𝑞 are both complements of 𝑝. 5

📘 Beran (1985) pages 30–31, 📃 Birkhoff and Neumann (1936) page 830 ⟨L74⟩, 📘 Cohen (1989) page 37 ⟨3B.13. Theorem⟩ version 0.30


Daniel J. Greenhoe


page 143

CHAPTER 9. ORTHOCOMPLEMENTED LATTICES

2. Proof that the right side lattice is multiply complemented: 𝑎, 𝑝, and 𝑞 are all complements of 𝑟.

✏

9.1.3 Characterization 6

Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an algebraic structure. ⟂ ⟂ ⟂ 1. (𝑧 ∧ 𝑦 ) ∨ 𝑥 = (𝑥 ∨ 𝑦) ∨ 𝑧 ⎧ ⎪ 𝙇 is an = 𝑥 ⟺ ⎨ 2. 𝑥 ∧ (𝑥 ∨ 𝑦) orthocomplemented lattice } ⟂ ⎪ = 𝑥 ⎩ 3. 𝑥 ∨ (𝑦 ∧ 𝑦 )

Theorem 9.2. t h m


and

∀𝑥,𝑦∈𝑋

and

∀𝑥,𝑦∈𝑋.

✎PROOF: 1. Proof that orthocomplemented lattice ⟹ 3 properties: ⟂

⟂

⟂

⟂ ⟂ ⟂ ⟂ (𝑧 ∧ 𝑦 ) ∨ 𝑥 = [(𝑧 ) ∨ (𝑦 ) ] ∨ 𝑥

by de Morgan property (Theorem 9.1 page 142)

= (𝑧 ∨ 𝑦) ∨ 𝑥

by involutory property (Definition 9.1 page 140)

= 𝑥 ∨ (𝑧 ∨ 𝑦)


= 𝑥 ∨ (𝑦 ∨ 𝑧)


= (𝑥 ∨ 𝑦) ∨ 𝑧

by associative property (Theorem 4.3 page 62)

𝑥 ∧ (𝑥 ∨ 𝑦) = 𝑥


𝑥 ∨ (𝑦 ∧ 𝑦⟂ ) = 𝑥 ∨ 0

by complemented property (Definition 9.1 page 140)

=𝑥 2. Proof that orthocomplemented lattice ⟸ 3 properties: (a) Proof that 𝙇 is meet-idempotent: 𝑥 ∧ 𝑥 = 𝑥 ∧ [𝑥 ∨ (𝑦 ∧ 𝑦⟂ )]

by (3)

⟂

= 𝑥 ∧ [𝑥 ∨ (𝑦 ∧ 𝑦 )] =𝑥

by (3) by (2)

(b) Define 0 ≜ 𝑥𝑥⟂ for some 𝑥 ∈ 𝑋. Proof that 0 is the greatest lower bound of 𝙇: The element 0 is the greatest lower bound if and only if 𝑥𝑥⟂ = 𝑦𝑦⟂ ∀𝑥, 𝑦 ∈ 𝑋… i. Proof that (𝑥𝑥⟂ )⟂⟂ = (𝑥𝑥⟂ )

∀𝑥 ∈ 𝑋:

(𝑥𝑥⟂ )⟂⟂ = (𝑥𝑥⟂ )⟂⟂ + (𝑥𝑥⟂ ) ⟂ ⟂

⟂ ⟂ ⟂

by (3) ⟂

= [(𝑥𝑥 ) (𝑥𝑥 ) ] + (𝑥𝑥 ) = [(𝑥𝑥⟂ ) + (𝑥𝑥⟂ )] + (𝑥𝑥⟂ ) ⟂

⟂

= [(𝑥𝑥 )] + (𝑥𝑥 ) = (𝑥𝑥⟂ ) 6

by item (2a) by (1) by (3) by (3)

📘 Beran (1985) pages 31–33, 📃 Beran (1976) pages 251–252


Daniel J. Greenhoe


version 0.30

page 144


ii. Proof that 𝑎 = (𝑥𝑥⟂ ) + 𝑎

∀𝑎, 𝑥 ∈ 𝑋:

𝑎 = 𝑎 + (𝑥𝑥⟂ )

by (3)

= [𝑎 + (𝑥𝑥⟂ )] + (𝑥𝑥⟂ )

by (3)

⟂ ⟂ ⟂

= [(𝑥𝑥⟂ )⟂ (𝑥𝑥 ) ] + 𝑎

by (1)

⟂ ⟂ ⟂

= [(𝑥𝑥 ) ] + 𝑎 = (𝑥𝑥⟂ ) + 𝑎 iii. Proof that (𝑥𝑥⟂ ) = (𝑦𝑦⟂ )

by item (2a) by item (2(b)i)

∀𝑥, 𝑦 ∈ 𝑋:

(𝑥𝑥⟂ ) = (𝑥𝑥⟂ ) + (𝑦𝑦⟂ )

by (3)

= (𝑦𝑦⟂ ) (c) Proof that 𝑥 + 0 = 0 + 𝑥 = 𝑥

by item (2(b)ii)

∀𝑥 ∈ 𝑋 (join identity):

𝑥 + 0 = 𝑥 + (𝑦𝑦⟂ )

by item (2(b)iii)

=𝑥

by (3)

0 + 𝑥 = (𝑢𝑢⟂ ) + 𝑥

by item (2(b)iii)

=𝑥 (d) Proof that 𝑥 + 𝑦 = (𝑦⟂ 𝑥⟂ )⟂

by item (2(b)ii)

∀𝑥, 𝑦 ∈ 𝑋:

(𝑦⟂ 𝑥⟂ )⟂ = (𝑦⟂ 𝑥⟂ )⟂ + 0

(e) Proof that 𝑥 + 𝑥 = 𝑥⟂⟂

by item (2c)

= (0 + 𝑥) + 𝑦

by (1)

=𝑥+𝑦

by item (2c)

∀𝑥 ∈ 𝑋: 𝑥 + 𝑥 = (𝑥⟂ 𝑥⟂ )⟂

by item (2d)

= (𝑥⟂ )⟂ (f) Proof that 𝑥 + 𝑦 = 𝑦 + 𝑥

by item (2a)

∀𝑥, 𝑦 ∈ 𝑋

(join-commutative):

𝑥 + 𝑦 = (𝑥 + 0) + 𝑦 ⟂ ⟂ ⟂

by item (2c)

= (0 𝑥 ) + 𝑦

by item (2d)

= (𝑦 + 𝑥) + 0

by (1)

=𝑦+𝑥

by item (2c)

(g) Proof that (𝑥 + 𝑦) + 𝑧 = 𝑥 + (𝑦 + 𝑧)

∀𝑥, 𝑦, 𝑧 ∈ 𝑋

(join-associative):

(𝑥 + 𝑦) + 𝑧 = (𝑧⟂ 𝑦⟂ )⟂ + 𝑥

(h) Proof that 𝑥⟂⟂ = 𝑥

∀𝑥 ∈ 𝑋

= (𝑦 + 𝑧) + 𝑥

by item (2d)

= 𝑥 + (𝑦 + 𝑧)

by item (2f)

(involutory):

𝑥⟂⟂ = (𝑥⟂ ) ⟂

by definition of 𝑥⟂⟂

= [𝑥⟂ (𝑥⟂ + 𝑥)] ⟂ ⟂

version 0.30

by (1)

by (2)

= [𝑥⟂ (𝑥⟂ 𝑥⟂⟂ ) ] ⟂

by item (2d)

= (𝑥⟂ 𝑥⟂⟂ ) + 𝑥

by item (2d)

= (0) + 𝑥

by item (2b)

=𝑥

by item (2c)


Daniel J. Greenhoe


page 145


(i) Proof of de Morgan's laws: (𝑥 + 𝑦)⟂ = (𝑦 + 𝑥)⟂

by item (2g)

⟂ ⟂ ⟂ ⟂

= [(𝑥 𝑦 ) ]

by item (2d)

= 𝑥 ⟂ 𝑦⟂

by item (2h)

(𝑥𝑦)⟂ = (𝑥⟂⟂ 𝑦⟂⟂ ) = 𝑦 ⟂ + 𝑥⟂ ⟂

=𝑥 +𝑦 (j) Proof that (𝑥𝑦)𝑧 = 𝑥(𝑦𝑧)

⟂

by item (2h) by item (2d)

⟂

by item (2g)

∀𝑥, 𝑦, 𝑧 ∈ 𝑋

(meet-commutative):

𝑥𝑦 = (𝑥𝑦)⟂⟂

by item (2h) ⟂

= (𝑥⟂ + 𝑦⟂ )

by item (2i)

⟂ ⟂

(k) Proof that (𝑥𝑦)𝑧 = 𝑥(𝑦𝑧)

= (𝑦⟂ + 𝑥 ) = 𝑦⟂⟂ 𝑥⟂⟂

by item (2g)

= 𝑦𝑥

by item (2i)

by item (2i)

∀𝑥, 𝑦, 𝑧 ∈ 𝑋

(meet-associative):

(𝑥𝑦)𝑧 = [(𝑥𝑦)𝑧]⟂ ⟂ ⟂

by item (2h)

⟂ ⟂

by item (2i)

= [(𝑥𝑦) + 𝑧 ]

= [(𝑥⟂ + 𝑦⟂ ) + 𝑧⟂ ]

⟂

by item (2i)

= [𝑥⟂ + (𝑦⟂ + 𝑧⟂ )]

⟂

by item (2g)

⟂⟂

⟂

⟂ ⟂

(𝑦 + 𝑧 ) = 𝑥⟂⟂ (𝑦⟂⟂ 𝑧⟂⟂ )

by item (2i)

= 𝑥(𝑦𝑧)

by item (2h)

=𝑥

by item (2i)

(l) Proof that 𝑥 + (𝑥𝑧) = 𝑥 (join-meet-absorptive): 𝑥 ∨ (𝑥𝑧) = [𝑥 + (𝑥𝑧)]⟂⟂

by item (2h)

⟂

= [𝑥⟂ (𝑥𝑧)⟂ ]

by item (2i) ⟂

= [𝑥⟂ (𝑥⟂ + 𝑧⟂ )] ⟂ ⟂

= [𝑥 ] =𝑥

by item (2i) by (2) by item (2h)

(m) Because 𝙇 is commutative (item (2f) and item (2j)), associative (item (2g) and item (2k)), and absorptive ((2) and item (2l)), and by Theorem 4.8 (page 68), 𝙇 is a lattice. (n) Define 1 ≜ 𝑥 + 𝑥⟂ for some 𝑥 ∈ 𝑋. Proof that 1 is the least upper bound of 𝙇: The element 1 is the least upper bound if and only if 𝑥 + 𝑥⟂ = 𝑦 + 𝑦⟂ ∀𝑥, 𝑦 ∈ 𝑋… 1 = (𝑥 + 𝑥 ⟂ )

by item (2h)

= (𝑥 𝑥)

⟂

by item (2h)

= (𝑥𝑥⟂ )

⟂

by item (2j)

= (𝑥 + 𝑥 ) ⟂

⟂ ⟂

= (𝑦𝑦 ) = 𝑦⟂ + 𝑦⟂⟂ ⟂

=𝑦 +𝑦 =𝑦+𝑦 2016 December 27 Tuesday UTC

by definition of 1

⟂ ⟂⟂

⟂

Daniel J. Greenhoe

by item (2(b)iii) by item (2i) by item (2h) by item (2f) Sets, Relations, and Order Structures

version 0.30

page 146


(o) Proof that 𝙇 is antitone: by Theorem 11.4 (page 162). (p) Proof that 𝙇 is complemented: by item (2(b)iii) and item (2n). (q) Because 𝙇 is a bounded (item (2b) and item (2n)) lattice (item (2m)), and because 𝙇 is complemented (item (2p)), is involutory (item (2h)), and is antitone (item (2o)), and by Definition 9.1 (page 140), 𝙇 is an orthocomplemented lattice.

✏

9.1.4

Restrictions resulting in Boolean algebras

Proposition 9.1. 7 Let 𝙇 = ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a lattice (Definition 4.3 page 61). p 1. 𝙇 is orthocomplemented ((Definition 9.1 page 140)) and 𝙇 is Boolean r ⟹ } { (Definition 8.1 page 115) } ((Definition 6.2 page 90)) p { 2. 𝙇 is distributive ✎PROOF:

To be a Boolean algebra, 𝙇 must satisfy the 8 requirements of boolean algebras (Definition 8.1 page 115):

1. Proof for commutative properties: These are true for all lattices (Definition 4.3 page 61). 2. Proof for join-distributive property: by hypothesis (2). 3. Proof for meet-distributive property: by join-distributive property and the Principle of duality 4.4 page 63) for lattices.

(Theorem

4. Proof for identity properties: because 𝙇 is a bounded lattice and by definitions of 1 (least upper bound), 0 (greatest lower bound), ∨, and ∧. 5. Proof for complemented properties: by hypothesis (1) and definition of orthocomplemented lattices (Definition 9.1 page 140).

✏ Proposition 9.2. Let 𝙇 = ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a lattice (Definition 4.3 page 61). p 𝙇 is 1. 𝙇 is orthocomplemented (Definition 9.1 page 140) and r ⟺ { { } 2. Every 𝑥 ∈ 𝙇 is in the center of 𝙇 (Definition 13.4 page 196) Boolean } p ✎PROOF: 1. Proof that (1,2) ⟹ Boolean: 𝙇 is Boolean because it satisfies Huntington's Fourth Set 133), as demonstrated by the following …

(Proposition 8.6 page

(a) Proof that 𝑥 ∨ 𝑥 = 𝑥 (idempotent): 𝙇 is a lattice (by definition of 𝙇), and all lattices are idempotent (Definition 4.3 page 61). (b) Proof that 𝑥 ∨ 𝑦 = 𝑦 ∨ 𝑥 (commutative): 𝙇 is a lattice (by definition of 𝙇), and all lattices are commutative (Definition 4.3 page 61). (c) Proof that (𝑥 ∨ 𝑦) ∨ 𝑧 = 𝑥 ∨ (𝑦 ∨ 𝑧) (associative): 𝙇 is a lattice (by definition of 𝙇), and all lattices are associative (Definition 4.3 page 61). 7

📘 Kalmbach (1983) page 22

version 0.30


Daniel J. Greenhoe


page 147


(d) Proof that (𝑥⟂ ∨ 𝑦⟂ )⟂ ∨ (𝑥⟂ ∨ 𝑦)⟂ = 𝑥 (Huntington's axiom): (𝑥⟂ ∨ 𝑦⟂ )⟂ ∨ (𝑥⟂ ∨ 𝑦)⟂ = (𝑥⟂ ⟂ ∧𝑦⟂ ⟂) ∨ (𝑥⟂ ⟂ ∧𝑦⟂ ) by de Morgan property (Theorem 9.1 page 142) = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦⟂ )

by involution property (Definition 9.1 page 140)

=𝑥

by definition of center


2. Proof that (1) ⟸ Boolean: (a) Proof that 𝑥 ∨ 𝑥⟂ = 1: by definition of Boolean algebras (Definition 8.1 page 115). (b) Proof that 𝑥 ∧ 𝑥⟂ = 0: by definition of Boolean algebras (Definition 8.1 page 115). (c) Proof that 𝑥⟂⟂ = 𝑥: by involutory property of Boolean algebra (Theorem 8.2 page 120). (d) Proof that 𝑥 ≤ 𝑦 ⟹ 𝑦⟂ ≤ 𝑥⟂ : 𝑦⟂ ≤ 𝑥 ⟂ ⟺ 𝑦 ⟂

= 𝑦 ⟂ ∧ 𝑥⟂

by Lemma 4.1 page 62 ⟂

⟺ 𝑦⟂⟂

= (𝑦⟂ ∧ 𝑥⟂ )

⟺ 𝑦⟂⟂

= 𝑦⟂⟂ ∨ 𝑥⟂⟂


⟺ 𝑦

=𝑦∨𝑥


⟺ 𝑦

=𝑦


3. Proof that (2) ⟸ Boolean: for all 𝑥, 𝑦 ∈ 𝙇 (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦⟂ ) = [(𝑥 ∧ 𝑦) ∨ 𝑥] ∧ [(𝑥 ∧ 𝑦) ∨ 𝑦⟂ ]


⟂

= 𝑥 ∧ [(𝑥 ∧ 𝑦) ∨ 𝑦 ] = 𝑥 ∧ [(𝑥 ∨ 𝑦⟂ ) ∧ (𝑦 ∨ 𝑦⟂ )]

by absorptive property (Theorem 8.2 page 120) by distributive property (Theorem 8.2 page 120)

⟂

= 𝑥 ∧ (𝑥 ∨ 𝑦 ) ∧ 1 =𝑥

by complement property (Theorem 8.2 page 120)

⟹ 𝑥Ⓒ𝑦



∀𝑥, 𝑦 ∈ 𝙇

⟹ 𝑥 is in the center of 𝙇 for all 𝑥 ∈ 𝙇 by Definition 13.4 page 196

✏ Example 9.6. The 𝑂6 lattice (Definition 9.2 page 140) illustrated to the left is orthocomplemented (Definition 9.1 page 140) but non-join-distributive (Definition 6.2 page 90),and hence non-Boolean. The lattice illustrated to e the right is orthocomplemented and distributive and hence also x Boolean (Proposition 9.1 page 146). Alternatively, the right side lattice is orthocomplemented and every element is in the center, and hence also Boolean (Proposition 9.2 page 146). Note that of the 5 lattices on 5 element sets (Example 4.11 page 67), the 15 lattices on 6 element sets (Example 4.12 page 67), and 53 lattices on 7 element sets (Example 4.13 page 68), none are uniquely complemented. ✎PROOF: 1. Proof that the 𝑂6 lattice is non-join-distributive: 𝑥 ∨ (𝑥⟂ ∧ 𝑧⟂ ) = 𝑥 ∨ 0 =𝑥 ≠ 𝑧⟂ = 1 ∧ 𝑧⟂ = (𝑥 ∨ 𝑥⟂ ) ∧ (𝑥 ∨ 𝑧⟂ ) 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 148

9.2. ORTHOMODULAR LATTICES

2. Proof that the 𝑂6 lattice is also non-meet-distributive: 𝑧⟂ ∧ (𝑥 ∨ 𝑧) = 𝑧⟂ ∧ 1 = 𝑧⟂ ≠𝑥 =𝑥∨1 = (𝑧⟂ ∧ 𝑥) ∨ (𝑧⟂ ∧ 𝑧)

✏

9.2 9.2.1

Orthomodular lattices Properties

Definition 9.3. 8 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an algebraic structure. d 𝙇 is an orthomodular lattice if e 1. 𝙇 is an orthocomplemented lattice and f ⟂ 2. 𝑥 ≤ 𝑦 ⟹ 𝑥 ∨ (𝑥 ∧ 𝑦) = 𝑦 ∀𝑥,𝑦∈𝑋 (orthomodular identity) Example 9.7. e The 𝑂6 lattice (Definition 9.2 page 140) is orthocomplemented, but non-orthomodular x (and hence, non-modular and non-Boolean).

𝙔 = 𝙓 ⊕ (𝙓 ⟂ ∩ 𝑌 )

Example 9.8. 9 Let 𝙃 be a Hilbert space and 𝟚𝙃 the set of closed linear subspaces of 𝙃 . 𝙃 e ( 𝟚 , ⊕, ∩, ∅, 𝙃 ; ⊆) x is an orthomodular lattice. This concept is illustrated to the right where 𝙓 , 𝙔 ∈ 𝟚𝙃 are linear subspaces of the linear space 𝙃 and 𝙓 ⊆𝙔 ⟹ 𝙔 = 𝙓 ⊕ (𝙓 ⟂ ∩ 𝑌 ).

?

𝙓⟂

-𝙓 @ I @

@ @

𝙓⟂ ∩ 𝙔

Theorem 9.3. 10 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a lattice. t 1. 𝙇 is orthomodular and h (𝑥, 𝑦, 𝑧) ∈ Ⓓ ⟹ } m 2. 𝑦Ⓒ𝑥 and 𝑧Ⓒ𝑥 8

📘 Kalmbach (1983) page 22, 📘 Lidl and Pilz (1998) page 90, 📃 Husimi (1937) 📘 Iturrioz (1985) pages 56–57 10 📘 Kalmbach (1983) page 25, 📃 Holland (1963) pages 69–70 ⟨THEOREM 3⟩, 📃 Foulis (1962) page 68 ⟨THEOREM 5⟩ 9

version 0.30


Daniel J. Greenhoe


page 149


9.2.2 Characterizations Theorem 9.4. 11 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an orthocomplemented lattice (Definition 9.1 page 140). Let Ⓜ and Ⓜ∗ be the modularity relation and dual modularity relation, respectively (Definition 5.1 page 79), ⟂ the orthogonality relation (Definition 13.1 page 191), and Ⓒ the commutes relation (Definition 13.2 page 193). The following statements are equivalent: 1. 𝙇 is orthomodular ⟺ 2. 𝑥 ≤ 𝑦 and 𝑦 ∧ 𝑥⟂ = 0 ⟹ 𝑥 = 𝑦 ⟺ 3. 𝙇 does not contain the 𝑂6 lattice t ⟺ 4. 𝑥Ⓒ𝑦 ⟺ 𝑦Ⓒ𝑥 (Ⓒ is symmetric) h m ⟺ 5. 𝑥Ⓜ𝑥⟂ ∀𝑥 ∈ 𝑋 ∗ ⟂ ⟺ 6. 𝑥Ⓜ 𝑥 ∀𝑥 ∈ 𝑋 ⟂ ⟺ 7. 𝑥 ∨ [𝑥 ∧ (𝑥 ∨ 𝑦)] = 𝑥 ∨ 𝑦 ∀𝑥, 𝑦 ∈ 𝑋 ⟺ 8. 𝑥 ≤ 𝑦 ⟹ ∃𝑝 ∈ 𝑋 such that 𝑥 ⟂ 𝑝 and 𝑥 ∨ 𝑝 = 𝑦 ✎PROOF: 1. Proof that orthomodular ⟺ symmetric: by Proposition 13.3 (page 194).

✏

9.2.3 Restrictions resulting in Boolean algebras Theorem 9.5. 12 Let 𝙇 = ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an algebraic structure. 𝙇 is an orthomodular lattice and 𝙇 is a ⎫ ⎧ t ⎧ ⎪ 𝑥 ∧ 𝑦⟂ ⟂ = 𝑦 ∨ 𝑥⟂ ∧ 𝑦⟂ ⎪ ⎪ ∀𝑥, 𝑦 ∈ 𝑋 h ⎨ (⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⟹ ) ( ) ⎬ ⎨ Boolean algebra m ⎪ ⎪ ⎪ ⎩ (Definition 8.1 page 115) ⎩ Elkan's law ⎭

⎫ ⎪ ⎬ ⎪ ⎭

Definition 9.4. 13 set ({0, 𝑥, 𝑦, 𝑥⟂ , 𝑦⟂ , 1}, ≤) with cover relation d The MO2 lattice is the ordered e ≺= {(0, 𝑥) , (0, 𝑦) , (0, 𝑥⟂ ) , (0, 𝑦⟂ ) , (𝑥, 1) , (𝑦, 1) , (𝑥⟂ , 1) , (𝑦⟂ , 1)} f This lattice is also called the Chinese lantern.

1 𝑥

𝑦

𝑦⟂

𝑥⟂

0

MO2 Theorem 9.6.

L2 14

MO2 ×L2

Let 𝙈 = ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an orthomodular lattice.

11

📘 Kalmbach (1983) page 22, 📘 Stern (1999) page 12, 📘 Nakamura (1957), 📘 Holland (1963), 📃 Foulis (1962), 📘 Maeda and Maeda (1970), page 132 ⟨Theorem 29.13⟩ 12 📃 Renedo et al. (2003) page 72 13 📘 Iturrioz (1985) page 57, 📘 Davey and Priestley (2002) pages 18–19 ⟨1.25 Products⟩ 14 📘 Iturrioz (1985) page 57, 📃 Carrega (1982) ⟨cf Iturrioz 1985 page 57⟩ 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 150

9.3. MODULAR ORTHOCOMPLEMENTED LATTICES

t 𝙈 is h ⟺ { m { Boolean }

9.3

1. 2.

𝙈 does not contain the MO2 lattice (Definition 9.4 page 149) 𝙈 does not contain the MO2 ×L2 lattice.

and

}

Modular orthocomplemented lattices

Definition 9.5. Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice (Definition 4.4 page 71). d 𝙇 is a modular orthocomplemeted lattice if e 1. 𝙇 is orthocomplemented (Definition 9.1 page 140) and f 2. 𝙇 is modular (Definition 5.3 page 80)

9.4

Relationships between orthocomplemented lattices

Theorem 9.7.

15

Let 𝙇 be a lattice. ⎧𝙇 is t ⎪modular h {𝙇 is ⟹ ⎨ } orthocomBoolean m ⎪ ⎩plemented

⎫ 𝙇 is ⎪ 𝙇 is ortho⎬ ⟹ {modular } ⟹ {orthocom- } ⎪ plemented ⎭

Remark 9.1. 16 Lattice number 8 in Example 9.2 (page 140) was originally introduced by Dilworth as a counterexample to Husimi's conjecture (1937). Kalmbach(1983) points out that this lattice was the first example of a finite orthomodular lattice.

15 16

📘 Kalmbach (1983) page 32 ⟨20.⟩, 📘 Iturrioz (1985) page 57 📃 Dilworth (1940), 📃 Dilworth (1990), 📘 Kalmbach (1983) page 9

version 0.30


Daniel J. Greenhoe


Part III Functions on Lattices

CHAPTER 10 VALUATIONS ON LATTICES

Definition 10.1. 1 Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice (Definition 4.3 page 61). d A function v ∈ ℝ𝙓 is a valuation on 𝙇 if e v(𝑥 ∨ 𝑦) + v(𝑥 ∧ 𝑦) = v(𝑥) + v(𝑦) ∀𝑥,𝑦∈𝑋 f Proposition 10.1. Let v ∈ ℝ𝙓 be a function on a lattice 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) (Definition 4.3 page 61). p r { 𝙇 is linear (Definition 4.3 page 61) } ⟹ { v is a valuation (Definition 10.1 page 153) } p ✎PROOF:

Let 𝑥, 𝑦 ∈ 𝑋 such that 𝑥 ≤ 𝑦 or 𝑦 ⪇ 𝑥. v(𝑥 ∨ 𝑦) + v(𝑥 ∧ 𝑦) = v(𝑥) + v(𝑦)

because 𝙇 is linear

✏ 2

Example 10.1. Consider the real valued lattice 𝙇 ≜ ( ℝ, ∨, ∧ ; ≤). The absolute value function |⋅| is a valuation on 𝙇. ✎PROOF:

𝙇 is linear

(Definition 4.3 page 61),

so v is a valuation by Proposition 10.1 (page 153).

✏

Definition 10.2. 3 Let 𝑋 be a set and ℝ⊢ the set of non-negative real numbers. 𝑋×𝑋 A function 𝖽 ∈ ℝ⊢ is a metric on 𝑋 if 1. 𝖽(𝑥, 𝑦) ≥ 0 ∀𝑥,𝑦∈𝑋 (non-negative) and d 2. 𝖽(𝑥, 𝑦) = 0 ⟺ 𝑥 = 𝑦 ∀𝑥,𝑦∈𝑋 (nondegenerate) and e 3. 𝖽(𝑥, 𝑦) = 𝖽(𝑦, 𝑥) ∀𝑥,𝑦∈𝑋 (symmetric) and f 4 4. 𝖽(𝑥, 𝑦) ≤ 𝖽(𝑥, 𝑧) + 𝖽(𝑧, 𝑦) ∀𝑥,𝑦,𝑧∈𝑋 (subadditive/triangle inequality). A metric space is the pair (𝑋, 𝖽). A metric is also called a distance function. 1

📘 Istrǎţescu (1987) page 127, 📘 Birkhoff (1967) page 230 ⟨Definition X.1(V1)⟩, 📘 Blyth (2005) page 58 ⟨Exercise 4.25⟩, 📘 Deza and Laurent (1997) page 105 ⟨(8.1.1)⟩, 📘 Deza and Deza (2006) page 143 ⟨§10.3⟩, 📘 Deza and Deza (2009) page 193 ⟨§10.3⟩ 2 📘 Khamsi and Kirk (2001) page 119 ⟨§5.7⟩ 3 📘 Dieudonné (1969), page 28, 📘 Copson (1968), page 21, 📘 Hausdorff (1937) page 109, 📘 Fréchet (1928), 📘 Fréchet (1906) page 30 4 📘 Euclid (circa 300BC) ⟨Book I Proposition 20⟩

page 154 Actually, it is possible to significantly simplify the definition of a metric to an equivalent statement requiring only half as many conditions. These equivalent conditions (a “characterization”) are stated in Theorem 10.1 (next). 𝑋×𝑋

Theorem 10.1 (metric characterization). 5 Let 𝖽 be a function in (ℝ⊢ ) t 1. 𝖽(𝑥, 𝑦) = 0 ⟺ 𝑥 = 𝑦 h 𝖽(𝑥, 𝑦) is a metric ⟺ { 2. 𝖽(𝑥, 𝑦) ≤ 𝖽(𝑧, 𝑥) + 𝖽(𝑧, 𝑦) m

.

∀𝑥,𝑦∈𝑋

and


Definition 10.3 (next) defines the open ball. In a metric space (Definition 10.2 page 153), sets are often specified in terms of an open ball; and an open ball is specified in terms of a metric. Definition 10.3. 6 Let (𝑋, 𝖽) be a metric space (Definition 10.2 page 153). An open ball centered at 𝑥 with radius 𝑟 is the set 𝖡(𝑥, 𝑟) ≜ {𝑦 ∈ 𝑋|𝖽(𝑥, 𝑦) < 𝑟}. d A closed ball centered at 𝑥 with radius 𝑟 is the set 𝖡 (𝑥, 𝑟) ≜ {𝑦 ∈ 𝑋|𝖽(𝑥, 𝑦) ≤ 𝑟}. e is the set 𝖡(𝑥, 1). f A unit ball centered at 𝑥 A closed unit ball centered at 𝑥 is the set 𝖡 (𝑥, 1). Let v ∈ ℝ𝙓 be a function on a lattice 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) (Definition 4.3 page 61). 𝖽(𝑥, 𝑦) ≜ ⎧ v(𝑥 ∨ 𝑦) + v(𝑥 ∧ 𝑦) = v(𝑥) + v(𝑦) ∀𝑥,𝑦∈𝑋 (valuation) and ⎪ ⟹ ⎨ v(𝑥 ∨ 𝑦) − v(𝑥 ∧ 𝑦) } 𝑥 ≤ 𝑦 ⟹ v(𝑥) ≤ v(𝑦) ∀𝑥,𝑦∈𝑋 (isotone) ⎪ ⎩ is a metric on 𝙇

Theorem 10.2. t h m

1. 2.

7

Definition 10.4. 8 Let v be a valuation (Definition 10.1 page 153) on a lattice 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) (Definition 4.3 page 61). Let 𝖽(𝑥, 𝑦) be the metric defined in Theorem 10.2 (page 154). d e The pair (𝙇, 𝖽) is called a metric lattice. f For finite modular lattices, the height function 𝗁(𝑥) (Definition 4.5 page 72) can serve as the isotone valuation that induces a metric (next proposition). Such a height function actually satisfies the stronger condition of being positive (rather than just being isotone)—all positive functions are also isotone. Proposition 10.2. 9 Let 𝗁(𝑥) be the height (Definition 4.5 page 72) of a point 𝑥 in a bounded lattice (Definition 4.4 page 71) 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤). 1. 𝙇 is modular and { 2. 𝙇 is finite } p 1. 𝗁(𝑥 ∨ 𝑦) + 𝗁(𝑥 ∧ 𝑦) = 𝗁(𝑥) + 𝗁(𝑦) ∀𝑥,𝑦∈𝑋 (valuation) and r ⟹ { 2. 𝑥 ⪇ 𝑦 ⟹ 𝗁(𝑥) ⪇ 𝗁(𝑦) } ∀𝑥,𝑦∈𝑋 (positive) p 1. 𝗁(𝑥 ∨ 𝑦) + 𝗁(𝑥 ∧ 𝑦) = 𝗁(𝑥) + 𝗁(𝑦) ∀𝑥,𝑦∈𝑋 (valuation) and ⟹ { 2. 𝑥 ≤ 𝑦 ⟹ 𝗁(𝑥) ≤ 𝗁(𝑦) } ∀𝑥,𝑦∈𝑋 (isotone) Theorem 10.3. 10 Let v be a valuation (Definition 10.1 page 153) on a lattice 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) (Definition 4.3 page 61). Let 𝖽(𝑥, 𝑦) be the metric defined in Theorem 10.2 (page 154). t (𝙇, 𝖽) is a metric lattice 𝙇 is modular h ⟹ { (Definition 5.3 page 80) } m { (Definition 10.4 page 154) } 5

📘 Michel and Herget (1993), page 264, 📘 Giles (1987), page 18 📘 Aliprantis and Burkinshaw (1998), page 35 7 📘 Deza and Laurent (1997) page 105 ⟨(8.1.2)⟩, 📘 Birkhoff (1967) pages 230–231 8 📘 Deza and Laurent (1997) page 105, 📘 Birkhoff (1967) page 231 ⟨§X.2⟩ 9 📘 Birkhoff (1967) page 230 10 📘 Birkhoff (1967) page 232Theorem X.2, 📘 Deza and Laurent (1997) pages 105–106, 📘 Blyth (2005) page 58 ⟨Exercise 4.25⟩ 6

version 0.30


Daniel J. Greenhoe


page 155

CHAPTER 10. VALUATIONS ON LATTICES

Example 10.2. The function 𝗁 on the Boolean (and thus also modular) lattice 𝙇32 illustrated to the right is a valuation (Definition 10.1 page 153) that is positive (and thus also isotone, Example 10.2 page 154). Therefore 𝖽(𝑥, 𝑦) ≜ 𝗁(𝑥 ∨ 𝑦) − 𝗁(𝑥 ∧ 𝑦) ∀𝑥,𝑦∈𝑋 is a metric (Definition 10.4 page 154) on 𝙇32 . For example, 𝖽(𝑏, 𝑞) ≜ 𝗁(𝑏 ∨ 𝑞) − 𝗁(𝑏 ∧ 𝑞) = 𝗁(1) − 𝗁(0) = 3 − 0 = 3 . The closed unit ball centered at 𝑏 (Definition 10.3 page 154) and illustrated with solid dots to the right is 𝖡(𝑏, 1) ≜ {𝑥 ∈ 𝑋|𝖽(𝑏, 𝑥) ≤ 1} = {𝑏, 𝑝, 𝑟, 0} Example 10.3. The height function 𝗁 (Definition 4.5 page 72) on the orthocomplemented but non-modular lattice O6 illustrated to the right is not a valuation because for example 𝗁(𝑎 ∨ 𝑐) + 𝗁(𝑎 ∧ 𝑐) = 𝗁(1) + 𝗁(0) = 3 + 0 = 3 ≠ 2 = 1 + 1 = 𝗁(𝑎) + 𝗁(𝑏). Moreover, we might expect the “distance” from 𝑎 to 𝑐 to be 2. However, if we attempt to use 𝗁(𝑥) to define a metric on O6 , then we get 𝖽(𝑎, 𝑐) ≜ 𝗁(𝑎 ∨ 𝑐) − 𝗁(𝑎 ∧ 𝑐) = 𝗁(1) − 𝗁(0) = 3 − 0 = 3 ≠ 2.


Daniel J. Greenhoe

𝗁(𝑝) = 2 𝗁(𝑎) = 1

𝗁(1) = 3 𝗁(𝑝) = 2 𝗁(𝑟) = 2 𝗁(𝑐) = 1 𝗁(𝑏) = 1 𝗁(0) = 0

𝗁(1) = 3 𝗁(𝑝) = 2 𝗁(𝑎) = 1


𝗁(𝑟) = 2 𝗁(𝑐) = 1 𝗁(0) = 0

version 0.30

page 156

version 0.30


Daniel J. Greenhoe


CHAPTER 11 NEGATION

When we say not-being, we speak , I think , not of something that is the opposite of being, but only of something different. …Then when we are told that the negative signifies the opposite, we shall not admit it; we shall admit only that the particle “not” indicates something different from the words to which it is prefixed, or rather from the things denoted by the words that follow the negative. ❞ ❝

Plato's the Sophist (circa 360 B.C.) 1

❝Clearly, then, it is a principle of this kind that is the most certain of all principles.…Let

us next state what this principle is. “It is impossible for the same attribute at once to belong and not to belong to the same thing and in the same relation”; …This is the most certain of all principles,…for it is impossible for anyone to suppose that the same thing is and is not,…it is by nature the starting-point of all the other axioms as well.❞ Aristotle (384BC–322BC), Greek philosopher 2

11.1 Definitions Definition 11.1. 3 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice (Definition 4.4 page 71). d A function ¬ ∈ 𝑋 𝑋 is a subminimal negation on 𝙇 if e 𝑥 ≤ 𝑦 ⟹ ¬𝑦 ≤ ¬𝑥 ∀𝑥,𝑦∈𝑋 (antitone) 4 f

1

📘 Plato (circa 360 B.C.) ⟨257b–257c⟩, 📘 Horn (2001), page 5 📘 Aristotle page 4.1005b 3 📘 Dunn (1996) pages 4–6, 📘 Dunn (1999) pages 24–26 ⟨2 THE KITE OF NEGATIONS⟩ 4 The antitone property may also be referred to as antitonic, order-reversing , or contrapositive. 2

page 158

11.1. DEFINITIONS

subminimal negation (Definition 11.1 page 157) (Example 11.7 page 165)

minimal negation (Definition 11.2 page 158) (Example 11.8 page 166)

de Morgan negation

fuzzy negation

(Definition 11.3 page 158) (Example 11.14 page 169)


Kleene negation

intuitionalistic negation



ortho negation (Definition 11.3 page 158) (Example 11.13 page 168)

orthomodular negation (Definition 11.3 page 158)

Figure 11.1: lattice of negations

Remark 11.1.

5

In the context of natural language, D. Devidi argues that, subminimal negation (Defis “difficult to take seriously as” a negation. He essentially gives this example: Let 𝑥 ≜“𝑝 is a fish” and 𝑦 ≜“𝑝 has gills”. Suppose “𝑝 is a fish” implies “𝑝 has gills” (𝑥 ≤ 𝑦). Now let 𝑝 ≜ “many dogs”. Then the antitone property and 𝑥 ≤ 𝑦 tells us ( ⟹ ) that “Not many dogs have gills” implies that “Not many dogs are fish”. inition 11.1 page 157)

Definition 11.2. 6 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice (Definition 4.4 page 71). A function ¬ ∈ 𝑋 𝑋 is a negation, or minimal negation, on 𝙇 if 1. 𝑥 ≤ 𝑦 ⟹ ¬𝑦 ≤ ¬𝑥 ∀𝑥,𝑦∈𝑋 (antitone) and 2. 𝑥 ≤ ¬¬𝑥 ∀𝑥∈𝑋 (weak double negation). d e A minimal negation ¬ is an intuitionistic negation if f 3. 𝑥 ∧ ¬𝑥 = 0 ∀𝑥,𝑦∈𝑋 (non-contradiction). A minimal negation ¬ is a fuzzy negation if 4. ¬1 = 0 (boundary condition).

Definition 11.3.

7

Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice (Definition 4.4 page 71).

5

📘 Devidi (2010) page 511, 📘 Devidi (2006) page 568 📘 Dunn (1996) pages 4–6, 📘 Dunn (1999) pages 24–26 ⟨2 THE KITE OF NEGATIONS⟩, 📘 TROELSTRA AND VAN DALEN (1988) PAGE 4 ⟨1.6 INTUITIONISM. (B)⟩, 📃 DE VRIES (2007) PAGE 11 ⟨DEFINITION 16⟩, 📘 GOTTWALD (1999) PAGE 21 ⟨DEFINITION 3.3⟩, 📘 NOVáK ET AL. (1999) PAGE 50 ⟨DEFINITION 2.26⟩, 📘 NGUYEN AND WALKER (2006) PAGES 98–99 ⟨5.4 NEGATIONS⟩, 📃 HöHLE (1978) ⟨???⟩, 📃 BELLMAN AND GIERTZ (1973) PAGES 155–156 ⟨(N1) ¬0 = 1 AND ¬1 = 0, (N3) ¬¬𝑥 = 𝑥⟩ 7 📘 Dunn (1999) pages 24–26 ⟨2 THE KITE OF NEGATIONS⟩, 📘 JENEI (2003) PAGE 283, 📘 KALMBACH (1983) PAGE 22, 📘 LIDL AND PILZ (1998) PAGE 90, 📃 HUSIMI (1937) 6

version 0.30


Daniel J. Greenhoe


page 159

CHAPTER 11. NEGATION

A minimal negation ¬ is a de Morgan negation if 5. 𝑥 = ¬¬𝑥 ∀𝑥∈𝑋 (involutory). A de Morgan negation ¬ is a Kleene negation if 6. 𝑥 ∧ ¬𝑥 ≤ 𝑦 ∨ ¬𝑦 ∀𝑥,𝑦∈𝑋 (Kleene condition). A de Morgan negation ¬ is an ortho negation if 7. 𝑥 ∧ ¬𝑥 = 0 ∀𝑥,𝑦∈𝑋 (non-contradiction). A de Morgan negation ¬ is an orthomodular negation if 8. 𝑥 ∧ ¬𝑥 = 0 ∀𝑥,𝑦∈𝑋 (non-contradiction) 9. 𝑥 ≤ 𝑦 ⟹ 𝑥 ∨ (𝑦 ∧ ¬𝑥) = 𝑦 ∀𝑥,𝑦∈𝑋 (orthomodular).

d e f

and

Remark 11.2. 8 The Kleene condition is basically a weakened form of the non-contradiction and excluded middle properties because 𝑥 ∧ ¬𝑥 = 0 ≤ 1⏟⏟⏟⏟⏟⏟⏟ = 𝑦 ∨ ¬𝑦 . ⏟⏟⏟⏟⏟⏟⏟ non-contradiction

excluded middle

Definition 11.4. 9 A minimal negation ¬ ∈ 𝑋 𝑋 is strict (¬ is a strict negation) if 1. 𝑥 ⪇ 𝑦 ⟹ ¬𝑦 ⪇ ¬𝑥 ∀𝑥,𝑦∈𝑋 (strictly antitone) and d e 2. ¬ is continuous f A strict negation ¬ is strong (¬ is a strong negation) if 3. ¬¬𝑥 = 𝑥 ∀𝑥∈𝑋 (involutory). Definition 11.5. Let 𝙇 ≜ ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤) be a bounded lattice (Definition 4.4 page 71) with a function ¬ in 𝑋 𝑋 . d e If ¬ is a minimal negation, then 𝙇 is a lattice with negation. f

11.2 Properties of negations Lemma 11.1. 10 Let ¬ ∈ 𝑋 𝑋 be a function on a bounded lattice 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) (Definition 4.4 page 71). l ¬𝑥 ∨ ¬𝑦 ≤ ¬(𝑥 ∧ 𝑦) ∀𝑥,𝑦∈𝑋 (conjunctive de Morgan ineq.) and e ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑥 ≤ 𝑦 ⟹ ¬𝑦 ≤ ¬𝑥} ⟹ { ¬(𝑥 ∨ 𝑦) ≤ ¬𝑥 ∧ ¬𝑦 ∀𝑥,𝑦∈𝑋 (disjunctive de Morgan ineq.) and m antitone

✎PROOF: 1. Proof that antitone ⟹ conjunctive de Morgan: 𝑥 ∧ 𝑦 ≤ 𝑥 and 𝑥 ∧ 𝑦 ≤ 𝑦


⟹ ¬(𝑥 ∧ 𝑦) ≥ ¬𝑥 and ¬(𝑥 ∧ 𝑦) ≥ ¬𝑦

by antitone

⟹ ¬(𝑥 ∧ 𝑦) ≥ ¬𝑥 ∨ ¬𝑦


8

📘 Cattaneo and Ciucci (2009) page 78 📓 Fodor and Yager (2000), pages 127–128, 📃 Bellman and Giertz (1973) 10 📘 Beran (1985) page 31 ⟨Theorem 1.2 Proof⟩, 📃 Fáy (1967) page 268 ⟨Lemma 1 Proof⟩, 📃 de Vries (2007) page 12 ⟨Theorem 18⟩ 9


Daniel J. Greenhoe


version 0.30

page 160

11.2. PROPERTIES OF NEGATIONS

2. Proof that antitone ⟹ disjunctive de Morgan: 𝑥 ≤ 𝑥 ∨ 𝑦 and 𝑦 ≤ 𝑥 ∨ 𝑦


⟹ ¬𝑥 ≥ ¬(𝑥 ∨ 𝑦) and ¬𝑦 ≥ ¬(𝑥 ∨ 𝑦)

by antitone

⟹ ¬𝑥 ∧ ¬𝑦 ≥ ¬(𝑥 ∨ 𝑦)


⟹ ¬(𝑥 ∨ 𝑦) ≤ ¬𝑥 ∧ ¬𝑦

✏ Lemma 11.2. 11 Let ¬ ∈ 𝑋 𝑋 be a function on a lattice 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) (Definition 4.3 page 61). If 𝑥 = (¬¬𝑥) for all 𝑥 ∈ 𝑋 (involutory), then ¬(𝑥 ∨ 𝑦) = ¬𝑥 ∧ ¬𝑦 ∀𝑥,𝑦∈𝑋 (disjunctive de Morgan) and l 𝑥 ≤ 𝑦 ⟹ ¬𝑦 ≤ ¬𝑥} ⟺ e ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ { ¬(𝑥 ∧ 𝑦) = ¬𝑥 ∨ ¬𝑦 ∀𝑥,𝑦∈𝑋 (conjunctive de Morgan) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ m antitone

de Morgan

✎PROOF: 1. Proof that antitone ⟹ de Morgan equalities: (a) Proof that ¬(¬𝑥 ∧ ¬𝑦) ≥ 𝑥 ∨ 𝑦: ¬(¬𝑥 ∧ ¬𝑦) ≥ ¬¬𝑥 ∨ ¬¬𝑦 =𝑥∨𝑦

by Lemma 11.1 by involutory property (Definition 11.5 page 159)

(b) Proof that ¬(¬𝑥 ∨ ¬𝑦) ≤ 𝑥 ∧ 𝑦: ¬(¬𝑥 ∨ ¬𝑦) ≤ ¬¬𝑥 ∧ ¬¬𝑦 =𝑥∧𝑦

by Lemma 11.1 by involutory property (Definition 11.5 page 159)

(c) Proof that ¬(𝑥 ∧ 𝑦) = ¬𝑥 ∨ ¬𝑦: ¬(𝑥 ∧ 𝑦) ≥ ¬𝑥 ∨ ¬𝑦

by Lemma 11.1

¬(𝑥 ∧ 𝑦) = ¬[¬¬𝑥 ∧ ¬¬𝑦]


≤ ¬𝑥 ∨ ¬𝑦

by item (1b)

(d) Proof that ¬(𝑥 ∨ 𝑦) = ¬𝑥 ∧ ¬𝑦: ¬(𝑥 ∨ 𝑦) ≥ ¬𝑥 ∧ ¬𝑦

by Lemma 11.1

¬(𝑥 ∨ 𝑦) = ¬[¬¬𝑥 ∨ ¬¬𝑦]


≤ ¬𝑥 ∧ ¬𝑦

by item (1a)

2. Proof that antitone ⟸ de Morgan: 𝑥 ≤ 𝑦 ⟹ ¬𝑦 = ¬(𝑥 ∨ 𝑦)


= ¬𝑥 ∧ ¬𝑦

by de Morgan

≤ ¬𝑥


✏ 11

📘 Beran (1985) pages 30–31 ⟨Theorem 1.2⟩, 📃 Fáy (1967) page 268 ⟨Lemma 1⟩, 📃 Nakano and Romberger (1971) ⟨cf Beran 1985⟩ version 0.30


Daniel J. Greenhoe


page 161


Lemma 11.3. Let ¬ ∈ 𝑋 𝑋 be a function on a lattice 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) (Definition 4.3 page 61). l 1. 𝑥 ≤ ¬¬𝑥 ∀𝑥∈𝑋 (weak double negation) and e ⟹ { ¬0 = 1 (boundary condition) } } (boundary condition) m { 2. ¬1 = 0 ✎PROOF: ¬0 = ¬¬1

by boundary condition hypothesis (2)

≥1

by weak double negation hypothesis (1)

⟹ ¬0 = 1

by upper bound property (Definition 4.4 page 71)

✏ Lemma 11.4. Let ¬ ∈ 𝑋 𝑋 be a function on a lattice 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) (Definition 4.3 page 61). l e { (𝑥 ∧ ¬𝑥 = 0 ∀𝑥∈𝑋 (non-contradiction) } ⟹ { ¬1 = 0 (boundary condition) } m ✎PROOF: 0 = 1 ∧ ¬1

by non-contradiction hypothesis

= ¬1

by definition of g.u.b. 1 and ∧

✏ Lemma 11.5. 12 Let ¬ ∈ 𝑋 𝑋 be a function on a bounded lattice 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) (Definition 4.4 page 71). l (A). ¬ is bijective and (1). ¬0 = 1 and e ⟹ { } { } (B). 𝑥 ≤ 𝑦 ⟹ ¬𝑦 ≤ ¬𝑥 ∀𝑥,𝑦∈𝑋 (antitone) (2). ¬1 = 0 m ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ boundary conditions

✎PROOF: 1. Proof that ¬0 = 1: 𝑥≤1

∀𝑥 ∈ 𝑋

by definition of l.u.b. 1

⟹ ¬1 ≤ ¬𝑥

∀𝑥 ∈ 𝑋

by antitone hypothesis

⟹ ¬1 ≤ 𝑦

∀𝑦 ∈ 𝑋

by bijective hypothesis

⟹ ¬1 = 0

by definition of g.l.b. 0

2. Proof that ¬0 = 1: 0≤𝑥

∀𝑥 ∈ 𝑋

by definition of g.l.b. 0

⟹ ¬𝑥 ≤ ¬0

∀𝑥 ∈ 𝑋

by antitone hypothesis

⟹ ¬𝑥 ≤ 𝑦

∀𝑦 ∈ 𝑋

by bijective hypothesis

⟹ ¬0 = 1

by definition of l.u.b. 1

✏ 12

📘 Varadarajan (1985) page 42


Daniel J. Greenhoe


version 0.30

page 162

11.2. PROPERTIES OF NEGATIONS

Theorem 11.1. Let ¬ ∈ 𝑋 𝑋 be a function on a bounded lattice 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) (Definition 4.4 page 71). t ¬ is an h (boundary condition) ⟹ { ¬1 = 0 } m { intuitionistic negation } ✎PROOF:

✏

This follows directly from Definition 11.5 (page 159) and Lemma 11.4 (page 161).

Theorem 11.2. Let ¬ ∈ 𝑋 𝑋 be a function on a bounded lattice 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) (Definition 4.4 page 71). t ¬ is a h ⟹ { ¬0 = 1 (boundary condition) } { fuzzy negation } m ✎PROOF:

✏


Theorem 11.3. Let ¬ ∈ 𝑋 𝑋 be a function on a bounded lattice 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) (Definition 4.4 page 71). ¬ is a ⎫ t ⎧ ¬𝑥 ∨ ¬𝑦 ≤ ¬(𝑥 ∧ 𝑦) ∀𝑥,𝑦∈𝑋 (conjunctive de Morgan inequality) and ⎪ ⎪ h ⎨ minimal ⎬ ⟹ { ¬(𝑥 ∨ 𝑦) ≤ ¬𝑥 ∧ ¬𝑦 ∀𝑥,𝑦∈𝑋 (disjunctive de Morgan inequality) } m ⎪ negation ⎪ ⎩ ⎭ ✎PROOF:

✏


Theorem 11.4. Let ¬ ∈ 𝑋 𝑋 be a function on a bounded lattice 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) (Definition 4.4 page 71). t ¬ is a ¬(𝑥 ∨ 𝑦) = ¬𝑥 ∧ ¬𝑦 ∀𝑥,𝑦∈𝑋 (disjunctive de Morgan) and h ⟹ { ¬(𝑥 ∧ 𝑦) = ¬𝑥 ∨ ¬𝑦 ∀𝑥,𝑦∈𝑋 (conjunctive de Morgan) m de Morgan negation } ✎PROOF:

✏


Theorem 11.5. page 71).

13

Let ¬ ∈ 𝑋 𝑋 be a function on a bounded lattice 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) (Definition 4.4

⎧ ⎪ ⎪ t ¬ is an ⎪ h ⟹ ⎨ m { ortho negation } ⎪ ⎪ ⎪ ⎩

1. 2. 3. 4. 5. 6.

¬0 ¬1 ¬(𝑥 ∨ 𝑦) ¬(𝑥 ∧ 𝑦) 𝑥 ∨ ¬𝑥 𝑥 ∧ ¬𝑥

= = = = = ≤

1 0 ¬𝑥 ∧ ¬𝑦 ¬𝑥 ∨ ¬𝑦 1 𝑦 ∨ ¬𝑦

(boundary condition)

and

(boundary condition)

and

∀𝑥,𝑦∈𝑋

(disjunctive de Morgan)

and

∀𝑥,𝑦∈𝑋

(conjunctive de Morgan)

and

∀𝑥∈𝑋

(excluded middle)

and

∀𝑥,𝑦∈𝑋

(Kleene condition).

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

✎PROOF: 1. Proof for 0 = ¬1 boundary condition: by Lemma 11.4 (page 161) 2. Proof for boundary conditions: 1 = ¬¬1 = ¬0

by involutory property by previous result

3. Proof for de Morgan properties: 13

📘 Beran (1985) pages 30–31, 📃 Birkhoff and Neumann (1936) page 830 ⟨L74⟩, 📘 Cohen (1989) page 37 ⟨3B.13. Theorem⟩ version 0.30


Daniel J. Greenhoe


page 163


(a) By Definition 11.5 (page 159), ortho negation is involutory and antitone. (b) Therefore by Lemma 11.2 (page 160), de Morgan properties hold. 4. Proof for excluded middle property: 𝑥 ∨ ¬𝑥 = (𝑥 ∨ ¬𝑥)¬¬ ¬¬

by involutory property of ortho negation (Definition 11.5 page 159)

= ¬(𝑥¬ ∧ 𝑥 )

by disjunctive de Morgan property

= ¬(¬𝑥 ∧ 𝑥)

by involutory property of ortho negation (Definition 11.5 page 159)

= ¬(𝑥 ∧ ¬𝑥)

by commutative property of lattices (Definition 4.3 page 61)

= ¬0

by non-contradiction property of ortho negation (Definition 11.5 page 159)

=1

by boundary condition (item (2) page 162) of minimal negation

5. Proof for Kleene condition: 𝑥 ∧ ¬𝑥 = 0

by non-contradiction property (Definition 11.5 page 159)

≤1

by definition of 0 and 1

= 𝑦 ∨ ¬𝑦

by excluded middle property (item (4) page 163)

✏

11.3 Examples Example 11.1 (discrete negation). 14 Let 𝙇 ≜ ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤) be a bounded lattice (Definition 4.4 𝑋 page 71) with a function ¬ ∈ 𝑋 . The function ¬𝑥 defined as 1 for 𝑥 = 0 e ¬𝑥 ≜ x { 0 otherwise is an intuitionistic negation (Definition 11.2 page 158) and a fuzzy negation (Definition 11.2 page 158). ✎PROOF:

To be an intuitionistic negation, ¬𝑥 must be antitone, have weak double negation, and have the non-contradiction property (Definition 11.2 page 158). To be a fuzzy negation, ¬𝑥 must be antitone, have weak double negation, and have the boundary condition ¬1 = 0. ¬𝑦 ≤ ¬𝑥 ⟺ ⎧ ⎪ ⎨ ¬𝑦 ≤ ¬𝑥 ⟺ ⎪ ⎩ ¬𝑦 ≤ ¬𝑥 ⟺ ¬¬𝑥 = ¬1 = 0 ≥ { ¬¬𝑥 = ¬0 = 1 ≥

1 ≤ 1 for 0 = 𝑥 = 𝑦 0 ≤ 1 for 0 = 𝑥 ⪇ 𝑦 0 ≤ 0 for 0 ≠ 𝑥 ≤ 𝑦

⎫ ⎪ ⎬ ⟹ ¬𝑥 is antitone ⎪ ⎭

0 = 𝑥 for 𝑥 = 0 ⟹ ¬𝑥 has weak double negation 𝑥 = 𝑥 for 𝑥 ≠ 0 }

𝑥 ∧ ¬𝑥 = 𝑥 ∧ 1 = 0 ∧ 0 = 0 for 𝑥 = 0 ⟹ ¬𝑥 has non-contradiction property { 𝑥 ∧ ¬𝑥 = 𝑥 ∧ 0 = 𝑥 ∧ 0 = 0 for 𝑥 ≠ 0 } ¬1 = 0 ⟹ ¬𝑥 has the boundary condition property

✏ 14

📘 Fodor and Yager (2000) page 128, 📃 Yager (1980) pages 256–257, 📃 Yager (1979) ⟨cf Fodor⟩


Daniel J. Greenhoe


version 0.30

page 164

11.3. EXAMPLES

Example 11.2 (dual discrete negation). 15 Let 𝙇 ≜ ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤) be a bounded lattice (Definition 𝑋 4.4 page 71) with a function ¬ ∈ 𝑋 . The function ¬𝑥 defined as 0 for 𝑥 = 1 ¬𝑥 ≜ e { 1 otherwise x is a subminimal negation (Definition 11.1 page 157) but it is not a minimal negation (Definition 11.2 page 158) (and not any other negation defined here). ✎PROOF: To be an subminimal negation, ¬𝑥 must be antitone (Definition 11.1 page 157). To be a minimal negation, ¬𝑥 must be antitone and have weak double negation (Definition 11.2 page 158). ¬𝑦 ≤ ¬𝑥 ⟺ ⎧ ⎪ ⎨ ¬𝑦 ≤ ¬𝑥 ⟺ ⎪ ⎩ ¬𝑦 ≤ ¬𝑥 ⟺ ¬¬𝑥 = ¬0 = { ¬¬𝑥 = ¬1 =

0 ≤ 0 for 𝑥 = 𝑦 = 1 0 ≤ 1 for 𝑥 ⪇ 𝑦 = 1 1 ≤ 1 for 𝑥 ≤ 𝑦 ≠ 1

⎫ ⎪ ⎬ ⟹ ¬𝑥 is antitone ⎪ ⎭

1 ≥ 𝑥 for 𝑥 = 1 ⟹ ¬𝑥 does not have weak double negation 0 ≤ 𝑥 for 𝑥 ≠ 1 }

✏ Example 11.3. 16 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice The function ¬𝑥 is an intuitionistic negation (Definition 11.2 page 158) if e 1 for 𝑥 = 0 x ¬𝑥 ≜ { 0 otherwise Example 11.4. 1 = ¬0

e x

The function ¬ illustrated to the right is an ortho negation (Definition 11.3 page 158).

0 = ¬1

✎PROOF: 1. Proof that ¬ is antitone: 0 ≤ 1

⟹ ¬1 = 0 ≤ 𝑥 = ¬0

⟹ ¬ is antitone over (0, 1)

2. Proof that ¬ is involutory: 1 = ¬0 = ¬¬1 3. Proof that ¬ has the non-contradiction property: 1 ∧ ¬1 = 1 ∧ 0 = 0 0 ∧ ¬0 = 0 ∧ 1 = 0

✏ Example 11.5. The functions ¬ illustrated to the right are not any negation defined here. In particular, they are not antitone.

e x

1 = ¬1

1 = ¬0

1 = ¬𝑎

𝑎 = ¬𝑎

𝑎 = ¬1

𝑎 = ¬0

0 = ¬0

0 = ¬𝑎

0 = ¬1

(a)

(b)

(c)

✎PROOF: 1. Proof that (a) is not antitone: 𝑎 ≤ 1 ⟹ ¬1 = 1 ≰ 𝑎 = ¬𝑎 2. Proof that (b) is not antitone: 𝑎 ≤ 1 ⟹ ¬1 = 𝑎 ≰ 0 = ¬𝑎 3. Proof that (c) is not antitone: 0 ≤ 𝑎 ⟹ ¬𝑎 = 1 ≰ 𝑎 = ¬0 15 16

📘 Fodor and Yager (2000) page 128, 📃 Ovchinnikov (1983) page 235 ⟨Example 4⟩ 📘 Fodor and Yager (2000) page 128

version 0.30


Daniel J. Greenhoe


page 165

CHAPTER 11. NEGATION 1 = ¬𝑎 = ¬0

1 = ¬0

1

1 = ¬0

𝑎 = ¬1

𝑎 = ¬𝑎

𝑎 = ¬0

𝑎

0

0 = ¬1

0 = ¬1 = ¬𝑎

0 = ¬1 = ¬𝑎

(A) minimal negation

(B) Kleene negation

(C) intuitionistic negation

(D) fuzzy and intuitionistic negation

(Example 11.8 page 166)




Figure 11.2: negations on 𝙇3 ✏ Example 11.6. 1 ¬0

e x

The function ¬ as illustrated to the right is not a subminimal negation 𝑐 ¬𝑑 ¬𝑐 𝑑 (it is not antitone) and so is not any negation defined here. Note however that the problem is not the O6 lattice—it is possible to define a 𝑎 ¬𝑏 ¬𝑎 𝑏 negation on an O6 lattice (Example 11.16 page 169). 0 ¬1

✎PROOF:

✏

Proof that ¬ is not antitone: 𝑎 ≤ 𝑐 ⟹ ¬𝑐 = 𝑑 ≰ 𝑏 = ¬𝑎

Remark 11.3. The concept of a complement (Definition 7.1 page 109) and the concept of a negation are fundamentally different. A complement is a relation (Definition 15.1 page 237) on a lattice 𝙇 and a negation is a function (Definition 15.8 page 249). In Example 11.6 (page 165), 𝑏 and 𝑑 are both complements of 𝑎, but yet ¬ is not a negation. In the right side lattice of Example 11.16 (page 169), both 𝑏 and 𝑑 are complements of 𝑎 (and so the lattice is multipy complemented), but yet only 𝑑 is equal to the negation of 𝑎 (𝑑 = ¬𝑎). It can also be said that complementation is a property of a lattice, whereas negation is a function defined on a lattice. Example 11.7. Each of the functions ¬ illustrated to the right is a subminimal negation (Defe inition 11.1 page 157); none of them is a x minimal negation (each fails to have weak double negation).

1

1

1 = ¬0 = ¬𝑎

𝑎 = ¬0

𝑎

0 = ¬1 = ¬𝑎

𝑎 = ¬1 = ¬𝑎 = ¬0 0

(a)

(b)

(c)

0 = ¬1

✎PROOF: 1. Proof that (a) ¬ is antitone: 𝑎 ≤ 1 0 ≤ 1 0 ≤ 𝑎

⟹ ¬1 = 0 ≤ 0 = ¬𝑎 ⟹ ¬1 = 0 ≤ 𝑎 = ¬0 ⟹ ¬𝑎 = 0 ≤ 𝑎 = ¬0

⟹ ¬ is antitone over (𝑎, 1) ⟹ ¬ is antitone over (0, 1) ⟹ ¬ is antitone over (0, 𝑎)

2. Proof that (a) ¬ fails to have weak double negation: 1 ≰ 𝑎 = ¬0 = ¬¬1 3. Proof that (b) ¬ is antitone: 𝑎 ≤ 1 0 ≤ 1 0 ≤ 𝑎

⟹ ¬1 = 𝑎 ≤ 𝑎 = ¬𝑎 ⟹ ¬1 = 𝑎 ≤ 𝑎 = ¬0 ⟹ ¬𝑎 = 𝑎 ≤ 𝑎 = ¬0


4. Proof that (b) ¬ fails to have weak double negation: 1 ≰ 𝑎 = ¬𝑎 = ¬¬1 5. (c) is a special case of the dual discrete negation (Example 11.2 page 164).


Daniel J. Greenhoe


version 0.30

page 166

11.3. EXAMPLES

Example 11.8. The function ¬ illustrated in Figure 11.2 page 165 (A) is a minimal negation (Definition 11.2 page 158); it is not an intuitionistic negation (it does not have the non-contradiction property), it is not a de Morgan negation (it is not involutory), and it is not a fuzzy negation (¬1 ≠ 0). ✎PROOF: 1. Proof that ¬ is antitone: 𝑎 ≤ 1 0 ≤ 1 0 ≤ 𝑎

⟹ ¬1 = 𝑎 ≤ 1 = ¬𝑎 ⟹ ¬1 = 𝑎 ≤ 1 = ¬0 ⟹ ¬𝑎 = 1 ≤ 1 = ¬0


2. Proof that ¬ is a weak double negation (and so is a minimal negation, but is not a de Morgan negation): 1 = 1 = ¬𝑎 = ¬¬1 ⟹ ¬ is involutory at 1 𝑎 = 𝑎 = ¬1 = ¬¬𝑎 ⟹ ¬ is involutory at 𝑎 0 ≤ 𝑎 = ¬1 = 0¬¬ ⟹ ¬ is a weak double negation at 0 3. Proof that ¬ does not have the non-contradiction property (and so is not an intuitionistic negation): 1 ∧ ¬1 = 1 ∧ 𝑎 = 𝑎 ≠ 0 4. Proof that ¬ is not a fuzzy negation: ¬1 = 𝑎 ≠ 0

✏ Example 11.9 (Łukasiewicz 3-valued logic/Kleene 3-valued logic/RM3 logic). 17 The function ¬ illustrated in Figure 11.2 page 165 (B) is a Kleene negation (Definition 11.3 page 158), and is also a fuzzy negation (Definition 11.2 page 158); but it is not an ortho negation and is not an intuitionistic negation (it does not have the non-contradiction property). ✎PROOF: 1. Proof that ¬ is antitone: 𝑎 ≤ 1 0 ≤ 1 0 ≤ 𝑎

⟹ ¬1 = 0 ≤ 𝑎 = ¬𝑎 ⟹ ¬1 = 0 ≤ 1 = ¬0 ⟹ ¬𝑎 = 𝑎 ≤ 1 = ¬0


2. Proof that ¬ is involutory (and so is a de Morgan negation): 1 = ¬0 = ¬¬1 ⟹ ¬ is involutory at 1 𝑎 = ¬𝑎 = ¬¬𝑎 ⟹ ¬ is involutory at 𝑎 0 = ¬0 = 0¬¬ ⟹ ¬ is involutory at 0 3. Proof that ¬ does not have the non-contradiction property (and so is not an ortho negation): 𝑥 ∧ ¬𝑥 = 𝑥 ∧ 𝑥 = 𝑥 ≠ 0 4. Proof that ¬ satisfies the Kleene condition (and so is a Kleene negation): 1 ∧ ¬1 = 1 ∧ 0 = 0 ≤ 𝑎 = 𝑎 ∨ 𝑎 = 𝑎 ∨ ¬𝑎 1 ∧ ¬1 = 1 ∧ 0 = 0 ≤ 1 = 0 ∨ 1 = 0 ∨ ¬0 𝑎 ∧ ¬𝑎 = 1 ∧ 𝑎 = 𝑎 ≤ 1 = 1 ∨ 0 = 1 ∨ ¬1 𝑎 ∧ ¬𝑎 = 1 ∧ 𝑎 = 𝑎 ≤ 1 = 0 ∨ 1 = 0 ∨ ¬0 0 ∧ ¬0 = 0 ∧ 1 = 0 ≤ 1 = 1 ∨ 0 = 1 ∨ ¬1 0 ∧ ¬0 = 0 ∧ 1 = 0 ≤ 𝑎 = 𝑎 ∨ 𝑎 = 𝑎 ∨ ¬𝑎

✏ 17

📃 Łukasiewicz (1920), 📃 Avron (1991) pages 277–278, 📃 Kleene (1938) page 153, 📘 Kleene (1952), pages 332–339 ⟨§64. The 3-valued logic⟩, 📃 Sobociński (1952) version 0.30


Daniel J. Greenhoe


page 167

CHAPTER 11. NEGATION 1 = ¬0 𝑎

1 = ¬0 𝑏 = ¬𝑏

1 = ¬0 𝑏 = ¬𝑎

𝑎 = ¬𝑏 0 = ¬1 = ¬𝑎

𝑏 = ¬𝑏 𝑎 = ¬𝑎

0 = ¬1

0 = ¬1

(A) fuzzy negation

(B) ortho negation

(C) de Morgan negation




Figure 11.3: negations on 𝙈2 Example 11.10. The function ¬ illustrated in Figure 11.2 page 165 (C) an intuitionistic negation (Definition 11.2 page 158); but it is not a fuzzy negation (1 ≠ ¬0), and it is not a de Morgan negation (it is not involutory). ✎PROOF: 1. Proof that ¬ is antitone: 𝑎 ≤ 1 0 ≤ 1 0 ≤ 𝑎

⟹ ¬1 = 0 ≤ 0 = ¬𝑎 ⟹ ¬1 = 0 ≤ 𝑎 = ¬0 ⟹ ¬𝑎 = 0 ≤ 𝑎 = ¬0

⟹ ¬ is antitone at (𝑎, 1) ⟹ ¬ is antitone at (0, 1) ⟹ ¬ is antitone at (0, 𝑎)

2. Proof that ¬ has weak double negation property (and so is a minimal negation, but not a de Morgan negation): 1 ≤ 𝑎 = ¬0 = ¬¬1 ⟹ ¬ has weak double negation at 1 𝑎 = ¬0 = ¬¬𝑎 ⟹ ¬ has weak double negation at 𝑎 0 = ¬𝑎 = 0¬¬ ⟹ ¬ is involutory at 0 3. Proof that ¬ has the non-contradiction property (and so is an intuitionistic negation): 1 ∧ ¬1 = 1 ∧ 0 = 0 𝑎 ∧ ¬𝑎 = 𝑎 ∧ 0 = 0 0 ∧ ¬0 = 0 ∧ 𝑎 = 0 4. Proof that ¬ is not a fuzzy negation: ¬1 ≠ 0

✏ Example 11.11 (Heyting 3-valued logic/Jaśkowski's first matrix). 18 The function ¬ illustrated in Figure 11.2 page 165 (D) is an intuitionistic negation (Definition 11.2 page 158), and is also a fuzzy negation (Definition 11.2 page 158), but it is not a de Morgan negation (it is not involutory). ✎PROOF:

This is simply a special case of the discrete negation (Example 11.1 page 163).

✏

Remark 11.4. There is only one linearly ordered (Definition 3.4 page 47) 3-element lattice (𝙇3 ) that is a fuzzy negation (Example 11.11 page 167). However, this lattice is also an intuitionistic negation. There are no 𝙇3 lattices that are fuzzy but yet not intuitionistic. In fact, there are only three linearly ordered 3element lattices with with 1 = ¬0 and 0 = ¬1. Of these three, only one is both fuzzy and intuitionistic (Example 11.11 page 167), one is Kleene but not fuzzy (Example 11.9 page 166), and one is subminimal but not fuzzy (Example 11.7 page 165). It can be claimed that the “simplist” fuzzy negation that is not de Morgan and not intuitionistic is the 𝙈2 lattice of Example 11.12 (next). Example 11.12. The function ¬ illustrated in Figure 11.3 page 167 (A) is a fuzzy negation (Definition 11.2 page 158). It is not an intuitionistic negation (it does not have the non-contradiction property) and it is not a de Morgan negation (it is not involutory). 18

📘 Karpenko (2006) page 45, 📘 Johnstone (1982) page 9 ⟨§1.12⟩, 📃 Heyting (1930a), 📃 Heyting (1930b), 📃 Heyting (1930c), 📃 Heyting (1930d), 📃 Jaskowski (1936), 📘 Mancosu (1998) 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 168

11.3. EXAMPLES 1 = ¬0

1 = ¬0 𝑏 = ¬𝑏

𝑎

✎PROOF:

Note that

𝑎

= 0 = ¬1 = ¬𝑎

≤ ≤ ≤ ≤ ≤

1 1 𝑎 1 𝑏

⟹ ⟹ ⟹ ⟹ ⟹

𝑎 = ¬𝑎

+

0 = ¬1 = ¬𝑎

0 = ¬1

fuzzy and intuitionistic

Kleene negation




1. Proof that ¬ is antitone: 𝑎 0 0 𝑏 0

1 = ¬0

¬1 ¬1 ¬𝑎 ¬1 ¬𝑏

= = = = =

0 0 0 0 𝑏

≤ ≤ ≤ ≤ ≤

0 1 1 𝑏 1

= = = = =

¬𝑎 ¬0 ¬0 ¬𝑏 ¬0

¬ is antitone at (𝑎, 1) ¬ is antitone at (0, 1) ¬ is antitone at (0, 𝑎) ¬ is antitone at (𝑏, 1) ¬ is antitone at (0, 𝑏)

⟹ ⟹ ⟹ ⟹ ⟹

2. Proof that ¬ has weak double negation property (and so is a minimal negation, but not a de Morgan negation): 1 = ¬0 = ¬¬1 ⟹ ¬ is involutory at 1 𝑎 ≤ 1 = ¬0 = ¬¬𝑎 ⟹ ¬ has weak double negation at 𝑎 0 = ¬1 = 0¬¬ ⟹ ¬ is involutory at 0 𝑏 = ¬𝑏 = ¬¬𝑏 = ⟹ ¬ is involutory at 𝑏 3. Proof that ¬ does not have the non-contradiction property (and so is not an intuitionistic negation): 𝑏 ∧ ¬𝑏 = 𝑏 ∧ 𝑏 = 𝑏 ≠ 0 4. Proof that ¬ is has boundary conditions (and so is a fuzzy negation): ¬1 = 0, ¬0 = 1

✏ Example 11.13. 11.3 page 158).

19

The function ¬ illustrated in Figure 11.3 page 167 (B) is an ortho negation (Definition

✎PROOF: 1. Proof that ¬ is antitone: 𝑎 0 0 𝑏 0

≤ ≤ ≤ ≤ ≤

1 1 𝑎 1 𝑏

⟹ ⟹ ⟹ ⟹ ⟹

¬1 ¬1 ¬𝑎 ¬1 ¬𝑏

= = = = =

0 0 𝑏 0 𝑎

≤ ≤ ≤ ≤ ≤

𝑏 1 1 𝑎 1

= = = = =

¬𝑎 ¬0 ¬0 ¬𝑏 ¬0

2. Proof that ¬ is involutory (and so is a de Morgan negation): 1 𝑎 𝑏 0

= = = =

¬0 ¬𝑎 ¬𝑏 ¬0

= = = =

¬¬1 ¬¬𝑎 ¬¬𝑏 0¬¬

3. Proof that ¬ is has the non-contradiction property (and so is an ortho negation): 1 ∧ ¬1 = 1 ∧ 0 = 0 𝑎 ∧ ¬𝑎 = 𝑎 ∧ 𝑏 = 0 𝑏 ∧ ¬𝑏 = 𝑏 ∧ 𝑎 = 0 0 ∧ ¬0 = 0 ∧ 1 = 0

✏ 19

📃 Belnap (1977) page 13 📘 Restall (2000) page 177 ⟨Example 8.44⟩, 📃 Pavičić and Megill (2008) page 28 ⟨Definition 2, classical implication⟩ version 0.30


Daniel J. Greenhoe


page 169


Example 11.14 (BN4 ). 20 The function ¬ illustrated in Figure 11.3 page 167 (C) is a de Morgan negation (Definition 11.3 page 158), but it is not a Kleene negation and not an ortho negation (it does not satisfy the Kleene condition). ✎PROOF: 1. Proof that ¬ is antitone: 𝑎 0 0 𝑏 0

≤ ≤ ≤ ≤ ≤

1 1 𝑎 1 𝑏

⟹ ⟹ ⟹ ⟹ ⟹

¬1 ¬1 ¬𝑎 ¬1 ¬𝑏

= = = = =

0 0 𝑎 0 𝑏

≤ ≤ ≤ ≤ ≤

𝑏 1 1 𝑏 1

= = = = =

¬𝑎 ¬0 ¬0 ¬𝑏 ¬0

2. Proof that ¬ is involutory (and so is a de Morgan negation): 1 𝑎 𝑏 0

= = = =

¬0 ¬𝑎 ¬𝑏 ¬0

= = = =

¬¬1 ¬¬𝑎 ¬¬𝑏 0¬¬

3. Proof that ¬ does not have the non-contradiction property (and so is not an ortho negation): 𝑎 ∧ ¬𝑎 = 𝑎 ∧ 𝑎 = 𝑎 ≠ 0 𝑏 ∧ ¬𝑏 = 𝑏 ∧ 𝑏 = 𝑏 ≠ 0 4. Proof that ¬ does not satisfy the Kleene condition (and so is a de Morgan negation): 𝑎 ∧ ¬𝑎 = 𝑎 ∧ 𝑎 = 𝑎 ≰ 𝑏 ∧ ¬𝑏 = 𝑏

✏ Example 11.15. The function ¬ illustrated to the left is a de Morgan negation (Definition 11.3 page 158), but it 1 = ¬0 is not a Kleene negation and not an ortho 1 = ¬0 ¬𝑎 = 𝑎 𝑐 = ¬𝑐 negation (it does not satisfy the Kleene condi- ¬𝑐 = 𝑎 𝑐 = ¬𝑎 𝑏 = ¬𝑏 𝑏 = ¬𝑏 tion). The negation illustrated to the right is 0 = ¬1 a Kleene negation (Definition 11.3 page 158), but it is 0 = ¬1 not an ortho negation (it does not have the noncontradiction property).

e x

Example 11.16. 1 ¬0

¬0 1 The function ¬ illustrated to the left is a de Morgan 𝑐 ¬𝑎 ¬𝑏 𝑑 negation (Definition 11.3 page 158); it is not a Kleene negation 𝑐 ¬𝑏 ¬𝑎 𝑑 (it does not satisfy the Kleene condition). The negation 𝑎 ¬𝑐 ¬𝑑 𝑏 𝑎 ¬𝑑 ¬𝑐 𝑏 illustrated to the right is an ortho negation (Definition 11.3 page 158). 0 ¬1 0 ¬1

e x

Example 11.17. 1 ¬0

1 ¬0 The function ¬ illustrated to the left is not antitone and 𝑐 ¬𝑑 ¬𝑐 𝑑 therefore is not a negation (Definition 11.2 page 158). The 𝑐 ¬𝑏 ¬𝑎 𝑑 function ¬ illustrated to the right is a Kleene negation 𝑎 ¬𝑏 ¬𝑎 𝑏 𝑎 ¬𝑑 ¬𝑐 𝑏 (Definition 11.3 page 158); it is not an ortho negation (it does not have the non-contradiction property). 0 ¬1 0 ¬1

e x

✎PROOF: 20

📃

📃 Cignoli (1975) page 270, 📘 Restall (2000) page 171 ⟨Example 8.39⟩, 📃 de Vries (2007) pages 15–16 ⟨Example 26⟩, Dunn (1976), 📃 Belnap (1977)


Daniel J. Greenhoe


version 0.30

page 170

11.3. EXAMPLES

1. Proof that left ¬ is not antitone: 𝑎 ≤ 𝑐 but ¬𝑐 ≰ ¬𝑎. 2. Proof that right ¬ satisfies the Kleene condition: 𝑏 for 𝑥 = 𝑏 𝑐 for 𝑦 = 𝑐 𝑥 ∧ ¬𝑥 = ∀𝑥∈𝑋 and 𝑦 ∧ ¬𝑦 = { 0 otherwise { 0 otherwise ⟹ 𝑥 ∧ ¬𝑥 ≤ 𝑦 ∨ ¬𝑦 ∀𝑥,𝑦∈𝑋

∀𝑦∈𝑋

3. Proof that right ¬ does not have the non-contradiction property: 𝑏 ∧ ¬𝑏 = 𝑏 ∧ 𝑐 = 𝑏 ≠ 0

✏ Example 11.18. 1 ¬0

e x

1 ¬0

The lattices illustrated to the left and right are 𝑒 ¬𝑏 Boolean (Definition 8.1 page 115). The function ¬ illustrated 𝑑 ¬𝑐 𝑒 ¬𝑏¬𝑎 𝑓 𝑑 ¬𝑎 ¬𝑐 𝑓 to the left is a Kleene negation (Definition 11.3 page 158), but it is not an ortho negation (it does not have the non- 𝑎 ¬𝑑 𝑎 ¬𝑑 ¬𝑓 𝑐 ¬𝑓 𝑐 𝑏 ¬𝑒 𝑏 ¬𝑒 contradiction property). The negation illustrated to the right is an ortho negation (Definition 11.3 page 158). 0 ¬1 0 ¬1

✎PROOF: 1. Proof that left side negation does not have non-contradiction property (and so is not an ortho negation): 𝑎 ∧ ¬𝑎 = 𝑎 ∧ 𝑑 = 𝑎 ≠ 0 2. Proof that left side negation does not satisfy Kleene condition (and so is not a Kleene negation): 𝑎 ∧ ¬𝑎 = 𝑎 ∧ 𝑑 = 𝑎 ≰ 𝑓 = 𝑐 ∨ 𝑓 = 𝑐 ∨ ¬𝑐

✏

version 0.30


Daniel J. Greenhoe


page 171


11.4 Projections Definition 11.6. 21 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an orthocomplemented lattice (Definition 9.1 page 140). A function 𝜙𝑥 ∈ 𝑋 𝑋 is a Sasaki projection on 𝑥 ∈ 𝑋 if d 𝜙𝑥 (𝑦) ≜ (𝑦 ∨ 𝑥⟂ ) ∧ 𝑥. e f The Sasaki projections 𝜙𝑥 and 𝜙𝑦 are permutable if 𝜙𝑥 ∘ 𝜙𝑦 (𝑢) = 𝜙𝑦 ∘ 𝜙𝑥 (𝑢) ∀𝑢 ∈ 𝑋. Proposition 11.1. Let 𝜙𝑥 (𝑦) be the Sasaki projection of 𝑦 onto 𝑥 (Definition 11.6 page 171) in an orthocomplemented lattice 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤). ⟹ 𝜙𝑥 (𝑦) = 𝑥 ∀𝑥,𝑦∈𝑋 p (1). 𝑥 ≤ 𝑦 r (2). 𝑦 ≤ 𝑥 ⟹ 𝑦 ≤ 𝜙𝑥 (𝑦) ≤ 𝑥 ∀𝑥,𝑦∈𝑋 p (3). 𝑦 ≤ 𝑥 and 𝙇 is Boolean ⟹ 𝜙𝑥 (𝑦) = 𝑦 ∀𝑥,𝑦∈𝑋

✎PROOF: 𝑥 ≤ 𝑦 ⟹ 𝜙𝑥 (𝑦) ≜ (𝑦 ∨ 𝑥⟂ ) ∧ 𝑥

by definition of Sasaki projection (Definition 11.6 page 171)

=1∧𝑥

by 𝑥 ≤ 𝑦 hypothesis and Proposition 13.1 page 191

=𝑥

by property of bounded lattices (Proposition 4.4 page 71)

𝑦≤𝑥 ⟹ 𝑦 =𝑦∧𝑥


≤ (𝑦 ∨ 𝑥⟂ ) ∧ 𝑥


= 𝜙𝑥 (𝑦)


≤ (𝑦 ∨ 𝑥⟂ ) ∧ 𝑥


≤ 𝑥


𝑦 ≤ 𝑥 and Boolean ⟹ 𝜙𝑥 (𝑦) = (𝑦 ∨ 𝑥⟂ ) ∧ 𝑥


⟂

= (𝑦 ∧ 𝑥) ∨ (𝑥 ∧ 𝑥) by distributive prop. of Boolean lattices (Theorem 8.2 page 120) = (𝑦 ∧ 𝑥) ∨ 0

by non-contradiction of Boolean lat.

= (𝑦 ∧ 𝑥)

by boundary prop. of bounded lattices (Proposition 4.4 page 71)

=𝑦

by 𝑦 ≤ 𝑥 hypothesis and definition of ∧ (Definition 3.22 page 58)

(Theorem 8.2 page 120)

✏ Proposition 11.2. Let 𝜙𝑥 (𝑦) be the Sasaki projection of 𝑦 onto 𝑥 (Definition 11.6 page 171) in an orthocomplemented lattice ( 𝑋, ∨, ∧, 0, 1 ; ≤). (1). 𝜙0 (𝑦) = 0 ∀𝑦∈𝑋 𝜙𝑥 (0) = 0 ∀𝑥∈𝑋 p (2). r (3). 𝜙1 (𝑦) = 1 ∀𝑦∈𝑋 p (4). 𝜙𝑥 (1) = 𝑥 ∀𝑥∈𝑋 ⟂ (5). 𝜙𝑥 (𝑥 ) = 0 ∀𝑥∈𝑋 21

📘 Nakamura (1957) pages 158–159 ⟨equation (S)⟩ 📘 Sasaki (1954) page 300 ⟨Def.5.1, cf Foulis 1962⟩ 📘 Kalmbach (1983) page 117


Daniel J. Greenhoe


version 0.30

page 172

11.4. PROJECTIONS

✎PROOF: 𝜙0 (𝑦) = 0

because 0 ≤ 𝑦 and by Proposition 11.1 page 171 ⟂

𝜙𝑥 (0) ≜ (0 ∨ 𝑥 ) ∧ 𝑥 = 𝑥⟂ ∧ 𝑥 =0

by definition of Sasaki projection (Definition 11.6 page 171) by property of bounded lattices (Proposition 4.4 page 71) by definition of orthocomplemented (Definition 9.1 page 140)

⟂

𝜙1 (𝑦) ≜ (𝑦 ∨ 1 ) ∧ 1 = (𝑦 ∨ 0) ∧ 1

by definition of Sasaki projection (Definition 11.6 page 171) by boundary condition (Theorem 11.5 page 162)

=𝑦∧1


=1


𝜙𝑥 (1) = 𝑥

because 𝑥 ≤ 1 and by Proposition 11.1 page 171 ⟂

⟂

⟂

𝜙𝑥 (𝑥 ) ≜ (𝑥 ∨ 𝑥 ) ∧ 𝑥 by definition of Sasaki projection (Definition 11.6 page 171) = 𝑥⟂ ∧ 𝑥 by idempotency of lattices (Theorem 4.3 page 62) =0

by non-contradiction property of orthocomplemented lattice (Definition 9.1 page 140)

✏ Example 11.19. Here are some examples of projections in the 𝑂6 lattice onto the element 𝑥: ≜ (𝑞 ∨ 𝑝⟂ ) ∧ 𝑝 = 𝑝⟂ ∧ 𝑝 = 0 (because 𝑝 ⟂ 𝑞) 𝜙𝑝 (𝑞) 𝜙𝑝 (𝑝⟂ ) ≜ (𝑝⟂ ∨ 𝑝⟂ ) ∧ 𝑝 = 𝑝⟂ ∧ 𝑝 = 0 (because 𝑝 ⟂ 𝑝⟂ ) e 𝜙𝑝 (𝑞 ⟂ ) ≜ (𝑞 ⟂ ∨ 𝑝⟂ ) ∧ 𝑝 = 1 ∧ 𝑝 = 𝑝 (because 𝑝 ≤ 𝑞⟂ ) 𝑞 ⟂ x 𝑝 𝜙𝑞⟂ (𝑝) ≜ (𝑝 ∨ 𝑞) ∧ 𝑞 ⟂ = 1 ∧ 𝑞 ⟂ = 𝑞 ⟂ (because 𝑞⟂ ≤ 1) ≜ (1 ∨ 𝑝⟂ ) ∧ 𝑝 = 1 ∧ 𝑝 = 𝑝 (because 𝑝 ≤ 1) 𝜙𝑝 (1) ≜ (0 ∨ 𝑝⟂ ) ∧ 𝑝 = 𝑝⟂ ∧ 𝑝 = 0 (because 𝑝 ⟂ 0) 𝜙𝑝 (0)

1

𝑝⟂ 𝑞

0

Example 11.20. Here are some examples of projections in lattice 5 of Example 9.2 (page 140): ≜ (𝑝 ∨ 𝑥⟂ ) ∧ 𝑥 = 1 ∧ 𝑥 = 𝑥 𝜙𝑥 (𝑝) ≜ (𝑦 ∨ 𝑥⟂ ) ∧ 𝑥 = 𝑥⟂ ∧ 𝑥 = 0 (because 𝑥 ⟂ 𝑦) 𝜙𝑥 (𝑦) 𝜙𝑥 (𝑧) ≜ (𝑧 ∨ 𝑥⟂ ) ∧ 𝑥 = 𝑥⟂ ∧ 𝑥 = 0 (because 𝑥 ⟂ 𝑧) ⟂ 𝜙𝑥 (𝑝 ) ≜ (𝑝⟂ ∨ 𝑥⟂ ) ∧ 𝑥 = 𝑝⟂ ∧ 𝑥 = 0 e x 𝜙𝑥 (𝑥⟂ ) ≜ (𝑥⟂ ∨ 𝑥⟂ ) ∧ 𝑥 = 𝑥⟂ ∧ 𝑥 = 0 (because 𝑥 ⟂ 𝑥⟂ ) 𝜙𝑥 (𝑦⟂ ) ≜ (𝑦⟂ ∨ 𝑥⟂ ) ∧ 𝑥 = 1 ∧ 𝑥 = 𝑥 (because 𝑥 ≤ 𝑦⟂ ) 𝜙𝑥 (𝑧⟂ ) ≜ (𝑧⟂ ∨ 𝑥⟂ ) ∧ 𝑥 = 1 ∧ 𝑥 = 𝑥 (because 𝑥 ≤ 𝑧⟂ ) 𝜙𝑥 (1) ≜ (1 ∨ 𝑥⟂ ) ∧ 𝑥 = 1 ∧ 𝑥 = 𝑥 (because 𝑥 ≤ 1) 𝜙𝑥 (0) ≜ (0 ∨ 𝑥⟂ ) ∧ 𝑥 = 𝑥⟂ ∧ 𝑥 = 0 (because 𝑥 ⟂ 0) Example 11.21.

e x

Let ℝ3 be the 3-dimensional Euclidean space (Example 9.3 page 141) with subspaces 𝙕 and 𝙑 . Then the projection operator 𝑃𝙕 ⟂ onto 𝙕 ⟂ is a sasaki projection 𝜙𝙕 ⟂ . In particular 𝐏𝙕 ⟂ 𝙑 ≜ 𝜙𝙕 ⟂ (𝙑 ) ≜ (𝙑 + 𝙕 ⟂⟂ ) ∩ 𝙕 ⟂ = (𝙑 + 𝙕 ) ∩ 𝙕 ⟂ as illustrated to the right.

version 0.30


Daniel J. Greenhoe

𝙕 +𝙑

𝙕 𝙑

𝐏𝙕 ⟂ 𝙑 𝙕⟂


CHAPTER 12 LOGIC

❝I

dare say that this is the last effort of the human mind, and when this project shall have been carried out, all that men will have to do will be to be happy, since they will have an instrument that will serve to exalt the intellect not less than the telescope serves to perfect their vision.❞ Gottfried Leibniz (1646–1716), German mathematician, sharing his thoughts regarding mathematical logic. 1

❝I

cannot forget or omit to record this day last week. I was sleeping as usual for the night at St. Michael's Hamlet. As I awoke in the morning, the sun was shining brightly into my room. There was a consciousness on my mind that I was the discoverer of the true logic of the future. For a few minutes I felt a delight such as one can seldom hope to feel. But it would not last long— I remembered only too soon how unworthy and weak an instrument I was for accomplishing so great a work , and how hardly could I expect to do it.❞ William Stanley Jevons (1835–1882), English economist and logician 2

12.1 Implications Arguably a logic is not a logic without the inclusion of an implication function →. The mathematical structure logic is formally defined in Definition 12.2 (page 179). But before defining a logic, this text offers a very general definition (a “weak” definition) of implication that can be used in defining a very wide class of logics—including non-Boolean ones. For Boolean logics, the classical 1

quote:

2

image: image: quote:

📘 Padoa (1912), page 21 📘 Cajori (1993) ⟨paragraph 541⟩ http://en.wikipedia.org/wiki/Gottfried_Leibniz, public domain http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Jevons.html 📘 Jevons (1886), page 219 ⟨1866 March 28 entry⟩

page 174

12.1. IMPLICATIONS

𝑐 implication function 𝑥 → 𝑦 (Example 12.1 page 175) is arguably adequate. Two key properties of classical implication on a Boolean logic are entailment and modus ponens. The following definition exploits weakened versions of these two properties to define implication. Note that the definition is at this time probably not standard in the literature. But without it, it is difficult to offer a complete definition of a logic.

Definition 12.1. Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice (Definition 4.4 page 71). 𝑋 d The function → in 𝑋 is an implication on 𝙇 if e 1. {𝑥 ≤ 𝑦} ⟹ 𝑥 → 𝑦 ≥ 𝑥 ∨ 𝑦 ∀𝑥,𝑦∈𝑋 (weak entailment) and f 2. 𝑥 ∧ (𝑥 → 𝑦) ≤ ¬𝑥 ∨ 𝑦 ∀𝑥,𝑦∈𝑋 (weak modus ponens) Proposition 12.1. Let → be an implication (Definition 12.1 page 174) on a bounded lattice 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) (Definition 4.4 page 71). p r {𝑥 ≤ 𝑦} ⟺ {𝑥 → 𝑦 ≥ 𝑥 ∨ 𝑦} ∀𝑥,𝑦∈𝑋 p ✎PROOF: 1. Proof for ⟹ case: by weak entailment property of implications (Definition 12.1 page 174). 2. Proof for ⟸ case: 𝑦 ≥ 𝑥 ∧ (𝑥 → 𝑦)

by right hypothesis

≥ 𝑥 ∧ (𝑥 ∨ 𝑦)

by modus ponens property of → (Definition 12.1 page 174)

=𝑥

by absorptive property of lattices (Definition 4.3 page 61)

✏ 3

Remark 12.1. Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice (Definition 4.4 page 71). In the context of ortho lattices, a more common (and stronger) definition of implication → might be 1. 𝑥 ≤ 𝑦 ⟹ 𝑥 → 𝑦 = 1 ∀𝑥,𝑦∈𝑋 (entailment / strong entailment) and 2. 𝑥 ∧ (𝑥 → 𝑦) ≤ 𝑦 ∀𝑥,𝑦∈𝑋 (modus ponens / strong modus ponens) This definition yields a result stronger than that of Proposition 12.1 (page 174): {𝑥 ≤ 𝑦} ⟺ {𝑥 → 𝑦 = 1} ∀𝑥,𝑦∈𝑋 The Heyting 3-valued logic (Example 12.6 page 182) and Sasaki hook logic (Example 12.9 page 183) have both strong entailment and strong modus ponens. However, for non-ortho logics in general, these two properties seem inappropriate to serve as a definition for implication. For example, the Kleene 3valued logic (Example 12.3 page 180), RM3 logic (Example 12.5 page 181), and BN4 logic (Example 12.10 page 183) do not have the strong entailment property; and the Kleene 3-valued logic, Łukasiewicz 3-valued logic (Example 12.4 page 181), and BN4 logic do not have the strong modus ponens property. ✎PROOF: 1. Proof for ⟹ case: by entailment property of implications (Definition 12.1 page 174). 2. Proof for ⟸ case: 𝑥→𝑦=1 ⟹ 𝑥∧1≤𝑦 ⟹ 𝑥≤𝑦 3

by modus ponens property (Definition 12.1 page 174) by definition of 1 (least upper bound) (Definition 3.21 page 58)

Hardegree (1979) page 59 ⟨(E),(MP),(E*)⟩, 📃 Kalmbach (1973) page 498, 📘 Kalmbach (1983) pages 238–239 ⟨Chapter 4 §15⟩, 📃 Pavičić and Megill (2008) page 24, 📘 Xu et al. (2003) page 27 ⟨Definition 2.1.1⟩, 📃 Xu (1999) page 25, 📃 Jun et al. (1998) page 54 📃

version 0.30


Daniel J. Greenhoe


page 175

CHAPTER 12. LOGIC

✏ Example 12.1. 4 Let 𝙇 ≜ ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤) be a lattice with negation (Definition 11.5 page 159). If 𝙇 is an orthomodular lattice (Definition 11.3 page 158), then the functions listed below are all examples of valid implication functions (Definition 12.1 page 174) on 𝙇. If 𝙇 is an ortho lattice, then 1–5 are implication relations. 𝑐 1. 𝑥 → 𝑦 ≜ ¬𝑥 ∨ 𝑦 ∀𝑥,𝑦∈𝑋 (classical implication/material implication/horseshoe) 𝑠 2. 𝑥 → 𝑦 ≜ ¬𝑥 ∨ (𝑥 ∧ 𝑦) ∀𝑥,𝑦∈𝑋 (Sasaki hook / quantum implication) e 𝑑 3. 𝑥 → 𝑦 ≜ 𝑦 ∨ (¬𝑥 ∧ ¬𝑦) ∀𝑥,𝑦∈𝑋 (Dishkant implication) x 𝑘 4. 𝑥 → 𝑦 ≜ (¬𝑥 ∧ 𝑦) ∨ (¬𝑥 ∧ ¬𝑦) ∨ (𝑥 ∧ (¬𝑥 ∨ 𝑦)) ∀𝑥,𝑦∈𝑋 (Kalmbach implication) 𝑛 5. 𝑥 → 𝑦 ≜ (¬𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦) ∨ ((¬𝑥 ∨ 𝑦) ∧ ¬𝑦) ∀𝑥,𝑦∈𝑋 (non-tollens implication) 𝑟 6. 𝑥 → 𝑦 ≜ (¬𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦) ∨ (¬𝑥 ∧ ¬𝑦) ∀𝑥,𝑦∈𝑋 (relevance implication) 𝑐 Moreover, if 𝙇 is a Boolean lattice, then all of these implications are equivalent to → , and all of them have strong entailment and strong modus ponens. 𝑑 𝑠 𝑛 𝑘 Note that ∀𝑥,𝑦∈𝑋 , 𝑥 → 𝑦 = ¬𝑦 → ¬𝑥 and 𝑥 → 𝑦 = ¬𝑦 → ¬𝑥. The values for the 6 implications on an orthocomplemented O6 lattice (Definition 9.2 page 140) are listed in Example 12.11 (page 183). ✎PROOF: 𝑐 1. Proofs for the classical implication → :

𝑐 (a) Proof that on an ortho lattice, → is an implication: 𝑐 𝑥≤𝑦 ⟹ 𝑥→ 𝑦 ≜ ¬𝑥 ∨ 𝑦

𝑐 by definition of →

≥ ¬𝑦 ∨ 𝑦

by 𝑥 ≤ 𝑦 and antitone prop. of ¬ (Definition 11.3 page 158)

=1

by excluded middle prop. of ¬ (Theorem 11.5 page 162)

⟹ strong entailment

by definition of strong entailment

𝑥 ∧ (¬𝑥 ∨ 𝑦) ≤ ¬𝑥 ∨ 𝑦


⟹ weak modus ponens by definition of weak modus ponens Note that in general for an ortho lattice, the bound cannot be tightened to strong modus ponens because, for example in the O6 lattice (Definition 9.2 page 140) illustrated to the right 𝑥 ∧ (¬𝑥 ∨ 𝑦) = 𝑥 ∧ 1 = 𝑥 ≰ 𝑦 ⟹ not strong modus ponens

1 ¬𝑦

𝑥 𝑦

¬𝑥 0

𝑐 (b) Proof that on a Boolean lattice, → is an implication:

𝑥 ∧ (¬𝑥 ∨ 𝑦) = (𝑥 ∧ ¬𝑥) ∨ (𝑥 ∧ 𝑦)

by distributive prop. of Boolean lat.


= 1 ∨ (𝑥 ∧ 𝑦)

by excluded middle property of Boolean lattices

=𝑥∧𝑦

by definition of 1

≤𝑦


⟹ strong modus ponens by definition of strong modus ponens 𝑠 2. Proofs for Sasaki implication → : 4

📃 Kalmbach (1973) page 499, 📃 Kalmbach (1974), 📃 Mittelstaedt (1970) ⟨Sasaki hook⟩, 📃 Finch (1970) page 102 ⟨Sasaki hook (1.1)⟩, 📘 Kalmbach (1983) page 239 ⟨Chapter 4 §15, 3. THEOREM⟩


Daniel J. Greenhoe


version 0.30

page 176

12.1. IMPLICATIONS

𝑠 (a) Proof that on an ortho lattice, → is an implication: 𝑠 𝑥≤𝑦 ⟹ 𝑥→ 𝑦

≜ ¬𝑥 ∨ (𝑥 ∧ 𝑦)

𝑘 by definition of →

= ¬𝑥 ∨ 𝑥


=1

by excluded middle prop. of ortho neg. (Theorem 11.5 page 162)



𝑠

𝑠 by definition of →

𝑥 ∧ (𝑥 → 𝑦) ≜ 𝑥 ∧ [¬𝑥 ∨ (𝑥 ∧ 𝑦)] ≤ [¬𝑥 ∨ (𝑥 ∧ 𝑦)]


≤ ¬𝑥 ∨ 𝑦


⟹ weak modus ponens 𝑠 𝑐 (b) Proof that on a Boolean lattice, → =→ : 𝑠 𝑥→ 𝑦 ≜ ¬𝑥 ∨ (𝑥 ∧ 𝑦)


= ¬𝑥 ∨ 𝑦

by Lemma 8.2 (page 121)

𝑐


=𝑥→𝑦 𝑑 3. Proofs for Dishkant implication → : 𝑑 𝑠 (a) Proof that 𝑥 → 𝑦 ≡ ¬𝑦 → ¬𝑥: 𝑑 𝑥→ 𝑦 ≜ 𝑦 ∨ (¬𝑥 ∧ ¬𝑦)

𝑑 by definition of →

= 𝑦 ∨ (¬𝑦 ∧ ¬𝑥)


= ¬¬𝑦 ∨ (¬𝑦 ∧ ¬𝑥)

by involutory property of ortho negations (Definition 11.3 page 158)

𝑠


≜ ¬𝑦 → ¬𝑥

𝑑 (b) Proof that on an ortho lattice, → is an implication: 𝑑 𝑥≤𝑦 ⟹ 𝑥→ 𝑦

≜ 𝑦 ∨ (¬𝑥 ∧ ¬𝑦)


= 𝑦 ∨ ¬𝑦

by 𝑥 ≤ 𝑦 hypoth. and antitone prop. (Definition 11.3 page 158)

=1

by excluded middle prop. of ortho neg.



𝑑

(Theorem 11.5 page 162)


𝑥 ∧ (𝑥 → 𝑦) ≜ 𝑦 ∨ (¬𝑥 ∧ ¬𝑦) = 𝑦 ∨ ¬𝑥


⟹ weak modus ponens 𝑑 𝑐 (c) Proof that on a Boolean lattice, → =→ : 𝑑 𝑥→ 𝑦 ≜ 𝑦 ∨ (¬𝑥 ∧ ¬𝑦)

= ¬𝑥 ∨ 𝑦 𝑐

=𝑥→𝑦


by Lemma 8.2 (page 121) 𝑐 by definition of →

𝑘 4. Proofs for the Kalmbach implication → : version 0.30


Daniel J. Greenhoe


page 177

CHAPTER 12. LOGIC 𝑘 (a) Proof that on an ortho lattice, → is an implication:

𝑘 𝑥≤𝑦 ⟹ 𝑥→ 𝑦

≜ (¬𝑥 ∧ 𝑦) ∨ (¬𝑥 ∧ ¬𝑦) ∨ [𝑥 ∧ (¬𝑥 ∨ 𝑦)]


= (¬𝑥 ∧ 𝑦) ∨ (¬𝑦) ∨ [𝑥 ∧ (¬𝑥 ∨ 𝑦)]

by antitone property (Definition 11.3 page 158)

= (¬𝑥 ∧ 𝑦) ∨ ¬𝑦 ∨ [𝑥 ∧ (1)] = (¬𝑥 ∧ 𝑦) ∨ (𝑥 ∨ ¬𝑦)


= ¬¬(¬𝑥 ∧ 𝑦) ∨ (𝑥 ∨ ¬𝑦)


= ¬(¬¬𝑥 ∨ ¬𝑦) ∨ (𝑥 ∨ ¬𝑦)


= ¬(𝑥 ∨ ¬𝑦) ∨ (𝑥 ∨ ¬𝑦)


=1



𝑘 𝑥 ∧ (𝑥 → 𝑦) ≜ 𝑥 ∧ [(¬𝑥 ∧ 𝑦) ∨ (¬𝑥 ∧ ¬𝑦) ∨ [𝑥 ∧ (¬𝑥 ∨ 𝑦)]]


≤ (¬𝑥 ∧ 𝑦) ∨ (¬𝑥 ∧ ¬𝑦) ∨ [𝑥 ∧ (¬𝑥 ∨ 𝑦)]


≤ (¬𝑥 ∧ 𝑦) ∨ (¬𝑥 ∧ ¬𝑦) ∨ (¬𝑥 ∨ 𝑦)


≤ 𝑦 ∨ (¬𝑥 ∧ ¬𝑦) ∨ ¬𝑥 ∨ 𝑦


= 𝑦 ∨ ¬𝑥 ∨ (¬𝑥 ∧ ¬𝑦)

by idempotent p. (Theorem 4.3 page 62)

≤ 𝑦 ∨ ¬𝑥 ∨ ¬𝑥


= ¬𝑥 ∨ 𝑦

by idempotent p. (Theorem 4.3 page 62)

⟹ weak modus ponens 𝑘 𝑐 (b) Proof that on a Boolean lattice, → =→ :

𝑘 𝑥→ 𝑦 ≜ (¬𝑥 ∧ 𝑦) ∨ (¬𝑥 ∧ ¬𝑦) ∨ [𝑥 ∧ (¬𝑥 ∨ 𝑦)]


= (¬𝑥 ∧ 𝑦) ∨ (¬𝑥 ∧ ¬𝑦) ∨ [(𝑥 ∧ ¬𝑥) ∨ (𝑥 ∧ 𝑦)]

by distributive property (Definition 8.1 page 115)

= (¬𝑥 ∧ 𝑦) ∨ (¬𝑥 ∧ ¬𝑦) ∨ [(0) ∨ (𝑥 ∧ 𝑦)]

by non-contradiction property

= (¬𝑥 ∧ 𝑦) ∨ (¬𝑥 ∧ ¬𝑦) ∨ (𝑥 ∧ 𝑦)

by bounded property (Definition 4.4 page 71)

= ¬𝑥 ∧ (𝑦 ∨ ¬𝑦) ∨ (𝑥 ∧ 𝑦)


= ¬𝑥 ∧ 1 ∨ (𝑥 ∧ 𝑦)

by excluded middle property

= ¬𝑥 ∨ (𝑥 ∧ 𝑦)


= ¬𝑥 ∨ 𝑦

by Lemma 8.2 (page 121)

𝑐


≜𝑥→𝑦 𝑛 5. Proofs for the non-tollens implication → :

𝑛 𝑘 (a) Proof that 𝑥 → 𝑦 ≡ ¬𝑦 → ¬𝑥:

𝑛 𝑥→ 𝑦 ≜ (¬𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦) ∨ [(¬𝑥 ∨ 𝑦) ∧ ¬𝑦]

𝑛 by definition of →

= (𝑦 ∧ ¬𝑥) ∨ (𝑦 ∧ 𝑥) ∨ [¬𝑦 ∧ (𝑦 ∨ ¬𝑥)] = (¬¬𝑦 ∧ ¬𝑥) ∨ (¬¬𝑦 ∧ ¬¬𝑥) ∨ [¬𝑦 ∧ (¬¬𝑦 ∨ ¬𝑥)] 𝑘 ≜ ¬𝑦 → ¬𝑥 2016 December 27 Tuesday UTC

Daniel J. Greenhoe

𝑘 by definition of → Sets, Relations, and Order Structures

version 0.30

page 178

12.1. IMPLICATIONS

𝑛 (b) Proof that on an ortho lattice, → is an implication: 𝑛 𝑥≤𝑦 ⟹ 𝑥→ 𝑦 𝑘 ≡ ¬𝑦 → ¬𝑥

by item (5a) page 177

=1


⟹ strong entailment 𝑛 𝑘 𝑥 ∧ (𝑥 → 𝑦) = 𝑥 ∧ (¬𝑦 → ¬𝑥)


≤ ¬¬𝑦 ∨ ¬𝑥


= 𝑦 ∨ ¬𝑥

by involutory property of ¬ (Definition 11.3 page 158)

= ¬𝑥 ∨ 𝑦


⟹ weak modus ponens 𝑛 𝑐 (c) Proof that on a Boolean lattice, → =→ : 𝑛 𝑘 𝑥→ 𝑦 = ¬𝑦 → ¬𝑥


= ¬¬𝑦 ∨ ¬𝑥

by item (4b) page 177

= 𝑦 ∨ ¬𝑥

by involutory property of ¬ (Definition 11.3 page 158)

= ¬𝑥 ∨ 𝑦


𝑐

≜𝑥→𝑦


𝑟 6. Proofs for the relevance implication → : 𝑟 (a) Proof that on an ortho lattice, → does not have weak entailment: In the ortho lattice to the right… 𝑟 𝑥≤𝑦 ⟹ 𝑥→ 𝑦

1

≜ (¬𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦) ∨ (¬𝑥 ∧ ¬𝑦)

𝑟

by definition of →

= 0 ∨ 𝑥 ∨ ¬𝑦

¬𝑦

𝑥 𝑦

¬𝑥 0

= 𝑥 ∨ ¬𝑦 ≠𝑥∨𝑦 𝑟 (b) Proof that on an orthomodular lattice, → does have strong entailment: 𝑟 𝑥≤𝑦 ⟹ 𝑥→ 𝑦

≜ (¬𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦) ∨ (¬𝑥 ∧ ¬𝑦)

𝑟 by definition of →

= (¬𝑥 ∧ 𝑦) ∨ 𝑥 ∨ (¬𝑥 ∧ ¬𝑦)


= (¬𝑥 ∧ 𝑦) ∨ 𝑥 ∨ ¬𝑦

by 𝑥 ≤ 𝑦 and antitone property (Definition 11.3 page 158)

= 𝑦 ∨ ¬𝑦

by orthomodular identity (Definition 9.3 page 148)

=1

by excluded middle property of ¬ (Theorem 11.5 page 162)

𝑟 (c) Proof that on an ortho lattice, → does have weak modus ponens: 𝑟 𝑥 ∧ (𝑥 → 𝑦) ≜ 𝑥 ∧ [(¬𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦) ∨ (¬𝑥 ∧ ¬𝑦)]


≤ [(¬𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦) ∨ (¬𝑥 ∧ ¬𝑦)]


≤ ¬𝑥 ∨ (𝑥 ∧ 𝑦) ∨ (¬𝑥 ∧ ¬𝑦)


≤ ¬𝑥 ∨ 𝑦 ∨ (¬𝑥 ∧ ¬𝑦)


≤ ¬𝑥 ∨ 𝑦


⟹ weak modus ponens version 0.30


Daniel J. Greenhoe


page 179

CHAPTER 12. LOGIC 𝑟 𝑐 (d) Proof that on a Boolean lattice, → =→ : 𝑟 𝑥→ 𝑦 ≜ (¬𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦) ∨ (¬𝑥 ∧ ¬𝑦)


= [¬𝑥 ∧ (𝑦 ∨ ¬𝑦)] ∨ (𝑥 ∧ 𝑦)


= [¬𝑥 ∧ 1] ∨ (𝑥 ∧ 𝑦)

by excluded middle property of ¬ (Theorem 11.5 page 162)

= ¬𝑥 ∨ (𝑥 ∧ 𝑦)

by definition of 1 and ∧ (Definition 3.22 page 58)

= ¬𝑥 ∨ 𝑦

by property of Boolean lattices (Lemma 8.2 page 121)

𝑐


≜𝑥→𝑦

✏

12.2 Logics logic fuzzy logic intuitionalistic logic de Morgan logic ortho logic Boolean logic / classic logic

Figure 12.1: lattice of logics Definition 12.2. 5 Let → be an implication (Definition 12.1 page 174) defined on a lattice with negation 𝙇 ≜ ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤) (Definition 11.5 page 159). ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤, →) is a logic if ¬ is a minimal negation. ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤, →) is a fuzzy logic if ¬ is a fuzzy negation. d ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤, →) is an intuitionalistic logic if ¬ is an intuitionalistic negation. e ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤, →) is a de Morgan logic if ¬ is a de Morgan negation. f ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤, →) is a Kleene logic if ¬ is a Kleene negation. ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤, →) is an ortho logic if ¬ is an ortho negation. ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤, →) is a Boolean logic if ¬ is an ortho negation and 𝙇 is Boolean. Definition 12.3. 6 Let 𝙇 ≜ ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤, →) be a logic (Definition 12.2 page 179). d The function ↔ in 𝑋 𝑋 is an equivalence on 𝙇 if e 𝑥 ↔ 𝑦 ≜ (𝑥 → 𝑦) ∧ (𝑦 → 𝑥) ∀𝑥,𝑦∈𝑋 f Example 12.2 (Aristotelian logic/classical logic). 5

7

Straßburger (2005) page 136 ⟨Definition 2.1⟩, 📃 de Vries (2007) page 11 ⟨Definition 16⟩ 📘 Novák et al. (1999) page 18 7 📘 Novák et al. (1999) pages 17–18 ⟨EXAMPLE 2.1⟩ 📃

6


Daniel J. Greenhoe


version 0.30

page 180

12.2. LOGICS

The classical bi-variate logic is defined below. It is a 2 element Boolean logic (Definition 12.2 page 179). with 𝙇 ≜ ( {1, 0}, ∧, ¬, 0, 1, ≤ ; ∨) and a classical implication → with strong entailment and strong modus ponens. The value 1 represents “true” and 0 represents “false”. → 1 0 1 = ¬0 ⎧ ⎫ 1 ∀𝑥 ≤ 𝑦 ⎪ ⎪ 𝑥→𝑦≜ =⎨ 1 1 0 ∀𝑥,𝑦∈𝑋 ⎬ = ¬𝑥 ∨ 𝑦 { 𝑦 otherwise } 0 = ¬1 ⎪ ⎪ ⎩ 0 1 1 ⎭

e x

✎PROOF: 1. Proof that ¬ is an ortho negation: by Definition 11.3 (page 158) 2. Proof that → is an implication with strong entailment and strong modus ponens: (a) 𝙇 is Boolean and therefore is orthocomplemented. 𝑐 (b) → is equivalent to the classical implication → (Example 12.1 page 175).

(c) By Example 12.1 (page 175), → has strong entailment and strong modus ponens.

✏ The classical logic (previous example) can be generalized in several ways. Arguably one of the simplest of these is the 3-valued logic due to Kleene (next example). Example 12.3 (Kleene 3-valued logic). 8 The Kleene 3-valued logic ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤, →) is defined below. The function ¬ is a Kleene negation (Definition 11.3 page 158, Example 11.9 page 166) defined on a 3 element linearly ordered lattice (Definition 3.4 page 47). The function → is the classical implication 𝑥 → 𝑦 ≜ ¬𝑥 ∨ 𝑦. The values 1 e represents “true”, 0 represents “false”, and 𝑛 represents “neutral” or “undecided”. x 1 = ¬0 ⎧ → 1 𝑛 0 ⎫ ⎪ 1 1 𝑛 0 ⎪ 𝑛 = ¬𝑛 𝑥 → 𝑦 ≜ { ¬𝑥 ∨ 𝑦 ∀𝑥 ∈ 𝑋 } = ⎨ ∀𝑥,𝑦∈𝑋 ⎬ 𝑛 1 𝑛 𝑛 ⎪ ⎪ 0 = ¬1 0 1 1 1 ⎩ ⎭ ✎PROOF: 1. Proof that ¬ is a Kleene negation: see Example 11.9 (page 166) 2. Proof that → is an implication: This follows directly from the definition of → and the definition of an implication (Definition 12.1 page 174). 3. Proof that → does not have strong entailment: 𝑛 → 𝑛 = 𝑛 = 𝑛 ∨ 𝑛 ≠ 1. 4. Proof that → does not have strong modus ponens: 𝑛 → 0 = 𝑛 = ¬𝑛 ∨ 0 ≰ 0.

✏ A lattice and negation alone do not uniquely define a logic. Łukasiewicz also introduced a 3-valued logic with identical lattice structure to Kleene, but with a different implication relation (next example). Historically, Łukasiewicz's logic was introduced before Kleene's. 8

📃

Kleene (1938) page 153, 📘 Kleene (1952), pages 332–339 ⟨§64. The 3-valued logic⟩, 📃 Avron (1991) page 277

version 0.30


Daniel J. Greenhoe


page 181

CHAPTER 12. LOGIC

Example 12.4 (Łukasiewicz 3-valued logic). 9 The Łukasiewicz 3-valued logic ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤, →) is defined to the right and below. The function ¬ is a Kleene negation (Definition 11.3 page 158) defined on a 3 element linearly ordered lattice (Definition 3.4 page 47). The implication has strong entailment but weak modus ponens. In the implication table below, values that differ from the classical 𝑥 → 𝑦 ≜ ¬𝑥 ∨ 𝑦 are shaded . ⎫ ⎧ → 1 𝑛 0 ⎪ ⎪ 1 1 𝑛 0 1 ∀𝑥 ≤ 𝑦 1 for 𝑥 = 𝑦 = 𝑛 ∀𝑥,𝑦∈𝑋 ⎬ = 𝑥→𝑦≜ =⎨ { ¬𝑥 ∨ 𝑦 otherwise } { ¬𝑥 ∨ 𝑦 otherwise } 𝑛 1 1 𝑛 ⎪ ⎪ 0 1 1 1 ⎩ ⎭

1 = ¬0 𝑛 = ¬𝑛 0 = ¬1

✎PROOF: 1. Proof that ¬ is a Kleene negation: see Example 11.9 (page 166) 2. Proof that → is an implication: This follows directly from the definition of → and the definition of an implication (Definition 12.1 page 174). 3. Proof that → does not have strong modus ponens: 𝑛 → 0 = 𝑛 = ¬𝑛 ∨ 0 ≰ 0.

✏ Example 12.5 (RM3 logic). 10 The RM3 logic ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤, →) is defined below. The function ¬ is a Kleene negation (Definition 11.3 page 158) defined on a 3 element linearly ordered lattice (Definition 3.4 page 47). The implication function has weak entailment by strong modus ponens. In the implication table e below, values that differ from the classical 𝑥 → 𝑦 ≜ ¬𝑥 ∨ 𝑦 are shaded . x 1 = ¬0 → 1 𝑛 0 ⎫ 1 ∀𝑥 < 𝑦 ⎫ ⎧ ⎧ ⎪ ⎪ 1 1 0 0 ⎪ ⎪ 𝑛 = ¬𝑛 𝑥 → 𝑦 ≜ ⎨ 𝑛 ∀𝑥 = 𝑦 ⎬ = ⎨ ∀𝑥,𝑦∈𝑋 ⎬ 𝑛 1 𝑛 0 ⎪ ⎪ ⎩ 0 ∀𝑥 > 𝑦 ⎪ ⎭ ⎪ 0 1 1 1 0 = ¬1 ⎩ ⎭ ✎PROOF: 1. Proof that ¬ is a Kleene negation: see Example 11.9 (page 166) 2. Proof that → is an implication: This follows directly from the definition of → and the definition of an implication (Definition 12.1 page 174). 3. Proof that → does not have strong entailment: 𝑛 → 𝑛 = 𝑛 = 𝑛 ∨ 𝑛 ≠ 1.

✏ In a 3-valued logic, the negation does not necessarily have to be as in the previous three examples. The next example offers a different negation. 9

📃

10

📃

Łukasiewicz (1920) page 17 ⟨II. The principles of consequence⟩, 📃 Avron (1991) page 277 ⟨Łukasiewicz.⟩ Avron (1991) pages 277–278 📃 Sobociński (1952) 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 182

12.2. LOGICS

Example 12.6 (Heyting 3-valued logic/Jaśkowski's first matrix). 11 The Heyting 3-valued logic ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤, →) is defined below. The negation ¬ is both intuitionistic and fuzzy (Definition 11.2 page 158), and is defined on a 3 element linearly ordered lattice (Definition 3.4 page 47). The implication function has both strong entailment and strong modus ponens. In the implication table below, values that differ from the classical 𝑥 → 𝑦 ≜ e ¬𝑥 ∨ 𝑦 are shaded . x 1 = ¬0 ⎧ → 1 𝑛 0 ⎫ ⎪ 1 1 𝑛 0 ⎪ 1 ∀𝑥 ≤ 𝑦 𝑛 𝑥→𝑦≜ =⎨ ∀𝑥,𝑦∈𝑋 ⎬ { 𝑦 otherwise } 𝑛 1 1 0 ⎪ ⎪ 0 = ¬𝑛 = ¬1 ⎩ 0 1 1 1 ⎭ ✎PROOF:

1. Proof that ¬ is a Kleene negation: see Example 11.11 (page 167) 2. Proof that → is an implication: by definition of implication (Definition 12.1 page 174)

✏ Of course it is possible to generalize to more than 3 values (next example). Example 12.7 (Łukasiewicz 5-valued logic). 12 The Łukasiewicz 5-valued logic ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤, →) is defined below. The implication function has strong entailment but weak modus ponens. In the implication table below, values that differ from the classical 𝑥 → 𝑦 ≜ ¬𝑥 ∨ 𝑦 are shaded . → 1 𝑝 𝑛 𝑚 0 ⎧ 1 = ¬0 e ⎪ 1 1 𝑝 𝑛 𝑚 0 x 𝑝 = ¬𝑚 ⎪ ⎪ 𝑝 1 1 𝑛 𝑚 𝑚 𝑛 = ¬𝑛 𝑥→𝑦≜⎨ ∀𝑥,𝑦∈𝑋 𝑚 = ¬𝑝 ⎪ 𝑛 1 1 1 𝑚 𝑛 ⎪ 𝑚 1 1 1 1 𝑝 0 = ¬1 ⎪ ⎩ 0 1 1 1 1 1 ✎PROOF: ✏ All the previous examples in this section are linearly ordered. The following examples employ logics that are not. Example 12.8 (Boolean 4-valued logic).

13

11

📘 Karpenko (2006) page 45, 📘 Johnstone (1982) page 9 ⟨§1.12⟩, 📃 Heyting (1930a), 📃 Heyting (1930b), 📃 Heyting (1930c), 📃 Heyting (1930d), 📃 Jaskowski (1936), 📘 Mancosu (1998) 12 📘 Xu et al. (2003) page 29 ⟨Example 2.1.3⟩ 📃 Jun et al. (1998) page 54 ⟨Example 2.2⟩ 13 📃 Belnap (1977) page 13, 📘 Restall (2000) page 177 ⟨Example 8.44⟩, 📃 Pavičić and Megill (2008) page 28 ⟨Definition 2, classical implication⟩, 📃 Mittelstaedt (1970), 📃 Finch (1970) page 102 ⟨(1.1)⟩, 📃 Smets (2006) page 270 version 0.30


Daniel J. Greenhoe


page 183

CHAPTER 12. LOGIC

e x

The Boolean 4-valued logic is defined below. The negation function ¬ is an ortho negation (Example 11.13 page 168) defined on an 𝑀2 lattice. The value 1 represents “true”, 0 represents “false”, and 𝑚 and 𝑛 represent some intermediate values. 1 = ¬0 ⎧ → 1 𝑏 𝑛 0 ⎪ 1 1 𝑏 𝑛 0 𝑏 = ¬𝑛 ⎪ 𝑥 → 𝑦 ≜ ¬𝑥 ∨ 𝑦 = ⎨ 𝑏 1 1 𝑛 𝑛 ∀𝑥,𝑦∈𝑋 𝑛 = ¬𝑏 ⎪ 𝑛 1 𝑏 1 𝑏 ⎪ 0 1 1 1 1 0 = ¬1 ⎩

Example 12.9 (Sasaki hook / quantum implication). 14 The Sasaki hook logic ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤, →) is defined below. negation are the same as in Example 12.8 (page 182). ⎧ → 1 𝑏 𝑛 1 = ¬0 e ⎪ 1 1 𝑏 𝑛 x 𝑏 = ¬𝑛 ⎪ 𝑥 → 𝑦 ≜ ¬𝑥 ∨ (𝑥 ∧ 𝑦) = ⎨ 𝑏 1 1 𝑛 𝑛 = ¬𝑏 ⎪ 𝑛 1 𝑏 1 0 = ¬1 ⎪ 0 1 1 1 ⎩

The order structure and 0 0 𝑛 𝑏 1

∀𝑥,𝑦∈𝑋

All the previous examples in this section are distributive; the previous example was Boolean. The next example is non-distributive, and de Morgan (but non-Boolean). Note for a given order structure, the method of negation may not be unique; in the previous and following examples both have identical lattices, but are negated differently. Example 12.10 (BN4 logic). 15 The BN4 logic is defined below. The function ¬ is a de Morgan negation (Example 11.14 page 169) defined on a 4 element M2 lattice. The value 1 represents “true”, 0 represents “false”, 𝑏 represents “both” (both true and false), and 𝑛 represents “neither”. In the implication table below, 𝑐 the values that differ from those of the classical implication → are shaded . e x ⎧ → 1 𝑛 𝑏 0 1 = ¬0 ⎪ 1 1 𝑛 0 0 𝑏 = ¬𝑏 ⎪ ∀𝑥,𝑦∈𝑋 𝑥→𝑦≜⎨ 𝑛 1 1 𝑛 𝑛 𝑛 = ¬𝑛 ⎪ 𝑏 1 𝑛 𝑏 0 0 = ¬1 ⎪ 0 1 1 1 1 ⎩ Example 12.11. The tables that follow are the 6 implications defined in Example 12.1 (page 175) on the O6 lattice with ortho negation (Definition 11.3 page 158), or the e O orthocomplemented lattice (Definition 9.2 page 140), illustrated to the right. 6 x In the tables, the values that differ from those of the classical implication 𝑐 → are shaded . 𝑐

→ 1 𝑑 𝑐 𝑏 𝑎 0

1 1 1 1 1 1 1

𝑑 𝑑 1 𝑑 1 𝑑 1

𝑐 𝑐 𝑐 1 𝑐 1 1

𝑏 𝑏 1 𝑏 1 𝑑 1

𝑎 𝑎 𝑎 1 𝑐 1 1

0 0 𝑎 𝑏 𝑐 𝑑 1

𝑠

→ 1 𝑑 𝑐 𝑏 𝑎 0

1 1 1 1 1 1 1

𝑑 𝑑 1 𝑏 1 1 1

𝑐 𝑐 𝑎 1 𝑐 1 1

𝑏 𝑏 1 𝑏 1 𝑑 1

𝑎 𝑎 𝑎 1 𝑐 1 1

0 0 𝑎 𝑏 𝑐 𝑑 1

𝑑 → 1 𝑑 𝑐 𝑏 𝑎 0

1 1 1 1 1 1 1

𝑑 𝑑 1 𝑑 1 𝑑 1

¬0 1 𝑐 ¬𝑏

¬𝑎 𝑑

𝑎 ¬𝑑

¬𝑐 𝑏

0 ¬1

𝑐 𝑐 𝑐 1 𝑐 1 1

𝑏 𝑏 1 𝑏 1 𝑏 1

𝑎 𝑎 𝑎 1 𝑎 1 1

0 0 𝑎 𝑏 𝑐 𝑑 1

14

📃 Pavičić and Megill (2008) page 28 ⟨Definition 2⟩, 📃 Mittelstaedt (1970), 📃 Finch (1970) page 102 ⟨(1.1)⟩, 📃 Smets (2006) page 270 15 📘 Restall (2000) page 171 ⟨Example 8.39⟩ 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 184 𝑘 → 1 𝑑 𝑐 𝑏 𝑎 0

12.3. CLASSICAL TWO-VALUED LOGIC

1 1 1 1 1 1 1

𝑑 𝑑 1 𝑏 1 𝑑 1

𝑐 𝑐 𝑎 1 𝑐 1 1

𝑏 𝑏 1 𝑏 1 𝑏 1

𝑎 𝑎 𝑎 1 𝑎 1 1

0 0 𝑎 𝑏 𝑐 𝑑 1

𝑛 → 1 𝑑 𝑐 𝑏 𝑎 0

1 1 1 1 1 1 1

𝑑 𝑑 1 𝑏 1 𝑑 1

𝑐 𝑐 𝑎 1 𝑐 1 1

𝑏 𝑏 1 𝑏 1 𝑏 1

𝑎 𝑎 𝑎 1 𝑎 1 1

0 0 𝑎 𝑏 𝑐 𝑑 1

𝑟 → 1 𝑑 𝑐 𝑏 𝑎 0

1 1 1 1 1 1 1

𝑑 𝑑 1 𝑏 1 𝑑 1

𝑐 𝑐 𝑎 1 𝑐 1 1

𝑏 𝑏 1 𝑏 1 𝑏 1

𝑎 𝑎 𝑎 1 𝑎 1 1

0 0 𝑎 𝑏 𝑐 𝑑 1

Example 12.12. 16 A 6 element logic is defined below. The function ¬ is a Kleene negation (Example 11.17 page 169). The implication has strong entailment but weak modus ponens. In the implication table 𝑐 below, the values that differ from those of the classical implication → are shaded . ⎧ → 1 𝑝 𝑞 𝑚 𝑛 0 1 ¬0 ⎪ 1 1 𝑝 𝑞 𝑚 𝑛 0 e ⎪ 𝑝 1 1 𝑞 𝑝 𝑞 𝑛 x 𝑝 ¬𝑛 ¬𝑚 𝑞 ⎪ ∀𝑥,𝑦∈𝑋 𝑥→𝑦≜⎨ 𝑞 1 𝑝 1 𝑚 𝑝 𝑚 𝑚 ¬𝑞 ¬𝑝 𝑛 ⎪ 𝑚 1 1 𝑞 1 𝑞 𝑞 ⎪ 𝑛 1 1 1 𝑝 1 𝑝 0 ¬1 ⎪ ⎩ 0 1 1 1 1 1 1 ✎PROOF: 1. Proof that ¬ is a Kleene negation: see Example 11.17 (page 169) 2. Proof that → is an implication: This follows directly from the definition of → and the definition of an implication (Definition 12.1 page 174). 3. Proof that → does not have strong modus ponens: ¬𝑝 ∧ (𝑝 → 𝑚) = 𝑛 ∧ 𝑝 = 𝑛 ≤ 𝑝 = ¬𝑝 ∨ 𝑚 ¬𝑛 ∧ (𝑛 → 𝑚) = 𝑛 ∧ 𝑝 = 𝑛 ≤ 𝑝 = ¬𝑝 ∨ 𝑚 ¬𝑝 ∧ (𝑝 → 0) = 𝑛 ∧ 𝑛 = 𝑛 ≤ 𝑛 = ¬𝑝 ∨ 0 ¬𝑛 ∧ (𝑛 → 0) = 𝑝 ∧ 𝑛 = 𝑛 ≤ 𝑝 = ¬𝑛 ∨ 0

≰ ≰ ≰ ≰

𝑚 𝑚 0 0

✏ For an example of an 8-valued logic, see 📃 Kamide (2013). For examples of 16-valued logics, see 📃 Shramko and Wansing (2005).

12.3

Classical two-valued logic

Definition 12.4 (Aristotelian logic/classical logic). 17 The classical 2-value logic is a 2 element lattice with ortho negation (Definition 11.3 page 158) 𝑐 ( {1, 0}, ∨, ∧, ¬, 0, 1 ; ≤, →) as illustrated below with values 1 representing “true”, 0 representd ing “false”, and with an implication connective ⟹ as specified below: e ⟹ 1 0 1 = ¬0 f ⎧ ⎫ 1 ∀𝑥 ≤ 𝑦 ⎪ ⎪ 𝑥 ⟹ 𝑦≜ =⎨ 1 ∀𝑥,𝑦∈𝑋 ⎬ = ¬𝑥 ∨ 𝑦 1 0 { } 𝑦 otherwise 0 = ¬1 ⎪ ⎪ 1 1 ⎩ 0 ⎭ 16 17

📘 Xu et al. (2003) pages 29–30 ⟨Example 2.1.4⟩ 📘 Novák et al. (1999) pages 17–18 ⟨EXAMPLE 2.1⟩

version 0.30


Daniel J. Greenhoe


page 185

CHAPTER 12. LOGIC

Theorem 12.1. 𝑐 If ( {1, 0}, ∨, ∧, ¬, 0, 1 ; ≤, → ) is the classical 2-value logic (Definition 12.4 page 184), then the logical OR ∨, logical AND ∧, and logical equivalence ⟺ operations t are defined as follows: h ∨ 1 0 ∧ 1 0 ⟺ 1 0 m 1 1 1 1 1 0 1 1 0 0 1 0 0 0 0 0 0 1 ✎PROOF: 1. Proof for logical OR operation ∨: This follows from the lattice (Definition 4.3 page 61) properties of 𝙇2 . 2. Proof for logical AND operation ∧: This follows from the lattice (Definition 4.3 page 61) properties of 𝙇2 . 3. Proof for logical if and only if operation ⟺ : This follows from the definition of ⟹ 184) and Definition 12.3 (page 179).

(Definition 12.4 page

✏ One of the most useful facts concerning propositional logic systems is that they form a Boolean algebra (next theorem). Because they are a Boolean algebra, a number of useful properties automatically follow (next theorem) from the properties of Boolean algebras (Theorem 8.2 page 120). Theorem 12.2 (Boolean algebra properties). 18 Let {0, 1} be the set of logical properties false and true (Axiom 1.1 page 3). Let ∨ be the logical or and ∧ the logical and operations (Definition 12.1 page 185). Let ⟹ be the logical implies relation (Definition 1.12 page 8). ( {0, 1}, ∨, ∧ ; ⟹ ) is a Boolean algebra. In particular for all 𝑥, 𝑦, 𝑧 ∈ {0, 1}, 𝑥∨𝑥 = 𝑥 𝑥∧𝑥 = 𝑥 (idempotent) 𝑥∨𝑦 = 𝑦∨𝑥 𝑥∧𝑦 = 𝑦∧𝑥 (commutative) 𝑥 ∧ (𝑦 ∧ 𝑧) = (𝑥 ∧ 𝑦) ∧ 𝑧 (associative) 𝑥 ∨ (𝑦 ∨ 𝑧) = (𝑥 ∨ 𝑦) ∨ 𝑧 𝑥 ∨ (𝑥 ∧ 𝑦) = 𝑥 𝑥 ∧ (𝑥 ∨ 𝑦) = 𝑥 (absorptive) t 𝑥 ∨ (𝑦 ∧ 𝑧) = (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) 𝑥 ∧ (𝑦 ∨ 𝑧) = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) (distributive) h 𝑥∨0 = 𝑥 𝑥∧1 = 𝑥 (identity) m 𝑥∧0 = 0 (bounded) 𝑥∨1 = 1 𝑥 ∨ 𝑥′ = 1 𝑥 ∧ 𝑥′ = 0 (complemented)19 ′ ′ = 𝑥 (uniquely comp.) (𝑥 ) (𝑥 ∨ 𝑦)′ = 𝑥 ′ ∧ 𝑦′ (𝑥 ∧ 𝑦)′ = 𝑥 ′ ∨ 𝑦′ (de Morgan's laws) property with emphasizing ∨

dual property emphasizing ∧

property name

✎PROOF:

This follows directly from the fact that the classical 2-valued logic (Definition 12.4 page 184) is a Boolean algebra (Definition 8.1 page 115) and from Theorem 8.2 (page 120). ✏

Definition 12.5 (additional logic operations). 20 Let ( {0, 1}, ⟹ , ∨, ∧, ¬, 0, 1) be a propositional logic system. Let 𝑥′ ≜ ¬𝑥 and 𝑦′ ≜ ¬𝑦.The following table defines additional operations on {0, 1} in 18

📘 MacLane and Birkhoff (1999) page 488, 📘 Givant and Halmos (2009) page 10, 📘 Müller (1909), pages 20–21, 📘 Schröder (1890), 📘 Whitehead (1898) pages 35–37, 📘 Peano (1889b), page 88 19 The property 𝑥 ∨ 𝑥′ = 1 is also called the law of the excluded middle. The property 𝑥 ∧ 𝑥′ = 0 is also called non-contradiction or explosion. References: 📘 Renedo et al. (2003), page 71 📘 Restall (2004) pages 73–75 📘 Restall (2001), pages 1–3 20 📘 Givant and Halmos (2009) page 32 ⟨disjunction, conjunction, negation⟩, 📘 Shiva (1998) page 83 ⟨inhibit, transfer⟩, 📘 Whitesitt (1995) pages 68–69 ⟨Sheffer stroke functions ↓=↑, | =↓⟩, 📘 Quine (1979) pages 45–48 ⟨joint denial ↓, alternate denial |⟩, 📃 Bernstein (1934) page 876 ⟨implication ⊃⟩ 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 186


terms of ∨, ∧, and ¬.

name joint denial inhibit x inhibit y complete disjunction alternative denial

symbol ↓ ⊖ − ⊕ |

𝑥↓𝑦 𝑥⊖𝑦 𝑥−𝑦 𝑥⊕𝑦 𝑥|𝑦

≜ ≜ ≜ ≜ ≜

definition 𝑥 ∧ 𝑦′ 𝑥′ ∧ 𝑦 𝑥 ∧ 𝑦′ ′ ′ (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦 ) ′ ′ 𝑥 ∨𝑦 ′

∀𝑥,𝑦∈{0, 1} ∀𝑥,𝑦∈{0, 1} ∀𝑥,𝑦∈{0, 1} ∀𝑥,𝑦∈{0, 1} ∀𝑥,𝑦∈{0, 1}

There are a total of 24 = 16 possible binary operations on the set of relations {0, 1} × {0, 1} → {0, 1}. The following table summarizes these 16 operations.21

−

⦹

⊕ | ∧ ⇔ ◧ ⇒ ◨ ⇐ ∨ 1

⦹

⦹

0 ↓ ⊖

⦹

name and symbol zero joint denial inhibit 𝑥 complement 𝑥 inhibit 𝑦 complement 𝑦 complete disjunction alternative denial conjunction equivalence transfer y implication transfer x implied by disjunction identity

logic operations (𝑥, 𝑦) = 11 10 01 00 operation in terms of ∨, ∧, and ¬ 0 0 0 0 0 = 𝑥 ∧ 𝑥′ ∀𝑥∈{0, 1} 0 0 0 1 𝑥 ↓ 𝑦 = 𝑥 ′ ∧ 𝑦′ ∀𝑥,𝑦∈{0, 1} ′ 0 0 1 0 𝑥⊖𝑦 = 𝑥 ∧𝑦 ∀𝑥,𝑦∈{0, 1} 0 0 1 1 𝑥 𝑦 = 𝑥′ ∀𝑥,𝑦∈{0, 1} 0 1 0 0 𝑥 − 𝑦 = 𝑥 ∧ 𝑦′ ∀𝑥,𝑦∈{0, 1} ′ 0 1 0 1 𝑥 𝑦 = 𝑦 ∀𝑥,𝑦∈{0, 1} 0 1 1 0 𝑥 ⊕ 𝑦 = (𝑥′ ∧ 𝑦) ∨ (𝑥 ∧ 𝑦′ ) ∀𝑥,𝑦∈{0, 1} 0 1 1 1 𝑥|𝑦 = 𝑥 ′ ∨ 𝑦′ ∀𝑥,𝑦∈{0, 1} 1 0 0 0 𝑥∧𝑦 = 𝑥∧𝑦 ∀𝑥,𝑦∈{0, 1} 1 0 0 1 𝑥 ⇔ 𝑦 = (𝑥 ∧ 𝑦) ∨ (𝑥′ ∧ 𝑦′ ) ∀𝑥,𝑦∈{0, 1} 1 0 1 0 𝑥◧𝑦 = 𝑦 ∀𝑥,𝑦∈{0, 1} 1 0 1 1 𝑥 ⇒ 𝑦 = 𝑥′ ∨ 𝑦 ∀𝑥,𝑦∈{0, 1} 1 1 0 0 𝑥◨𝑦 = 𝑥 ∀𝑥,𝑦∈{0, 1} ′ 1 1 0 1 𝑥⇐𝑦 = 𝑥∨𝑦 ∀𝑥,𝑦∈{0, 1} 1 1 1 0 𝑥∨𝑦 = 𝑥∨𝑦 ∀𝑥,𝑦∈{0, 1} 1 1 1 1 1 = 𝑥 ∨ 𝑥′ ∀𝑥∈{0, 1}

The 16 logic operations of propositional logic can all be represented using the logic operations of disjunction ∨, conjunction ∧, and negation ¬. Using these representations, all 16 operations can be generalized to Boolean algebras using the equivalent Boolean algebra/lattice operations of join, meet, and complement.22 In addition to Boolean algebras, the 16 operations can also have equivalent operations on algebra of sets where the logic operations essentially define the set operations as in 𝐴∪𝐵 𝐴∩𝐵 𝐴⧵𝐵 𝐴 △𝐵 𝐴𝖼 21 22

= {𝑥 ∈ 𝑋|(𝑥 ∈ 𝐴) ∨ (𝑥 ∈ 𝐵)} = {𝑥 ∈ 𝑋|(𝑥 ∈ 𝐴) ∧ (𝑥 ∈ 𝐵)} = {𝑥 ∈ 𝑋|(𝑥 ∈ 𝐴) ⊖ (𝑥 ∈ 𝐵)} = {𝑥 ∈ 𝑋|(𝑥 ∈ 𝐴) ⊕ (𝑥 ∈ 𝐵)} = {𝑥 ∈ 𝑋|¬(𝑥 ∈ 𝐴)}

📘 Shiva (1998) page 83 📘 Givant and Halmos (2009), page 32

version 0.30


Daniel J. Greenhoe


page 187

CHAPTER 12. LOGIC

Computer science also makes use of some of the 16 logic operations, where disjunction becomes OR, and conjunction becomes AND. So, there are four fields (Boolean algebra, logic, set theory, computer science) that all use essentially the same operations, but sometimes call them by different names. The following table attempts to identify to these terms across the four fields:23

−

∅ ↓ ⊖ 𝖼𝑥 ⧵ 𝖼𝑦 △

| ∩ ⇔ ◧ ⇒ ◨ ÷ ∪ 𝑋

algebra of sets empty set rejection inhibit 𝑥 complement 𝑥 difference complement 𝑦 symmetric difference Sheffer stroke intersection equivalence projection 𝑦 implication projection 𝑥 adjunction union universal set

computer science 0 zero ↓ nor ⊖ inhibit 𝑥 not 𝑥 − difference not 𝑦 ⊕ exclusive-or | nand ∧ and ⇔ equivalence ◧ projection 𝑦 ⇒ implication ◨ projection 𝑥 ÷ adjunction ∨ or 1 one

⦹

⊕ | ∧ ⇔ ◧ ⇒ ◨ ⇐ ∨ 1

false joint denial inhibit 𝑥 negation 𝑥 inhibit 𝑦 negation 𝑦 complete disjunction alternate denial conjuction equivalence transfer 𝑦 implication transfer 𝑥 implied by disjunction true

⦹

0 ↓ ⊖ ⦹

⦹

terminology logic

⦹

⦹

0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

Boolean algebra 0 bottom ↓ rejection ⊖ inhibit 𝑥 complement 𝑥 − exception complement 𝑦 △ Boolean addition | Sheffer stroke ∧ meet ⇔ biconditional ◧ projection 𝑦 ⇒ implication ◨ projection 𝑥 ÷ adjunction ∨ join 1 top

❝I spent September in extending his [Peano's] methods to the logic of relations.…The

time was one of intellectual intoxication. My sensations resembled those one has after climbing a mountain in a mist, when, on reaching the summit, the mist suddenly clears, and the country becomes visible for forty miles in every direction.…Suddenly, in the space of a few weeks, I discovered what appeared to be definitive answers to the problems which had baffled me for years. And in the course of discovering these answers, I was introducing a new mathematical technique, by which regions formerly abandoned to the vaguenesses of philosophers were conquered for the precision of exact formulae. Intellectually, the month of September 1900 was the highest point of my life. I went about saying to myself that now at last I had done something worth doing, and I had the feeling that I must be careful not to be run over in the street before I had written it down.❞ Bertrand Russell (1872–1970), British mathematician,

23 24

24

http://groups.google.com/group/sci.math/browse_thread/thread/c1e9a7beb9a82311 quote: 📘 Russell (1951), pages 217–218 image: http://en.wikipedia.org/wiki/File:Russell1907-2.jpg, public domain


Daniel J. Greenhoe


version 0.30

page 188

version 0.30



Daniel J. Greenhoe


Part IV Relations

CHAPTER 13 RELATIONS ON LATTICES WITH NEGATION

The relations in this chapter are typically defined on an orthocomplemented lattice (Definition 9.1 page 140). Here, some relations are generalized to a lattice with negation (Definition 11.5 page 159). A lattice (Definition 4.3 page 61) with an ortho negation negation successfully defined on it is an orthocomplemented lattice (Definition 9.1 page 140). In many cases, these relations only work well on an orthocomplemented lattice, and thus many results are restricted to orthocomplemented lattices.

13.1 Orthogonality Proposition 13.1. Let ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an orthocomplemented lattice (Definition 9.1 page 140). p 𝑥⟂ ∨ 𝑦 = 1 and r 𝑥≤𝑦 ⟹ ∀𝑥,𝑦∈𝑋 { 𝑥 ∧ 𝑦⟂ = 0 } p ✎PROOF: 𝑥 ≤ 𝑦 ⟹ 𝑥 ∨ 𝑥⟂ ≤ 𝑦 ∨ 𝑥 ⟂ ⟹ 1≤𝑦∨𝑥

⟂

by excluded middle property of ortho lattices (Definition 9.1 page 140)

⟹ 𝑥⟂ ∨ 𝑦 = 1 ⟂

𝑥≤𝑦 ⟹ 𝑥∧𝑦 ≤𝑦∧𝑦

by monotone property of lattices (Proposition 4.1 page 63) by upper bounded property of bounded lattices (Definition 4.4 page 71)

⟂

by monotone property of lattices (Proposition 4.1 page 63)

⟂

by non-contradiction property of ortho lattices (Definition 9.1 page 140)

⟂

by lower bounded property of bounded lattices (Definition 4.4 page 71)

⟹ 𝑥∧𝑦 ≤0 ⟹ 𝑥∧𝑦 =0

✏ Definition 13.1. 1 Let ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤) be a lattice with negation (Definition 11.5 page 159). The orthogonality relation ⟂∈ 𝟚𝑋𝑋 is defined as d def e 𝑥⟂𝑦 ⟺ 𝑥 ≤ ¬𝑦 f If 𝑥 ⟂ 𝑦, we say that 𝑥 is orthogonal to 𝑦. 1

📘 Stern (1999) page 12, 📘 Loomis (1955) page 3

page 192

13.1. ORTHOGONALITY

Lemma 13.1. Let ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤) be a lattice with negation (Definition 11.5 page 159). l e { 𝑥 ⟂ 𝑦 (orthogonal Definition 13.1 page 191) } ⟹ { 𝑦 ⟂ 𝑥 (symmetric) } m ✎PROOF: 𝑥 ⟂ 𝑦 ⟹ 𝑥 ≤ ¬𝑦

by definition of ⟂ (Definition 13.1 page 191)

⟹ (¬¬𝑦) ≤ ¬𝑥


⟹ 𝑦 ≤ ¬𝑥

by weak double negation property of negation (Definition 11.2 page 158)

⟹ 𝑦⟂𝑥

by definition of ⟂ (Definition 13.1 page 191)

✏ Lemma 13.2. 2 Let ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an orthocomplemented lattice (Definition 9.1 page 140). l 1. 𝑥 ∧ 𝑦 = 0 and e 𝑥⏟ ⟂𝑦 ⟹ ⟂ ⟂ { 2. 𝑥 ∨ 𝑦 = 1 } m orthogonal (Definition 13.1 page 191)

✎PROOF: 𝑥 ⟂ 𝑦 ⟹ 𝑥 ≤ 𝑦⟂

by definition of ⟂ (Definition 13.1 page 191) ⟂

⟹ 𝑥∧𝑦≤𝑦 ∧𝑦 ⟹ 𝑥∧𝑦≤𝑦∧𝑦

by monotone property of lattices (Proposition 4.1 page 63)

⟂


⟹ 𝑥∧𝑦≤0

by non-contradiction property of ortho negation (Definition 11.3 page 158)

⟹ 𝑥∧𝑦=0

by lower bound property of bounded lattices (Definition 4.4 page 71)

𝑥 ⟂ 𝑦 ⟹ 𝑥 ≤ 𝑦⟂ ⟂

by definition of ⟂ (Definition 13.1 page 191) ⟂

⟹ 𝑥 ∨𝑥≤𝑥 ∨𝑦

⟂

⟹ 𝑥 ∨ 𝑥 ⟂ ≤ 𝑥 ⟂ ∨ 𝑦⟂ ⟂

⟹ 1≤𝑥 ∨𝑦 ⟂

⟹ 𝑥 ∨𝑦

⟂

⟂

by monotone property of lattices (Proposition 4.1 page 63) by commutative property of lattices (Theorem 4.3 page 62) by excluded middle property of ortho lattices (Theorem 11.5 page 162) by upper bound property of bounded lattices (Definition 4.4 page 71)

✏ Remark 13.1. In an orthocomplemented lattice 𝙇, the orthogonality relation ⟂ is in general nonassociative. That is 𝑥 ⟂ 𝑦 and ⟹ 𝑥⟂𝑧 / { 𝑦 ⟂ 𝑧 } Consider the 𝙇42 Boolean lattice in Example 9.2 (page 140). 𝑎⟂ ⟂ 𝑝 because 𝑎⟂ ≤ 𝑝⟂ . 𝑝 ⟂ 𝑟 because 𝑝 ≤ 𝑟⟂ . But yet 𝑎⟂ is not orthogonal to 𝑟 because 𝑎⟂ ≰ 𝑟⟂ .

✎PROOF:

✏

Example 13.1. 6! In the 𝑂6 lattice (Definition 9.2 page 140), there are a total of (62) = (6−2)!2! = 6×5 = 15 distinct un2 ordered (the ⟂ relation is symmetric by Lemma 13.1 page 192 so the order doesn't matter) pairs e of elements. x 𝑥 ⟂ 𝑦 𝑥 ⟂ 0 𝑦⟂ ⟂ 0 Of these 15 pairs, 8 are orthogonal to each ⟂ other, and 0 is orthogonal to itself, making a 𝑥 ⟂ 𝑥 𝑦 ⟂ 0 1 ⟂ 0 total of 9 orthogonal pairs: 𝑦 ⟂ 𝑦⟂ 𝑥⟂ ⟂ 0 0 ⟂ 0 2

📘 Holland (1963), page 67

version 0.30


Daniel J. Greenhoe


page 193

CHAPTER 13. RELATIONS ON LATTICES WITH NEGATION

Example 13.2. In lattice 5 of Example 9.2 (page 140), there are a total of (10 2) unordered pairs of elements. 𝑝 ⟂ 𝑝⟂ 𝑥 ⟂ 𝑥⟂ Of these 45 pairs, 18 are orthogoe 𝑝 ⟂ 𝑥⟂ 𝑥 ⟂ 𝑦 x nal to each other, and 0 is orthog𝑝 ⟂ 𝑦 𝑥 ⟂ 𝑧 onal to itself, making a total of 19 𝑝 ⟂ 𝑧 𝑥 ⟂ 0 orthogonal pairs: 𝑝 ⟂ 0 𝑦 ⟂ 𝑦⟂

=

10! (10−2)!2!

𝑦 𝑦 𝑧 𝑧 𝑝⟂

⟂ ⟂ ⟂ ⟂ ⟂

𝑧 0 𝑧⟂ 0 0

10×9 2

=

𝑥⟂ 𝑦⟂ 𝑧⟂ 0

= 45 distinct

⟂ ⟂ ⟂ ⟂

0 0 0 0

Example 13.3. In the ℝ3 Euclidean space illustrated in Example 9.3 (page 141), 𝙓 ⊆ 𝙔⟂ ⟹ 𝙓 ⟂ 𝙔 𝙔 ⊆ 𝙓⟂ ⟹ 𝙔 ⟂ 𝙓 e x 𝙓 ⊆ 𝙕⟂ ⟹ 𝙓 ⟂ 𝙕 𝙔 ⊆ 𝙕⟂ ⟹ 𝙔 ⟂ 𝙕 𝙓 ∧𝙔 =𝙓 ∧𝙕 =𝙔 ∧𝙕 =𝟬

13.2 Commutativity The commutes relation is defined next. Motivation for the name “commutes” is provided by Proposition 13.4 (page 196) which shows that if 𝑥 commutes with 𝑦 in a lattice 𝙇, then 𝑥 and 𝑦 commute in the Sasaki projection 𝜙𝑥 (𝑦) on 𝙇. Definition 13.2. 3 Let 𝙇 ≜ ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤) be a lattice with negation (Definition 11.5 page 159). The commutes relation Ⓒ is defined as def

⟺ 𝑥 = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ¬𝑦) 𝑥Ⓒ𝑦 ∀𝑥,𝑦∈𝑋 , in which case we say, “𝑥 commutes with 𝑦 in 𝙇”. That is, Ⓒ is a relation in 𝟚𝑋𝑋 such that Ⓒ ≜ {(𝑥, 𝑦) ∈ 𝑋 2 |𝑥 = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ¬𝑦)}

d e f

Proposition 13.2. 4 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an orthocomplemented lattice. 𝑥Ⓒ𝑦 ⟺ 𝑥Ⓒ𝑦⟂ ∀𝑥,𝑦∈𝑋 p 𝑥Ⓒ0 and 0Ⓒ𝑥 ∀𝑥∈𝑋 r 𝑥Ⓒ1 and 1Ⓒ𝑥 ∀𝑥∈𝑋 𝑥 ≤ 𝑦 ⟹ 𝑥Ⓒ𝑦 ∀𝑥,𝑦∈𝑋 p 𝑥Ⓒ𝑥 ∀𝑥∈𝑋 𝑥 ⟂ 𝑦 ⟹ 𝑥Ⓒ𝑦 ∀𝑥,𝑦∈𝑋 ✎PROOF: (𝑥 ∧ 0) ∨ (𝑥 ∧ 0⟂ ) = 0 ∨ (𝑥 ∧ 0⟂ )


= 0 ∨ (𝑥 ∧ 1)

by boundary condition of ortho negation (Theorem 11.5 page 162)

= 0 ∨ (𝑥)

by upper bound property of bounded lattices (Definition 4.4 page 71)

=𝑥


⟹ 𝑥Ⓒ0 (0 ∧ 𝑥) ∨ (0 ∧ 𝑥⟂ ) = 0 ∨ (0) =0 ⟹ 0Ⓒ𝑥

by definition of Ⓒ relation (Definition 13.2 page 193) by lower bound property of bounded lattices (Definition 4.4 page 71) by lower bound property of bounded lattices (Definition 4.4 page 71) by definition of Ⓒ relation (Definition 13.2 page 193)

3

📘 Kalmbach (1983) page 20, 📘 Holland (1970), page 79 ⟨A. Commutativity⟩, 📘 Maeda (1958), page 227 ⟨Hilfssatz (Lemma) XII.1.2⟩, 📃 Sasaki (1954) page 301 ⟨Def.5.2, cf Foulis 1962⟩, 📃 Birkhoff (1936b) page 833 ⟨“𝑎 = (𝑎 ∩ 𝑥) ∪ (𝑎 ∩ 𝑥′ )”⟩ 4

📘 Holland (1963), page 67


Daniel J. Greenhoe


version 0.30

page 194

13.2. COMMUTATIVITY

(𝑥 ∧ 1) ∨ (𝑥 ∧ 1⟂ ) = 𝑥 ∨ (𝑥 ∧ 1⟂ )


= 𝑥 ∨ (𝑥 ∧ 0)

by boundary condition of ortho negation (Theorem 11.5 page 162)

= (𝑥) ∨ (0)


=𝑥


⟹ 𝑥Ⓒ1

by definition of Ⓒ relation (Definition 13.2 page 193)

(1 ∧ 𝑥) ∨ (1 ∧ 𝑥⟂ ) = (𝑥) ∨ (𝑥⟂ )

by non-contradiction prop. of ortho negation (Definition 11.3 page 158)

=1

by excluded middle property of ortho negation (Theorem 11.5 page 162)

⟹ 1Ⓒ𝑥

by definition of Ⓒ relation (Definition 13.2 page 193)

⟂

⟂

(𝑥 ∧ 𝑥) ∨ (𝑥 ∧ 𝑥 ) = 𝑥 ∨ (𝑥 ∧ 𝑥 ) = 𝑥 ∨ (0)

by idempotent property of lattices (Theorem 4.3 page 62) by non-contradiction prop. of ortho negation (Definition 11.3 page 158)

=𝑥


⟹ 𝑥Ⓒ𝑥

by definition of Ⓒ relation (Definition 13.2 page 193) ⟂

𝑥Ⓒ𝑦 ⟹ (𝑥 ∧ 𝑦 ) ∨ (𝑥 ∧ 𝑦 = (𝑥 ∧ 𝑦⟂ ) ∨ (𝑥 ∧ 𝑦)

⟂⟂

= (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦⟂ )


=𝑥

by 𝑥Ⓒ𝑦 hypothesis and Definition 13.2 page 193

⟹ 𝑥Ⓒ𝑦⟂ 𝑥Ⓒ𝑦

⟂

) by definition of Ⓒ (Definition 13.2 page 193) by involution property of ⟂ (Definition 9.1 page 140)


⟹ (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦 ) = (𝑥 ∧ 𝑦⟂⟂ ) ∨ (𝑥 ∧ 𝑦⟂ ) ⟂

= (𝑥 ∧ 𝑦 ) ∨ (𝑥 ∧ 𝑦 =𝑥

⟂⟂

)

by definition of Ⓒ (Definition 13.2 page 193) by involution property of ⟂ (Definition 9.1 page 140) by commutative property of lattices (Definition 4.3 page 61) by 𝑥Ⓒ𝑦⟂ hypothesis and Definition 13.2 page 193

⟹ 𝑥Ⓒ𝑦


𝑥 ≤ 𝑦 ⟹ (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦 ) = 𝑥 ∨ (𝑥 ∧ 𝑦⟂ ) =𝑥

by definition of Ⓒ (Definition 13.2 page 193) by 𝑥 ≤ 𝑦 hypothesis by absorptive property (Theorem 4.3 page 62)

⟹ 𝑥Ⓒ𝑦

by definition of Ⓒ (Definition 13.2 page 193) ⟂

𝑥 ⟂ 𝑦 ⟹ (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦 ) = 0 ∨ (𝑥 ∧ 𝑦 ⟂ )

by definition of Ⓒ (Definition 13.2 page 193) by Lemma 13.2 page 192

=0∨𝑥

by 𝑥 ⟂ 𝑦 hypothesis (𝑥 ⟂ 𝑦 ⟹ 𝑥 ≤ 𝑦⟂ )

=𝑥∨0


=𝑥

by identity property of bounded lattices

⟹ 𝑥Ⓒ𝑦

by definition of Ⓒ (Definition 13.2 page 193)

✏ Definition 13.3. Let Ⓒ be the commutes relation (Definition 13.2 page 193) on a lattice with negation 𝙇 ≜ ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤) (Definition 11.5 page 159). d 𝙇 is symmetric if e 𝑥Ⓒ𝑦 ⟹ 𝑦Ⓒ𝑥 ∀𝑥,𝑦∈𝑋 f In general, the commutes relation is not symmetric. But Proposition 13.3 (next) describes some conditions under which it is symmetric. Proposition 13.3. 5

5

Let ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an orthocomplemented lattice (Definition 9.1 page 140).

📘 Holland (1963) page 68, 📃 Nakamura (1957) page 158

version 0.30


Daniel J. Greenhoe


page 195


p r p

{𝑥Ⓒ𝑦 ⟹ 𝑦Ⓒ𝑥} ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ Ⓒ is symmetric at (𝑥, 𝑦) (1)

⟂ {𝑥 ≤ 𝑦 ⟹ 𝑦 = 𝑥 ∨ (𝑥 ∧ 𝑦)} ⟂ {𝑥 ≤ 𝑦 ⟹ 𝑥 = 𝑦 ∧ (𝑥 ∨ 𝑦 )} ⟂ {𝑦 = (𝑥 ∧ 𝑦) ∨ [𝑦 ∧ (𝑥 ∧ 𝑦) ]} ⟂ {𝑥 = (𝑥 ∨ 𝑦) ∧ [𝑥 ∨ (𝑥 ∨ 𝑦) ]}

⟺ ⟺ ⟺ ⟺

(orthomodular identity)

(2)

(𝑥 = 𝜙𝑦 (𝑥) (Sasaki projection) )

(3) (4) (5)

✎PROOF: 1. Proof that (2) ⟺ (3): 𝑥 ≤ 𝑦 ⟹ 𝑦⟂ ≤ 𝑥⟂


⟹ 𝑥⟂ = 𝑦⟂ ∨ (𝑦⟂⟂ ∧ 𝑥⟂ ) ⟂ ⟂

⟂

⟹ (𝑥 ) = [𝑦 ∨ (𝑦

⟂⟂

by left hypothesis ⟂

⟂

∧ 𝑥 )]

⟹ 𝑥 = [𝑦⟂ ∨ (𝑦⟂⟂ ∧ 𝑥⟂ )]

⟂


⟂ ⟂

= 𝑦⟂⟂ ∧ (𝑦⟂⟂ ∧ 𝑥 )


⟂ ⟂

= 𝑦 ∧ (𝑦 ∧ 𝑥 ) = 𝑦 ∧ (𝑦⟂ ∨ 𝑥⟂⟂ )


⟂



= 𝑦 ∧ (𝑦 ∨ 𝑥 ) = 𝑦 ∧ (𝑥 ∨ 𝑦 ⟂ )


𝑥 ≤ 𝑦 ⟹ 𝑦⟂ ≤ 𝑥⟂


⟹ 𝑦⟂ = 𝑥⟂ ∧ (𝑦⟂ ∨ 𝑥⟂⟂ ) ⟂ ⟂

⟂

by right hypothesis

⟂

⟂⟂

⟂⟂

⟂

⟹ (𝑦 ) = [𝑥 ∧ (𝑦 ∨ 𝑥 ⟂

⟂

⟹ 𝑦 = [𝑥 ∧ (𝑦 ∨ 𝑥 = 𝑥⟂⟂ ∨ (𝑦⟂ ∨ 𝑥

)]

⟂

)]


⟂⟂ ⟂


)

⟂


⟂

∧𝑥 ) = 𝑥 ∨ (𝑦 ∧ 𝑥 ⟂ )


= 𝑥 ∨ (𝑥⟂ ∧ 𝑦)


= 𝑥 ∨ ( 𝑦⟂ ∨ 𝑥 ) = 𝑥 ∨ (𝑦

⟂⟂


2. Proof that (2) ⟺ (4): (𝑥𝑦) ∨ [𝑦(𝑥𝑦)⟂ ] = 𝑢 ∨ [𝑦𝑢⟂ ] ⟂

where 𝑢 ≜ 𝑥𝑦 ≤ 𝑦

= 𝑢 ∨ [𝑢 𝑦]


=𝑦

by left hypothesis

𝑥 ≤ 𝑦 ⟹ 𝑥 ∨ (𝑥⟂ 𝑦) = 𝑥𝑦 ∨ [(𝑥𝑦)⟂ 𝑦]


= 𝑥𝑦 ∨ [𝑦(𝑥𝑦)⟂ ]


=𝑦

by right hypothesis

3. Proof that (3) ⟺ (5): (𝑥 ∨ 𝑦)[𝑥 ∨ (𝑥 ∨ 𝑦)⟂ ] = 𝑢[𝑥 ∨ 𝑢⟂ ]

where 𝑥 ≤ 𝑢 ≜ 𝑥 ∨ 𝑦

=𝑥

by left hypothesis

𝑥 ≤ 𝑦 ⟹ 𝑦(𝑥 ∨ 𝑦⟂ ) = (𝑥 ∨ 𝑦)[𝑥 ∨ (𝑥 ∨ 𝑦)⟂ ] =𝑥 2016 December 27 Tuesday UTC

Daniel J. Greenhoe

by 𝑥 ≤ 𝑦 hypothesis by right hypothesis


version 0.30

page 196

13.3. CENTER

4. Proof that (1) ⟹ (2): 𝑥 ≤ 𝑦 ⟹ 𝑥Ⓒ𝑦

by Proposition 13.2 page 193

⟹ 𝑦Ⓒ𝑥

by symmetry hypothesis (left hypothesis)

⟹ 𝑦 = (𝑦 ∧ 𝑥) ∨ (𝑦 ∧ 𝑥⟂ )


⟂

⟹ 𝑦 = 𝑥 ∨ (𝑦 ∧ 𝑥 ) ⟹ 𝑦 = 𝑥 ∨ (𝑥⟂ ∧ 𝑦)

by 𝑥 ≤ 𝑦 hypothesis by commutative property of lattices (Theorem 4.3 page 62)

5. Proof that (2) ⟹ (4): (a) lemma: proof that 𝑥Ⓒ𝑦 ⟹ 𝑥⟂ 𝑦 = (𝑥𝑦)⟂ 𝑦: ⟂

𝑥Ⓒ𝑦 ⟹ 𝑥⟂ 𝑦 = (𝑥𝑦 ∨ 𝑥𝑦⟂ ) 𝑦 ⟂


⟂ ⟂

= (𝑥𝑦) (𝑥𝑦 ) 𝑦

by de Morgan's law (Theorem 11.4 page 162) ⟂

= (𝑥𝑦)⟂ [(𝑥⟂ ∨ 𝑦⟂ )𝑦]

by de Morgan's law (Theorem 11.4 page 162)

= (𝑥𝑦)⟂ [(𝑥⟂ ∨ 𝑦)𝑦]

by involutory's property (Definition 9.1 page 140)

⟂

= (𝑥𝑦) 𝑦


(b) Completion of proof for (2) ⟹ (4): 𝑥Ⓒ𝑦 ⟹ 𝑥𝑦 ∨ 𝑦(𝑥𝑦)⟂ = 𝑥𝑦 ∨ (𝑥𝑦)⟂ 𝑦 ⟂

= 𝑥𝑦 ∨ 𝑥 𝑦

by commutative property (Theorem 4.3 page 62) by 𝑥Ⓒ𝑦 hypothesis and item (5a)

⟂

= (𝑦𝑥) ∨ [𝑦𝑥 ] ⟹ 𝑦Ⓒ𝑥

by commutative property (Theorem 4.3 page 62) by definition of Ⓒ (Definition 13.2 page 193)

✏ Theorem 13.1. 6 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an orthocomplemented lattice (Definition 9.1 page 140). ⟂ t {𝑥Ⓒ𝑐 ∀𝑥 ∈ 𝑋} ⟺ {𝙇 is isomorphic to [0 ∶ 𝑐] × [0 ∶ 𝑐 ]} h with isomorphism 𝜃(𝑥) ≜ ([0 ∶ 𝑐], [0 ∶ 𝑐 ⟂ ]). m Proposition 13.4. p r 𝑥Ⓒ𝑦 ⟺ p

13.3

7

Let ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an orthomodular lattice. 𝜙𝑥 (𝑦) = 𝜙𝑦 (𝑥) = 𝑥 ∧ 𝑦

∀𝑥,𝑦∈𝑋

Center

An element in an orthocomplemented lattice (Definition 9.1 page 140) is in the center of the lattice if that element commutes (Definition 13.2 page 193) with every other element in the lattice (next definition). All the elements of an orthocomplemented lattice are in the center if and only if that lattice is Boolean (Proposition 9.2 page 146). Definition 13.4. 8 Let Ⓒ be the commutes relation (Definition 13.2 page 193) on a lattice with negation 𝙇 ≜ ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤) (Definition 11.5 page 159). d The center of 𝙇 is defined as e {𝑥 ∈ 𝑋|𝑥Ⓒ𝑦 ∀𝑦 ∈ 𝑋} f 6

📘 Kalmbach (1983) page 20, 📃 MacLaren (1964) 📃 Foulis (1962) page 66, 📃 Sasaki (1954) ⟨cf Foulis 1962⟩ 8 📘 Holland (1970), page 80 7

version 0.30


Daniel J. Greenhoe


page 197


Proposition 13.5. Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an orthocomplemented lattice (Definition 9.1 page 140). p r 0 and 1 are in the center of 𝙇. p ✎PROOF:

✏

This follows directly from Definition 13.2 (page 193) and Proposition 13.2 (page 193).

Theorem 13.2. 9 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an orthocomplemented lattice (Definition 9.1 page 140). t h The center of 𝙇 is boolean (Definition 8.1 page 115). m Example 13.4. The center of the 𝑂6 lattice (Definition 9.2 page 140) is the set {0, 𝑥, 𝑧, 1}. The ⟂ ⟂ e elements 𝑥 and 𝑧 are not in the center of 𝙇. The 𝑂6 lattice is illusx trated to the right, with the center elements as solid dots. Note that the center is the Boolean lattice 𝙇22 (Proposition 9.2 page 146).

1 𝑧⟂ 𝑥

𝑥⟂ 𝑧 0

✎PROOF: 1. Proof that 0 and 1 are in the center of 𝙇: by Proposition 13.5 (page 197). 2. Proof that 𝑥 is in the center of 𝙇: (𝑥 ∧ 𝑥) ∨ (𝑥 ∧ 𝑥⟂ ) = 𝑥 ∨ 0

=𝑥

⟹ 𝑥Ⓒ𝑥

(𝑥 ∧ 𝑧) ∨ (𝑥 ∧ 𝑧⟂ ) = 0 ∨ 𝑥

=𝑥

⟹ 𝑥Ⓒ𝑧

𝑥Ⓒ𝑥, 𝑥Ⓒ𝑥⟂ , 𝑥Ⓒ𝑧⟂ , 𝑥Ⓒ0, and 𝑥Ⓒ1 by Proposition 13.2 (page 193). 3. Proof that 𝑧 is in the center of 𝙇: (𝑧 ∧ 𝑧) ∨ (𝑧 ∧ 𝑧⟂ ) = 𝑧 ∨ 0

=𝑧

⟹ 𝑧Ⓒ𝑧

(𝑧 ∧ 𝑥) ∨ (𝑧 ∧ 𝑥⟂ ) = 0 ∨ 𝑧

=𝑧

⟹ 𝑧Ⓒ𝑥

𝑧Ⓒ𝑧, 𝑧Ⓒ𝑥⟂ , 𝑧Ⓒ𝑧⟂ , 𝑧Ⓒ0, and 𝑧Ⓒ1 by Proposition 13.2 (page 193). 4. Proof that 𝑥⟂ and 𝑧⟂ are not in the center of 𝙇: (𝑥⟂ ∧ 𝑦) ∨ (𝑥⟂ ∧ 𝑦⟂ ) = 𝑦 ∨ 0

=𝑦

⟹ 𝑥⟂ Ⓒ𝑦 /

(𝑧⟂ ∧ 𝑥) ∨ (𝑧⟂ ∧ 𝑥⟂ ) = 𝑥 ∨ 0

=𝑥

⟹ 𝑧⟂ Ⓒ𝑥 /

✏ Example 13.5. The center the lattice illustrated to the right (Example 9.2 page 140), with center elements as solid dots, is the set {0, 1, 𝑝, 𝑦, 𝑧, 𝑥⟂ , 𝑦⟂ , 𝑧⟂ , }. The elements 𝑥 and 𝑝⟂ are not in the center of 𝙇. Note that the center is the Boolean lattice 𝙇32 (Proposition 9.2 page 146).

e x

1 ⟂

𝑧

𝑝 𝑥

𝑦⟂ 𝑦

𝑥⟂ 𝑝⟂ 𝑧

0

✎PROOF: 9

📃

Jeffcott (1972) page 645 ⟨§5. Main theorem⟩


Daniel J. Greenhoe


version 0.30

page 198

13.3. CENTER

1. Proof that 0 and 1 are in the center of 𝙇: by Proposition 13.5 (page 197). 2. Proof that 𝑥 is in the center of 𝙇: (𝑥 ∧ 𝑝) ∨ (𝑥 ∧ 𝑝⟂ ) = 𝑥 ∨ 0

=𝑥

⟹ 𝑥Ⓒ𝑝

(𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦⟂ ) = 0 ∨ 𝑥

=𝑥

⟹ 𝑥Ⓒ𝑦

=𝑥

⟹ 𝑥Ⓒ𝑧

⟂

(𝑥 ∧ 𝑧) ∨ (𝑥 ∧ 𝑧 ) = 0 ∨ 𝑥

𝑥Ⓒ𝑥, 𝑥Ⓒ𝑥⟂ , 𝑥Ⓒ𝑝⟂ , 𝑥Ⓒ𝑦⟂ , 𝑥Ⓒ𝑧⟂ , 𝑥Ⓒ0, and 𝑥Ⓒ1 by Proposition 13.2 (page 193). 3. Proof that 𝑦 is in the center of 𝙇: (𝑦 ∧ 𝑥) ∨ (𝑦 ∧ 𝑥⟂ ) = 0 ∨ 𝑦

=𝑦

⟹ 𝑦Ⓒ𝑥

(𝑦 ∧ 𝑝) ∨ (𝑦 ∧ 𝑝⟂ ) = 0 ∨ 𝑦

=𝑦

⟹ 𝑦Ⓒ𝑝

=𝑦

⟹ 𝑦Ⓒ𝑧

⟂

(𝑦 ∧ 𝑧) ∨ (𝑦 ∧ 𝑧 ) = 0 ∨ 𝑦

𝑦Ⓒ𝑦, 𝑦Ⓒ𝑥⟂ , 𝑦Ⓒ𝑝⟂ , 𝑦Ⓒ𝑦⟂ , 𝑦Ⓒ𝑧⟂ , 𝑦Ⓒ0, and 𝑦Ⓒ1 by Proposition 13.2 (page 193). 4. Proof that 𝑧 is in the center of 𝙇: (𝑧 ∧ 𝑥) ∨ (𝑧 ∧ 𝑥⟂ ) = 0 ∨ 𝑧

=𝑧

⟹ 𝑧Ⓒ𝑥

⟂

=𝑧

⟹ 𝑧Ⓒ𝑝

⟂

=𝑧

⟹ 𝑧Ⓒ𝑦

(𝑧 ∧ 𝑝) ∨ (𝑧 ∧ 𝑝 ) = 0 ∨ 𝑧 (𝑧 ∧ 𝑦) ∨ (𝑧 ∧ 𝑦 ) = 0 ∨ 𝑧

𝑧Ⓒ𝑧, 𝑧Ⓒ𝑥⟂ , 𝑧Ⓒ𝑝⟂ , 𝑧Ⓒ𝑦⟂ , 𝑧Ⓒ𝑧⟂ , 𝑧Ⓒ0, and 𝑧Ⓒ1 by Proposition 13.2 (page 193). 5. Proof that 𝑥⟂ is in the center of 𝙇: (𝑝⟂ ∧ 𝑥) ∨ (𝑝⟂ ∧ 𝑥⟂ ) = 0 ∨ 𝑝⟂

= 𝑝⟂

⟹ 𝑝⟂ Ⓒ𝑥

(𝑝⟂ ∧ 𝑦) ∨ (𝑝⟂ ∧ 𝑦⟂ ) = 𝑦 ∨ 𝑧

= 𝑝⟂

⟹ 𝑝⟂ Ⓒ𝑦

(𝑝⟂ ∧ 𝑧) ∨ (𝑝⟂ ∧ 𝑧⟂ ) = 𝑧 ∨ 𝑦

= 𝑝⟂

⟹ 𝑝⟂ Ⓒ𝑧

𝑝⟂ Ⓒ𝑥⟂ , 𝑝⟂ Ⓒ𝑝⟂ , 𝑝⟂ Ⓒ𝑦⟂ , 𝑝⟂ Ⓒ𝑧⟂ , 𝑝⟂ Ⓒ0, and 𝑝⟂ Ⓒ1 by Proposition 13.2 (page 193). 6. Proof that 𝑦⟂ is in the center of 𝙇: (𝑦⟂ ∧ 𝑥) ∨ (𝑦⟂ ∧ 𝑥⟂ ) = 𝑥 ∨ 𝑧

= 𝑦⟂

⟹ 𝑦⟂ Ⓒ𝑥

(𝑦⟂ ∧ 𝑝) ∨ (𝑦⟂ ∧ 𝑝⟂ ) = 𝑝 ∨ 𝑧

= 𝑦⟂

⟹ 𝑦⟂ Ⓒ𝑝 /

(𝑦⟂ ∧ 𝑧) ∨ (𝑦⟂ ∧ 𝑧⟂ ) = 𝑧 ∨ 𝑝

= 𝑦⟂

⟹ 𝑦⟂ Ⓒ𝑧

𝑝⟂ Ⓒ𝑥⟂ , 𝑝⟂ Ⓒ𝑝⟂ , 𝑝⟂ Ⓒ𝑦⟂ , 𝑝⟂ Ⓒ𝑧⟂ , 𝑝⟂ Ⓒ0, and 𝑝⟂ Ⓒ1 by Proposition 13.2 (page 193). 7. Proof that 𝑧⟂ is in the center of 𝙇: (𝑧⟂ ∧ 𝑥) ∨ (𝑧⟂ ∧ 𝑥⟂ ) = 𝑥 ∨ 𝑦

= 𝑧⟂

⟹ 𝑧⟂ Ⓒ𝑥

(𝑧⟂ ∧ 𝑝) ∨ (𝑧⟂ ∧ 𝑝⟂ ) = 𝑝 ∨ 𝑦

= 𝑧⟂

⟹ 𝑦⟂ Ⓒ𝑝 /

(𝑧⟂ ∧ 𝑦) ∨ (𝑧⟂ ∧ 𝑦⟂ ) = 𝑧 ∨ 𝑝

= 𝑧⟂

⟹ 𝑦⟂ Ⓒ𝑧

𝑧⟂ Ⓒ𝑥⟂ , 𝑧⟂ Ⓒ𝑝⟂ , 𝑧⟂ Ⓒ𝑦⟂ , 𝑧⟂ Ⓒ𝑧⟂ , 𝑧⟂ Ⓒ0, and 𝑧⟂ Ⓒ1 by Proposition 13.2 (page 193). version 0.30


Daniel J. Greenhoe


page 199


8. Proof that 𝑝 and 𝑥⟂ are not in the center of 𝙇: (𝑝 ∧ 𝑥) ∨ (𝑝 ∧ 𝑥⟂ ) = 𝑥 ∨ 0 ⟂

⟂

⟂

(𝑥 ∧ 𝑝) ∨ (𝑥 ∧ 𝑝 ) = 0 ∨ 𝑝

⟂

=𝑥 =𝑝

⟂

⟹ 𝑝Ⓒ𝑥 / ⟹ 𝑥⟂ Ⓒ𝑝 /

✏ Example 13.6. 1 ⟂

e x

The center of the lattice illustrated to the right is illustrated with solid dots. Note that the center is the Boolean lattice 𝙇22 (Proposition 9.2 page 146).

𝑧

𝑝 𝑥

𝑦⟂ 𝑦

𝑥⟂ 𝑝⟂ 𝑧

0

Example 13.7.

e x

In a Boolean lattice, such as the one illustrated to the right, every element is in the center (Proposition 9.2 page 146).

𝑧⟂ 𝑥

1 𝑦⟂ 𝑦

𝑥⟂ 𝑧

0


Daniel J. Greenhoe


version 0.30

page 200

version 0.30

13.3. CENTER


Daniel J. Greenhoe


CHAPTER 14 SET STRUCTURES

14.1 General set structures Similar to the definition of a relation on a set 𝑋 as being any subset of the Cartesian product 𝑋 × 𝑋 𝑋 (Definition 15.1 page 237), a set structure on a set 𝑋 is simply any subset of the power set 𝟚 (Definition 14.3 page 201) of the set 𝑋 (next definition and Figure 14.1 page 202). Definition 14.1. 1 Let 𝟚𝑋 be the power set (Definition 14.3 page 201) of a set 𝑋. d A set 𝒮(𝑋) is a set structure on 𝑋 if 𝒮(𝑋) ⊆ 𝟚𝑋 . e f A set structure 𝒬(𝑋) is a paving on 𝑋 if ∅ ∈ 𝒬(𝑋). Definition 14.2. 2 Let 𝒬(𝑋) be a paving (Definition 14.1 page 201) on a set 𝑋. Let 𝑌 be a set containing the element 0. d A function 𝗆 ∈ 𝑌 𝒬(𝑋) is a set function if e 𝗆(∅) = 0. f

14.2 Operations on the power set 14.2.1 Standard operations Definition 14.3. d The power set 𝟚𝑋 on a set 𝑋 is defined as e 𝟚𝑋 ≜ {𝐴|𝐴 ⊆ 𝑋} (the set of all subsets of 𝑋) f Definition 14.4. page 249). 1

3

Let 𝟚𝑋 be a set. Let | 𝑋 | be a function in the function space [0 ∶ +∞]𝑋

(Definition 15.8

📘 Molchanov (2005) page 389, 📘 Pap (1995) page 7, 📘 Hahn and Rosenthal (1948) page 254 📘 Pap (1995) page 8 ⟨Definition 2.3: extended real-valued set function⟩, 📘 Halmos (1950) page 30 ⟨§7. MEASURE ON RINGS⟩, 📘 Hahn and Rosenthal (1948), 📃 Choquet (1954) 3 📘 Tao (2011) page 12 ⟨Example 3.6⟩, 📘 Tao (2010) page 7 ⟨Example 1.1.14⟩ 2

page 202

14.2. OPERATIONS ON THE POWER SET

set structures (Definition 14.1 page 201)

paving

partition



topology

σ-ring



topology on finite set

σ-algebra

ring of sets




algebra of sets (Definition 14.9 page 214)

power set (Definition 14.3 page 201)

Figure 14.1: some standard set structures 𝑋

𝑋

𝐵

𝐴

𝑋

𝐵

𝐴 △𝐵

𝐵

𝐴

𝐴𝖼

∅

𝐴

𝑋

𝐵

𝐴∩𝐵

𝐵

𝐴

𝐵𝖼

𝐴⧵𝐵 𝑋

𝐴

𝑋

𝑋

𝐵

𝐴

𝐴∪𝐵

𝑋

𝐵

𝐴

𝑋

Figure 14.2: Venn diagrams for standard set operations (Definition 14.5 page 202) | 𝑋 | is the cardinality or order of 𝑋 if number of elements in 𝑋 if 𝑋 is finite |𝑋 | ≜ { +∞ otherwise

d e f

Definition 14.5 (next) introduces seven standard set operations: two nullary operations, one unary operation, and four binary operations (Definition 15.9 page 250). Definition 14.5. 4 Let 𝟚𝑋 be the power set (Definition 14.3 page 201) on a set 𝑋. Let ¬ represent the logical not operation, ∨ represent the logical or operation, ∧ represent the logical and operation (Definition 12.2 page 179), and ⊕ represent the logical exclusive-or operation (Definition 12.5 page 185). 4

📘 Aliprantis and Burkinshaw (1998), pages 2–4

version 0.30


Daniel J. Greenhoe


page 203

CHAPTER 14. SET STRUCTURES

d e f

name/symbol emptyset universal set complement union intersection difference symmetric difference

∅ 𝑋 𝖼 ∪ ∩ ⧵ △

arity 0 0 1 2 2 2 2

∅ 𝑋 𝐴𝖼 𝐴∪𝐵 𝐴∩𝐵 𝐴⧵ 𝐵 𝐴 △𝐵

definition domain ≜ {𝑥 ∈ 𝑋 | 𝑥≠𝑥 } ≜ {𝑥 ∈ 𝑋 | 𝑥=𝑥 } ≜ {𝑥 ∈ 𝑋 | ¬(𝑥 ∈ 𝐴) } ∀𝐴∈𝟚𝑋 ≜ {𝑥 ∈ 𝑋 | (𝑥 ∈ 𝐴) ∨ (𝑥 ∈ 𝐵)} ∀𝐴,𝐵∈𝟚𝑋 ≜ {𝑥 ∈ 𝑋 | (𝑥 ∈ 𝐴) ∧ (𝑥 ∈ 𝐵)} ∀𝐴,𝐵∈𝟚𝑋 ≜ {𝑥 ∈ 𝑋 | (𝑥 ∈ 𝐴) ∧ ¬(𝑥 ∈ 𝐵)} ∀𝐴,𝐵∈𝟚𝑋 ≜ {𝑥 ∈ 𝑋 | (𝑥 ∈ 𝐴) ⊕ (𝑥 ∈ 𝐵)} ∀𝐴,𝐵∈𝟚𝑋

With regards to the standard seven set operations only, Theorem 14.1 (next) expresses each of the set operations in terms of pairs of other operations. Theorem 14.1. 𝑋 = ∅ = 𝑋 = 𝖼 𝐴 = t h 𝐴∪𝐵 = m 𝐴∩𝐵 = 𝐴⧵𝐵 = 𝐴△𝐵 = =

∅𝖼 𝑋 𝖼 = (𝐴 ∪ 𝐴𝖼 )𝖼 = 𝐴 ∩ 𝐴𝖼 𝐴 ∪ 𝐴𝖼 = (𝐴 ∩ 𝐴𝖼 )𝖼 𝑋 ⧵𝐴 = 𝑋 △𝐴 𝖼 𝖼 𝖼 = (𝐴△𝐵)△(𝐴 ∩ 𝐵) (𝐴 ∩ 𝐵 ) 𝖼 𝖼 𝖼 = (𝐴 ∪ 𝐵)△𝐴△𝐵 (𝐴 ∪ 𝐵 ) 𝖼 𝖼 𝐴 ∪ 𝐵 = 𝐴 ∩ 𝐵𝖼 ( ) 𝖼 𝖼 𝖼 𝖼 [(𝐴 ∪ 𝐵 ) ] ∪ [(𝐴 ∪ 𝐵 ) ] (𝐴⧵𝐵) ∪ (𝐵⧵𝐴)

= 𝐴⧵𝐴

= = = =

= 𝐴 △𝐴

(𝐴⧵𝐵)△𝐵 𝐴⧵(𝐴⧵𝐵) (𝐴 ∪ 𝐵)△𝐵 = (𝐴△𝐵) ∩ 𝐴 𝖼 𝖼 𝖼 𝖼 [(𝐴 ∩ 𝐵 ) ] ∩ (𝐴 ∩ 𝐵)

Proposition 14.1. Let 𝑋 be a set and 𝟚𝑋 the power set of 𝑋. Let 𝑹 ⊆ 𝑋 such that 𝑹 is closed with respect to the set symmetric difference operator △. (𝑹, △) is a group. In particular, p 1. ∅△𝐴 = 𝐴△∅ = 𝐴 ∀𝐴∈𝑹 (∅ is the identity element) r 2. 𝐴△𝐴 = ∅ ∀𝐴∈𝑹 (𝐴 is the inverse of 𝐴) p 3. 𝐴△(𝐵 △𝐶) = (𝐴△𝐵)△𝐶 ∀𝐴,𝐵,𝐶∈𝑹 (associative) ✎PROOF: Proof that ∅ is the identity element: 1a. Proof that ∅ ∈ 𝑹: ∅ = 𝐴 △𝐴

△

closed with respect to 𝑹

∈𝑹 1b. Proof that ∅△𝐴 = 𝐴: ∅△𝐴 = {𝑥 ∈ 𝑋|(𝑥 ∈ ∅) ⊕ (𝑥 ∈ 𝐴)} = {𝑥 ∈ 𝑋|(𝑥 ∈ {𝑥 ∈ 𝑋|𝑥 ≠ 𝑥}) ⊕ (𝑥 ∈ 𝐴)}

by definition of △ page 202 by definition of △ page 202

= {𝑥 ∈ 𝑋|0 ⊕ (𝑥 ∈ 𝐴)} = {𝑥 ∈ 𝑋|(𝑥 ∈ 𝐴)}

by definition of ⊕ (Definition 12.1 page 185)

=𝐴 1c. Proof that 𝐴△∅ = 𝐴: 𝐴△∅ = {𝑥 ∈ 𝑋|(𝑥 ∈ 𝐴) ⊕ (𝑥 ∈ ∅)} = {𝑥 ∈ 𝑋|(𝑥 ∈ 𝐴) ⊕ (𝑥 ∈ {𝑥 ∈ 𝑋|𝑥 ≠ 𝑥})}

by definition of △ page 202 by definition of △ page 202

= {𝑥 ∈ 𝑋|(𝑥 ∈ 𝐴) ⊕ 0} = {𝑥 ∈ 𝑋|(𝑥 ∈ 𝐴)}

by definition of ⊕ (Definition 12.1 page 185)

=𝐴 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 204


2. Proof that 𝐴△𝐴:

𝐴△𝐴 = {𝑥 ∈ 𝑋|(𝑥 ∈ 𝐴) ⊕ (𝑥 ∈ 𝐴)}

by definition of △ page 202

= {𝑥 ∈ 𝑋|0}


=∅


3. Proof that 𝐴△(𝐵 △𝐶) = (𝐴△𝐵)△𝐶:

𝐴△(𝐵 △𝐶) = {𝑥 ∈ 𝑋|(𝑥 ∈ 𝐴) ⊕ [𝑥 ∈ (𝐵 △𝐶)]}


= {𝑥 ∈ 𝑋|(𝑥 ∈ 𝐴) ⊕ [(𝑥 ∈ 𝐵) ⊕ (𝑥 ∈ 𝐶)]}


= {𝑥 ∈ 𝑋|[(𝑥 ∈ 𝐴) ⊕ (𝑥 ∈ 𝐵)] ⊕ (𝑥 ∈ 𝐶)} = (𝐴△𝐵)△𝐶

✏

14.2.2

Non-standard operations

Two subsets 𝐴 and 𝐵 of a set 𝑋 that are intersecting but yet one is not contained in the other, partition the set 𝑋 into four regions: 𝑋

𝐵

𝐴

𝑋

𝐴

𝐵

𝑋

𝐵

𝐴

𝑋

𝐴

𝐵

𝐴∩𝐵 𝐴 ∩ 𝐵𝖼 𝐴𝖼 ∩ 𝐵 𝐴𝖼 ∩ 𝐵 𝖼 Because there are four regions, the number of ways we can select one or more of them is 24 = 16. Therefore, a binary operator on sets 𝐴 and 𝐵 can likewise result in one of 24 = 16 possibilities. Definition 14.6 (page 204) presents 7 set operations. Therefore, there should be an additional 16 − 7 = 9 operations. Definition 14.6 (next definition) attempts to define these additional operations. Some definitions are adapted from logic (Table 12.3 page 186). But in general these definitions are non-standard definitions with respect to set theory. The 16 set operations under the inclusion relation ⊆ form a lattice; this lattice is illustrated by a Hasse diagram in Figure 14.3 (page 205).

Definition 14.6.

5

standard ops: rejection:

version 0.30

5

Let 𝟚𝑋 be the power set on a set 𝑋. For any sets 𝐴, 𝐵 ∈ 𝟚𝑋 , let 𝐴𝐵 ≜ (𝐴 ∩ 𝐵).

📘 Aliprantis and Burkinshaw (1998) pages 2–4 Sets, Relations, and Order Structures

Daniel J. Greenhoe


page 205


𝑋

𝐵

𝐴 𝑋

𝑋

𝐵

𝐴

𝑋

𝐵

𝑋

𝐵

𝐴

𝐴𝖼𝑦 𝐵 ≡ 𝐵 𝖼

𝐴◨𝐵 = 𝐴

𝐴 𝐴⧵𝐵

𝐵

𝐴

𝑋

𝐴⇒𝐵

𝐵

𝐴

𝐴⇔𝐵

𝑋

𝑋

𝐵

𝐵 𝐴 𝐴 ↓ 𝐵 = 𝐴𝖼 ∩ 𝐵 𝖼

𝐵 𝐴∩𝐵

𝑋

𝐵

𝐴 𝐴𝖼𝑥 𝐵 ≡ 𝐴𝖼

𝐴◧𝐵 = 𝐵

𝑋

𝐴

𝐵

𝐴

𝐴△𝐵

𝑋

𝐵

𝐴

𝑋

𝐵

𝐴

𝑋

𝐴∪𝐵

𝐴|𝐵

𝑋

𝐴

𝐵

𝐴

𝐴÷𝐵

𝑋

𝑋

𝐵 𝐴 𝐴 ⊖ 𝐵 = 𝐵⧵𝐴 = 𝐵⧵𝐴

𝑋

∅

Figure 14.3: lattice of set operations


Daniel J. Greenhoe


version 0.30

page 206


name/symbol empty set rejection inhibit 𝑥 complement 𝑥 difference complement 𝑦 symmetric difference Sheffer stroke intersection equivalence projection 𝑦 implication projection 𝑥 adjunction union universal set

d e f

14.2.3

⊘ ↓ ⊖ 𝖼𝑥 ⧵ 𝖼𝑦 △

| ∩ ⇔ ◧ ⇒ ◨ ÷ ∪ ⊗

arity 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

𝐴⊘𝐵 𝐴↓𝐵 𝐴⊖𝐵 𝐴 𝖼𝑥 𝐵 𝐴⧵ 𝐵 𝐴 𝖼𝑦 𝐵 𝐴△𝐵 𝐴 | 𝐵 𝐴∩𝐵 𝐴⇔𝐵 𝐴◧𝐵 𝐴⇒𝐵 𝐴◨𝐵 𝐴÷𝐵 𝐴∪𝐵 𝐴⊗𝐵

≜ ≜ ≜ ≜ ≜ ≜ ≜ ≜ ≜ ≜ ≜ ≜ ≜ ≜ ≜ ≜

definition ∅

domain ∀𝐴,𝐵∈𝟚𝑋 𝖼

𝖼

∀𝐴,𝐵∈𝟚𝑋

𝐴𝖼 𝐵 𝐴𝖼 𝐵 ∪ 𝐴 𝖼 𝐵 𝖼


𝐴𝐵 𝖼 𝐴𝐵 𝖼 ∪ 𝐴𝖼 𝐵 𝖼 𝐴𝐵 𝖼 ∪ 𝐴𝖼 𝐵 𝐴𝐵 𝖼 ∪ 𝐴𝖼 𝐵 ∪ 𝐴𝖼 𝐵 𝖼


𝐴𝐵 ∪ 𝐴𝐵 ∪ 𝐴𝖼 𝐵 𝖼 𝐴𝐵 ∪ 𝐴𝖼 𝐵 𝐴𝐵 ∪ 𝐴 𝖼 𝐵 ∪ 𝐴𝖼 𝐵 𝖼 𝐴𝐵 ∪ 𝐴𝐵 𝖼 𝐴𝐵 ∪ 𝐴𝐵 𝖼 ∪ 𝐴𝖼 𝐵 𝖼 𝖼 𝖼 𝐴𝐵 ∪ 𝐴𝐵 ∪ 𝐴 𝐵 𝐴𝐵 ∪ 𝐴𝐵 𝖼 ∪ 𝐴𝖼 𝐵 ∪ 𝐴𝖼 𝐵 𝖼


𝐴𝐵

∀𝐴,𝐵∈𝟚𝑋 ∀𝐴,𝐵∈𝟚𝑋 ∀𝐴,𝐵∈𝟚𝑋 ∀𝐴,𝐵∈𝟚𝑋 ∀𝐴,𝐵∈𝟚𝑋 ∀𝐴,𝐵∈𝟚𝑋 ∀𝐴,𝐵∈𝟚𝑋 ∀𝐴,𝐵∈𝟚𝑋 ∀𝐴,𝐵∈𝟚𝑋 ∀𝐴,𝐵∈𝟚𝑋 ∀𝐴,𝐵∈𝟚𝑋

Generated operations

Definition 14.5 (page 202) defines set operations in terms of logical operations. However, it is also possible to express set operations in terms of two or more other set operations. When all the set operations can be expressed in terms of a set of operations, then that set of operations is functionally complete (next definition, but see also Definition 8.3 page 126). Definition 14.7. 6 Let 𝑺 be a set structure. d A set of operations 𝛷 is functionally complete in 𝑺 if e ∪, ∩, 𝖼, ∅, and 𝑋 f can all be expressed in terms of elements of 𝛷. Example 14.1. Here are some examples of functionally complete sets: {↓} (rejection) {|} (Sheffer stroke) {÷, ∅} (adjunction and ∅) {⧵, 𝑋} (set difference and 𝑋) e {∪, 𝖼} (union and complement) x {∩, 𝖼} (intersection and complement) {△, ∩, 𝑋} (symmetric difference, intersection, and 𝑋) {△, ∪, 𝑋} (symmetric difference, union, and 𝑋) {△, ⧵, 𝖼} (symmetric difference, set difference, and complement) The five theorems that follow demonstrate which operations can be generated by sets of generating operations: 2 generators, (72) = 21 possibilities, Proposition 14.2 page 207 3 generators, (73) = 35 possibilities, Proposition 14.3 page 207 4 generators, (74) = 35 possibilities, Proposition 14.4 page 208 5 generators, (75) = 21 possibilities, Proposition 14.5 page 209 possibilities, Proposition 14.6 page 209 6 generators, (76) = 7 6

📘 Whitesitt (1995) page 69

version 0.30


Daniel J. Greenhoe


page 207


Starting with any two subsets 𝐴 and 𝐵 and using all the operations of a functionally complete set of operations, an algebra of sets (Definition 14.9 page 214) is produced. Thus, a functionally complete set of set operations induces an algebra of sets. Other less powerful sets of operations generate fewer operations and induce only a ring of sets (Definition 14.10 page 215). And some sets of operations, such as {∪, ∩}, generate no set operations but themselves. Proposition 14.2 (2 generators). The following table demonstrates the “standard” operations generated by sets of 2 operations. generators 1. ∅ 𝑋 2. ∅ 𝖼 3. ∅ ∪ 4. ∅ ∩ 5. ∅ ⧵ 6. ∅ △ 7. 𝑋 𝖼 8. 𝑋 ∪ 9. 𝑋 ∩ 10. 𝑋 ⧵ 11. 𝑋 △ 12. 𝖼 ∪ 13. 𝖼 ∩ 14. 𝖼 ⧵ 15. 𝖼 △ 16. ∪ ∩ 17. ∪ ⧵ 18. ∪ △ 19. ∩ ⧵ 20. ∩ △ 21. ⧵ △

generated operations ∅ 𝑋 ∅ 𝑋 𝖼 ∅ ∪ ∅ ∩ ∅ ⧵ ∅ △ ∅ 𝑋 𝖼 𝑋 ∪ 𝑋 ∩ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ⧵ ∅ 𝑋 𝖼 △ ∪ ∩ ∅ ∪ ∩ ⧵ △ ∅ ∪ ∩ ⧵ △ ∅ ∩ ⧵ ∅ ∪ ∩ ⧵ △ ∅ ∪ ∩ ⧵ △

induced set structure

algebra of sets algebra of sets algebra of sets

ring of sets ring of sets ring of sets ring of sets

Proposition 14.3 (3 generators). The following table demonstrates the “standard” operations generated by sets of 3 operations. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

generators ∅ 𝑋 𝖼 ∅ 𝑋 ∪ ∅ 𝑋 ∩ ∅ 𝑋 ⧵ ∅ 𝑋 △ ∅ 𝖼 ∪ ∅ 𝖼 ∩ ∅ 𝖼 ⧵ ∅ 𝖼 △ ∅ ∪ ∩ ∅ ∪ ⧵ ∅ ∪ △ ∅ ∩ ⧵ ∅ ∩ △ ∅ ⧵ △ 𝑋 𝖼 ∪


generated operations ∅ 𝑋 𝖼 ∅ 𝑋 ∪ ∅ 𝑋 ∩ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ⧵ ∅ 𝑋 𝖼 △ ∅ ∪ ∩ ∅ ∪ ∩ ⧵ △ ∅ ∪ ∩ ⧵ △ ∅ ∩ ⧵ ∅ ∪ ∩ ⧵ △ ∅ ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △

Daniel J. Greenhoe

induced set structure

algebra of sets algebra of sets algebra of sets

ring of sets ring of sets ring of sets ring of sets algebra of sets


version 0.30

page 208


17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

𝑋 𝑋 𝑋 𝑋 𝑋 𝑋 𝑋 𝑋 𝑋 𝖼 𝖼 𝖼 𝖼 𝖼 𝖼 ∪ ∪ ∪ ∩

𝖼 𝖼 𝖼 ∪ ∪ ∪ ∩ ∩ ⧵ ∪ ∪ ∪ ∩ ∩ ⧵ ∩ ∩ ⧵ ⧵

∩ ⧵ △

∩ ⧵ △

⧵ △ △

∩ ⧵ △

⧵ △ △

⧵ △ △ △

∅ 𝑋 ∅ 𝑋 ∅ 𝑋 𝑋 ∅ 𝑋 ∅ 𝑋 ∅ 𝑋 ∅ 𝑋 ∅ 𝑋 ∅ 𝑋 ∅ 𝑋 ∅ 𝑋 ∅ 𝑋 ∅ 𝑋 ∅ 𝑋 ∅ ∅ ∅ ∅

𝖼 𝖼 𝖼

∪ ∩ ⧵ ∪ ∩ ⧵

∪ ∩ 𝖼 ∪ ∩ ⧵ 𝖼 ∪ ∩ ⧵ 𝖼 ∪ ∩ ⧵ 𝖼 ∪ ∩ ⧵ 𝖼 ∪ ∩ ⧵ 𝖼 ∪ ∩ ⧵ 𝖼 ∪ ∩ ⧵ 𝖼 ∪ ∩ ⧵ 𝖼 ∪ ∩ ⧵ 𝖼 ∪ ∩ ⧵ 𝖼 ∪ ∩ ⧵ ∪ ∩ ⧵ ∪ ∩ ⧵ ∪ ∩ ⧵ ∪ ∩ ⧵

△ △

algebra of sets algebra of sets

△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △

algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets ring of sets ring of sets ring of sets ring of sets

Proposition 14.4 (4 generators). The following table demonstrates the “standard” operations generated by sets of 4 operations. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. version 0.30

generators ∅ 𝑋 𝖼 ∪ ∅ 𝑋 𝖼 ∩ ∅ 𝑋 𝖼 ⧵ ∅ 𝑋 𝖼 △ ∅ 𝑋 ∪ ∩ ∅ 𝑋 ∪ ⧵ ∅ 𝑋 ∪ △ ∅ 𝑋 ∩ ⧵ ∅ 𝑋 ∩ △ ∅ 𝑋 ⧵ △ ∅ 𝖼 ∪ ∩ ∅ 𝖼 ∪ ⧵ ∅ 𝖼 ∪ △ ∅ 𝖼 ∩ ⧵ ∅ 𝖼 ∩ △ ∅ 𝖼 ⧵ △ ∅ ∪ ∩ ⧵ ∅ ∪ ∩ △ ∅ ∪ ⧵ △ ∅ ∩ ⧵ △ 𝑋 𝖼 ∪ ∩ 𝑋 𝖼 ∪ ⧵ 𝑋 𝖼 ∪ △ 𝑋 𝖼 ∩ ⧵ 𝑋 𝖼 ∩ △ 𝑋 𝖼 ⧵ △ 𝑋 ∪ ∩ ⧵

generated operations ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 △ ∅ 𝑋 ∪ ∩ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ ∪ ∩ ⧵ △ ∅ ∪ ∩ ⧵ △ ∅ ∪ ∩ ⧵ △ ∅ ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △


induced set structure algebra of sets algebra of sets algebra of sets pre-topology algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets ring of sets ring of sets ring of sets ring of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets

Daniel J. Greenhoe


page 209


28. 29. 30. 31. 32. 33. 34. 35.

𝑋 𝑋 𝑋 𝖼 𝖼 𝖼 𝖼 ∪

∪ ∪ ∩ ∪ ∪ ∪ ∩ ∩

∩ ⧵ ⧵ ∩ ∩ ⧵ ⧵ ⧵

∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅

△ △ △

⧵ △ △ △ △

𝑋 𝑋 𝑋 𝑋 𝑋 𝑋 𝑋 𝑋

𝖼 𝖼 𝖼 𝖼 𝖼 𝖼 𝖼 𝖼

∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪

∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩

⧵ ⧵ ⧵ ⧵ ⧵ ⧵ ⧵ ⧵

△ △ △ △ △ △ △ △

algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets

Proposition 14.5 (5 generators). The following table demonstrates the “standard” operations generated by sets of 5 operations. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ 𝑋 𝑋 𝑋 𝑋 𝑋 𝖼

generators 𝑋 𝖼 ∪ 𝑋 𝖼 ∪ 𝑋 𝖼 ∪ 𝑋 𝖼 ∩ 𝑋 𝖼 ∩ 𝑋 𝖼 ⧵ 𝑋 ∪ ∩ 𝑋 ∪ ∩ 𝑋 ∪ ⧵ 𝑋 ∩ ⧵ 𝖼 ∪ ∩ 𝖼 ∪ ∩ 𝖼 ∪ ⧵ 𝖼 ∩ ⧵ ∪ ∩ ⧵ 𝖼 ∪ ∩ 𝖼 ∪ ∩ 𝖼 ∪ ⧵ 𝖼 ∩ ⧵ ∪ ∩ ⧵ ∪ ∩ ⧵

∩ ⧵ △

⧵ △ △

⧵ △ △ △

⧵ △ △ △ △

⧵ △ △ △ △ △

generated operations ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △

induced set structure algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets ring of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets

Proposition 14.6 (6 genererators). The following table demonstrates the “standard” operations generated by sets of 6 operations. 1. 2. 3. 4. 5. 6. 7.

∅ ∅ ∅ ∅ ∅ ∅ 𝑋

generators 𝑋 𝖼 ∪ ∩ 𝑋 𝖼 ∪ ∩ 𝑋 𝖼 ∪ ⧵ 𝑋 𝖼 ∩ ⧵ 𝑋 ∪ ∩ ⧵ 𝖼 ∪ ∩ ⧵ 𝖼 ∪ ∩ ⧵


⧵ △ △ △ △ △ △

generated operations ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △ ∅ 𝑋 𝖼 ∪ ∩ ⧵ △

Daniel J. Greenhoe

induced set structure algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets algebra of sets


version 0.30

page 210

14.2.4


Set multiplication

The Cartesian product operation × (Definition 1.7 page 6) is a kind of set multiplication operation. Theorem 14.2 (next theorem) demonstrates how this set operation interacts with certain other set operations. The Cartesian product is of critical importance in general because, for example, relations (Definition 15.1 page 237) and functions (Definition 15.8 page 249) are subsets of Cartesian products. Theorem 14.2. 7 Let 𝑋, 𝑌 , 𝑍 be sets. 𝑋 × (𝑌 ∪ 𝑍) = (𝑋 × 𝑌 ) ∪ (𝑋 × 𝑍) 𝑋 × (𝑌 ∩ 𝑍) = (𝑋 × 𝑌 ) ∩ (𝑋 × 𝑍) t h 𝑋 × (𝑌 ⧵𝑍) = (𝑋 × 𝑌 )⧵(𝑋 × 𝑍) m (𝑋 × 𝑌 ) ∩ (𝑌 × 𝑋) = (𝑋 ∩ 𝑌 ) × (𝑌 ∩ 𝑋) (𝑋 × 𝑋) ∩ (𝑌 × 𝑌 ) = (𝑋 ∩ 𝑌 ) × (𝑋 ∩ 𝑌 )

(× distributes over ∪) (× distributes over ∩) (× distributes over⧵)

✎PROOF: 𝑋 × (𝑌 ∪ 𝑍) = {(𝑎, 𝑏) |(𝑎 ∈ 𝑋) ∧ (𝑏 ∈ 𝑌 ∪ 𝑍)}

by Definition 1.5

= {(𝑎, 𝑏) |(𝑎 ∈ 𝑋) ∧ [(𝑏 ∈ 𝑌 ) ∨ (𝑏 ∪ 𝑍)]}

by Definition 14.5

= {(𝑎, 𝑏) | [(𝑎 ∈ 𝑋) ∧ (𝑏 ∈ 𝑌 )] ∨ [(𝑎 ∈ 𝑋) ∧ (𝑏 ∈ 𝑍)]}

by Theorem 12.2

{(𝑎, 𝑏) | [(𝑎 ∈ 𝑋) ∧ (𝑏 ∈ 𝑌 )]} ∪ {(𝑎, = ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑏) | [(𝑎 ∈ 𝑋) ∧ (𝑏 ∈ 𝑍)]} ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

by Definition 14.5

𝑋×𝑌

𝑋×𝑍

= (𝑋 × 𝑌 ) ∪ (𝑋 × 𝑍)

by Definition 1.5

𝑋 × (𝑌 ∩ 𝑍) = {(𝑎, 𝑏) |(𝑎 ∈ 𝑋) ∧ (𝑏 ∈ 𝑌 ∩ 𝑍)}

by Definition 1.5

= {(𝑎, 𝑏) |(𝑎 ∈ 𝑋) ∧ [(𝑏 ∈ 𝑌 ) ∧ (𝑏 ∪ 𝑍)]}

by Definition 14.5

= {(𝑎, 𝑏) | [(𝑎 ∈ 𝑋) ∧ (𝑏 ∈ 𝑌 )] ∧ [(𝑎 ∈ 𝑋) ∧ (𝑏 ∈ 𝑍)]} = {(𝑎, 𝑏) | [(𝑎 ∈ 𝑋) ∧ (𝑏 ∈ 𝑌 )]} ∩ {(𝑎, 𝑏) | [(𝑎 ∈ 𝑋) ∧ (𝑏 ∈ 𝑍)]} ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑋×𝑌

by Definition 14.5

𝑋×𝑍

= (𝑋 × 𝑌 ) ∩ (𝑋 × 𝑍)

by Definition 1.5

𝑋 × (𝑌 ⧵𝑍) = {(𝑎, 𝑏) |(𝑎 ∈ 𝑋) ∧ (𝑏 ∈ 𝑌 ⧵𝑍)}

by Definition 1.5

𝖼

= {(𝑎, 𝑏) |(𝑎 ∈ 𝑋) ∧ (𝑏 ∈ 𝑌 ∩ 𝑍 )}

by Theorem 14.1 𝖼

= {(𝑎, 𝑏) |(𝑎 ∈ 𝑋) ∧ [(𝑏 ∈ 𝑌 ) ∧ (𝑏 ∈ 𝑍 )]} = {(𝑎, 𝑏) | [(𝑎 ∈ 𝑋) ∧ (𝑏 ∈ 𝑌 )] ∧ [(𝑎 ∈ 𝑋) ∧ (𝑏 ∈ 𝑍 𝖼 )]}

by Definition 14.5

(𝑎, 𝑏) | [(𝑎 ∈ 𝑋) ∧ (𝑏 ∈ 𝑍 𝖼 )]} = {(𝑎, 𝑏) | [(𝑎 ∈ 𝑋) ∧ (𝑏 ∈ 𝑌 )]} ∩ { ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

by Definition 14.5

𝑋×𝑌

𝑋 × 𝑍𝖼 𝖼

= (𝑋 × 𝑌 ) ∩ (𝑋 × 𝑍 )

by Definition 1.5

≠ (𝑋 × 𝑌 )⧵(𝑋 × 𝑍) (𝑋 × 𝑌 ) ∩ (𝑌 × 𝑋) = {(𝑎, 𝑏) |(𝑎 ∈ 𝑋) ∧ (𝑏 ∈ 𝑌 )} ∩ {(𝑎, 𝑏) |(𝑎 ∈ 𝑌 ) ∧ (𝑏 ∈ 𝑋)} = {(𝑎, 𝑏) |[(𝑎 ∈ 𝑋) ∧ (𝑏 ∈ 𝑌 )] ∧ [(𝑎 ∈ 𝑌 ) ∧ (𝑏 ∈ 𝑋)]}

by Definition 1.5 by Definition 14.5

= {(𝑎, 𝑏) |[(𝑎 ∈ 𝑋) ∧ (𝑎 ∈ 𝑌 )] ∧ [(𝑏 ∈ 𝑌 ) ∧ (𝑏 ∈ 𝑋)]} = {(𝑎, 𝑏) |(𝑎 ∈ 𝑋 ∩ 𝑌 ) ∧ (𝑏 ∈ 𝑌 ∩ 𝑋)} = (𝑋 ∩ 𝑌 ) × (𝑌 ∩ 𝑋) 7

by Definition 1.5

📘 Menini and Oystaeyen (2004), page 50, 📘 Halmos (1960) page 25

version 0.30


Daniel J. Greenhoe


page 211


(𝑋 × 𝑋) ∩ (𝑌 × 𝑌 ) = {(𝑎, 𝑏) |(𝑎 ∈ 𝑋) ∧ (𝑏 ∈ 𝑋)} ∩ {(𝑎, 𝑏) |(𝑎 ∈ 𝑌 ) ∧ (𝑏 ∈ 𝑌 )} = {(𝑎, 𝑏) |[(𝑎 ∈ 𝑋) ∧ (𝑏 ∈ 𝑋)] ∧ [(𝑎 ∈ 𝑌 ) ∧ (𝑏 ∈ 𝑌 )]}

by Definition 1.5 by Definition 14.5

= {(𝑎, 𝑏) |[(𝑎 ∈ 𝑋) ∧ (𝑎 ∈ 𝑌 )] ∧ [(𝑏 ∈ 𝑋) ∧ (𝑏 ∈ 𝑌 )]} = {(𝑎, 𝑏) |(𝑎 ∈ 𝑋 ∩ 𝑌 ) ∧ (𝑏 ∈ 𝑋 ∩ 𝑌 )} by Definition 1.5

= (𝑋 ∩ 𝑌 ) × (𝑋 ∩ 𝑌 )

✏

14.3 Standard set structures Set structures are typically designed to satisfy some special properties— such as being closed with respect to certain set operations. Examples of commonly occurring set structures include power set (Definition 14.3 page 201) topologies (Definition 14.8 page 211) algebra of sets (Definition 14.9 page 214) ring of sets (Definition 14.10 page 215) partitions (Definition 14.11 page 216)

14.3.1 Topologies Definition 14.8. 8 Let 𝛤 be a set with an arbitrary (possibly uncountable) number of elements. Let 𝟚𝑋 be the power set of a set 𝑋. A family of sets 𝑻 ⊆ 𝟚𝑋 is a topology on a set 𝑋 if 1. ∅ ∈ 𝑻 (∅ is in 𝑻 ) and 2. 𝑋 ∈ 𝑻 (𝑋 is in 𝑻 ) and 3. 𝑈 , 𝑉 ∈ 𝑻 ⟹ 𝑈 ∩ 𝑉 ∈ 𝑻 (the intersection of a finite number of open sets is open) and 4. {𝑈𝛾 |𝛾 ∈ 𝛤 } ⊆ 𝑻 ⟹ 𝑈 ∈ 𝑻 (the union of an arbitrary number of open sets is open). ⋃ 𝛾 d 𝛾∈𝛤 e (𝑋, A topological space is the pair 𝑻 ). An open set is any member of 𝑻 . f A closed set is any set 𝐷 such that 𝐷𝖼 is open. The set of topologies on a set 𝑋 is denoted 𝒯 (𝑋). That is, 𝒯 (𝑋) ≜ {𝑻 ⊆ 𝟚𝑋 |𝑻 is a topology}. If 𝑋 is finite, then 𝑻 is a topology on a finite set, and (4.) can be replaced by 𝑈, 𝑉 ∈ 𝑻 ⟹ 𝑈 ∪𝑉 ∈ 𝑻. Example 14.2. 9 Let 𝒯 (𝑋) be the set of topologies on a set 𝑋 and 𝟚𝑋 the power set 201) on 𝑋. (indiscrete topology or trivial topology) e {∅, 𝑋} is a topology in 𝒯 (𝑋) x 𝟚𝑋 is a topology in 𝒯 (𝑋) (discrete topology) Example 14.3.

10

(Definition 14.3 page

There are four topologies on the set 𝑋 ≜ {𝑥, 𝑦}:

8

📘 Munkres (2000) page 76, 📘 Riesz (1909), 📘 Hausdorff (1914), 📃 Tietze (1923) ⟨cited by Thron page 18⟩, 📘 Hausdorff (1937) page 258 9 📘 Munkres (2000), page 77, 📘 Kubrusly (2011) page 107 ⟨Example 3.J⟩, 📘 Steen and Seebach (1978) pages 42–43 ⟨II.4⟩, 📘 DiBenedetto (2002) page 18 10 📘 Isham (1999), page 44, 📘 Isham (1989), page 1515 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 212

14.3. STANDARD SET STRUCTURES

topologies on {𝑥, 𝑦} corresponding closed sets 𝑻0 = {∅, 𝑋} {∅, 𝑋} e {𝑥}, {𝑦}, 𝑻 = {∅, 𝑋 } {∅, 𝑋 } 1 x {𝑦}, 𝑋 } 𝑻2 = {∅, {∅, {𝑥}, 𝑋} 𝑻3 = {∅, {𝑥}, {𝑦}, 𝑋 } {∅, {𝑥}, {𝑦}, 𝑋 } The topologies (𝑋, 𝑻1 ) and (𝑋, 𝑻2 ), as well as their corresponding closed set topological spaces, are all Serpiński spaces. Example 14.4. There are a total of 29 topologies (Definition 14.8 page 211) on the set 𝑋 ≜ {𝑥, 𝑦, 𝑧}: 𝑻00 𝑻01 𝑻02 𝑻04 𝑻10 𝑻20 𝑻40 𝑻11 𝑻21 𝑻41 𝑻12 𝑻22 𝑻42 𝑻14 𝑻24 𝑻44 𝑻31 𝑻52 𝑻64 𝑻13 𝑻25 𝑻46 𝑻33 𝑻53 𝑻35 𝑻65 𝑻56 𝑻66 𝑻77

= = = = = = = = = = = = = = = = = = = = = = = = = = = = =

topologies on {𝑥, 𝑦, 𝑧} {∅, 𝑋} {∅, {𝑥}, 𝑋} {𝑦}, {∅, 𝑋} {𝑧}, {∅, 𝑋} {𝑥, 𝑦}, {∅, 𝑋} {𝑥, 𝑧}, {∅, 𝑋} {𝑦, 𝑧}, 𝑋 } {∅, {𝑥, 𝑦}, {∅, {𝑥}, 𝑋} {𝑥, 𝑧}, {∅, {𝑥}, 𝑋} {𝑦, 𝑧}, 𝑋 } {∅, {𝑥}, {𝑦}, {𝑥, 𝑦}, {∅, 𝑋} {𝑦}, {𝑥, 𝑧}, {∅, 𝑋} {𝑦}, {𝑦, 𝑧}, 𝑋 } {∅, {𝑧}, {𝑥, 𝑦}, {∅, 𝑋} {𝑧}, {𝑥, 𝑧}, {∅, 𝑋} {𝑧}, {𝑦, 𝑧}, 𝑋 } {∅, {𝑥, 𝑦}, {𝑥, 𝑧}, {∅, {𝑥}, 𝑋} {𝑦}, {𝑥, 𝑦}, {𝑦, 𝑧}, 𝑋 } {∅, {𝑧}, {𝑥, 𝑧}, {𝑦, 𝑧}, 𝑋 } {∅, {𝑥, 𝑦}, {∅, {𝑥}, {𝑦}, 𝑋} {𝑧}, {𝑥, 𝑧}, {∅, {𝑥}, 𝑋} {𝑦}, {𝑧}, {𝑦, 𝑧}, 𝑋 } {∅, {𝑥, 𝑦}, {𝑥, 𝑧}, {∅, {𝑥}, {𝑦}, 𝑋} {𝑥, 𝑦}, {𝑦, 𝑧}, 𝑋 } {∅, {𝑥}, {𝑦}, {𝑧}, {𝑥, 𝑦}, {𝑥, 𝑧}, {∅, {𝑥}, 𝑋} {𝑧}, {𝑥, 𝑧}, {𝑦, 𝑧}, 𝑋 } {∅, {𝑥}, {𝑦}, {𝑧}, {𝑥, 𝑦}, {𝑦, 𝑧}, 𝑋 } {∅, {𝑦}, {𝑧}, {𝑥, 𝑧}, {𝑦, 𝑧}, 𝑋 } {∅, {∅, {𝑥}, {𝑦}, {𝑧}, {𝑥, 𝑦}, {𝑥, 𝑧}, {𝑦, 𝑧}, 𝑋 }

corresponding closed sets {∅, 𝑋} {𝑦, 𝑧}, 𝑋 } {∅, {𝑥, 𝑧} {∅, 𝑋} {𝑥, 𝑦}, {∅, 𝑋} {𝑧}, {∅, 𝑋} {𝑦}, {∅, 𝑋} {∅, {𝑥}, 𝑋} {𝑧}, {𝑦, 𝑧}, 𝑋 } {∅, {𝑦} {𝑦, 𝑧}, 𝑋 } {∅, {𝑦, 𝑧}, 𝑋 } {∅, {𝑥}, {𝑧}, {𝑥, 𝑧} {∅, 𝑋} {𝑦}, {𝑥, 𝑧}, {∅, 𝑋} {𝑥, 𝑧}, {∅, {𝑥}, 𝑋} {𝑧}, {𝑥, 𝑦}, {∅, 𝑋} {𝑦}, {𝑥, 𝑦}, {∅, 𝑋} {𝑥, 𝑦}, {∅, {𝑥}, 𝑋} {𝑦}, {𝑧}, {𝑦, 𝑧}, 𝑋 } {∅, {𝑧}, {𝑥, 𝑧}, {∅, {𝑥}, 𝑋} {𝑥, 𝑦}, {∅, {𝑥}, {𝑦}, 𝑋} {𝑧}, {𝑥, 𝑧}, {𝑦, 𝑧}, 𝑋 } {∅, {𝑦}, {𝑥, 𝑦}, {𝑦, 𝑧}, 𝑋 } {∅, {𝑥, 𝑦}, {𝑥, 𝑧}, {∅, {𝑥}, 𝑋} {𝑦}, {𝑧}, {𝑥, 𝑧}, {𝑦, 𝑧}, 𝑋 } {∅, {𝑧}, {𝑥, 𝑧}, {𝑦, 𝑧}, 𝑋 } {∅, {𝑥}, {𝑦}, {𝑧}, {𝑥, 𝑦}, {𝑦, 𝑧}, 𝑋 } {∅, {𝑥, 𝑦}, {𝑦, 𝑧}, 𝑋 } {∅, {𝑥}, {𝑦}, {𝑧}, {𝑥, 𝑦}, {𝑥, 𝑧}, {∅, {𝑥}, 𝑋} {𝑥, 𝑦}, {𝑥, 𝑧}, {∅, {𝑥}, {𝑦}, 𝑋} {∅, {𝑥}, {𝑦}, {𝑧}, {𝑥, 𝑦}, {𝑥, 𝑧}, {𝑦, 𝑧}, 𝑋 }

Theorem 14.3. Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. t h 𝑻 is a topology ( 𝑻 , ∪, ∩ ; ⊆) is a distributive lattice ⟹ m ✎PROOF: 1. By Proposition 14.15 (page 224), (𝑺, ⊆) is an ordered set. 2. By Proposition 14.16 (page 224), ∪ is least upper bound operation on (𝑺, ⊆). and ∩ is greatest lower bound operation on (𝑺, ⊆). 3. Therefore, by Definition 4.3 (page 61), ( 𝑺, ∪, ∩ ; ⊆) is a lattice. 4. By Theorem 4.3 (page 62), ( 𝑺, ∪, ∩ ; ⊆) is idempotent, commutative, associative, and absorptive. 5. Proof that ( 𝑺, ∪, ∩ ; ⊆) is distributive: version 0.30


Daniel J. Greenhoe


page 213


(a) Proof that 𝐴 ∩ (𝐵 ∪ 𝐶) = (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶): 𝐴 ∩ (𝐵 ∪ 𝐶) = {𝑥 ∈ 𝑋|𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∪ 𝐶)}

by definition of ∩ (Definition 14.5 page 202)

= {𝑥 ∈ 𝑋|𝑥 ∈ 𝐴 ∧ 𝑥 ∈ {𝑥 ∈ 𝑋|𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶}}

by definition of ∪ (Definition 14.5 page 202)

= {𝑥 ∈ 𝑋|𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)} = {𝑥 ∈ 𝑋|(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶)}


= {𝑥 ∈ 𝑋|𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵} ∪ {𝑥 ∈ 𝑋|𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶}

by definition of ∪ (Definition 14.5 page 202)

= (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶)

by definition of ∩ (Definition 14.5 page 202)

(b) Proof that 𝐴 ∪ (𝐵 ∩ 𝐶) = (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶): This follows from the fact that ( 𝑺, ∪, ∩ ; ⊆) is a lattice (item (3) page 212), that ∩ distributes over ∪ (item (5) page 212), and by Theorem 6.1 (page 90).

✏ Example 14.5. There are five unlabeled lattices on a five element set (Proposition 4.2 page 67). Of these five, three are distributive (Proposition 6.3 page 107). The following illustrates that the distributive lattices are isomorphic to topologies, while the non-distributive lattices are not. non-distributive/not topologies e x

{𝑥, 𝑦, 𝑧} {𝑥} {𝑦}

{𝑧}

distributive/are topologies

{𝑥, 𝑦, 𝑧} {𝑥, 𝑦} {𝑥}

∅

{𝑧}

{𝑥}

∅

{𝑥, 𝑦, 𝑧} {𝑥, 𝑦} {𝑦} ∅

{𝑥, 𝑧}

{𝑥, 𝑦, 𝑧} {𝑥, 𝑦} {𝑥} ∅

{𝑤, 𝑥, 𝑦, 𝑧} {𝑤, 𝑥, 𝑦} {𝑤, 𝑥} {𝑤} ∅

✎PROOF: 1. The first two lattices are non-distributive by Birkhoff distributivity criterion (Theorem 6.2 page 94). (a) This lattice is not a topology because, for example, {𝑥} ∨ {𝑦} = {𝑥, 𝑦, 𝑧} ≠ {𝑥, 𝑦} = {𝑥} ∪ {𝑦}. That is, the set union operation ∪ is not equivalent to the order join operation ∨. (b) This lattice is not a topology because, for example, {𝑥} ∨ {𝑦} = {𝑦} ≠ {𝑥, 𝑦} = {𝑥} ∪ {𝑦} 2. The last three lattices are distributive by Birkhoff distributivity criterion (Theorem 6.2 page 94). (a) This lattice is the topology 𝑻13 of Example 14.4 (page 212). On the set {𝑥, 𝑦, 𝑧}, there are a total of three topologies that have this order structure (see Example 14.4): 𝑻13 = { ∅, {𝑥}, {𝑦}, {𝑥, 𝑦}, {𝑥, 𝑦, 𝑧} } 𝑻25 = { ∅, {𝑥}, {𝑧}, {𝑥, 𝑧}, {𝑥, 𝑦, 𝑧} } 𝑻46 = { ∅, {𝑦}, {𝑧}, {𝑦, 𝑧}, {𝑥, 𝑦, 𝑧} } (b) This lattice is the topology 𝑻31 of Example 14.4 (page 212). On the set {𝑥, 𝑦, 𝑧}, there are a total of three topologies that have this order structure (see Example 14.4): 𝑻31 = { ∅, {𝑥}, {𝑥, 𝑦}, {𝑥, 𝑧}, {𝑥, 𝑦, 𝑧} } 𝑻52 = { ∅, {𝑦}, {𝑥, 𝑦}, {𝑦, 𝑧}, {𝑥, 𝑦, 𝑧} } 𝑻64 = { ∅, {𝑧}, {𝑥, 𝑧}, {𝑦, 𝑧}, {𝑥, 𝑦, 𝑧} } (c) This lattice is a topology by Definition 14.8 (page 211).


Daniel J. Greenhoe


version 0.30

page 214

14.3.2


Algebras of sets

Definition 14.9. 11 Let 𝑋 be a set with power set 𝟚𝑋 (Definition 14.3 page 201). 𝑨 ⊆ 𝟚𝑋 is an algebra of sets on 𝑋 if 1. 𝐴 ∈ 𝑨 ⟹ 𝐴𝖼 ∈ 𝑨 (closed under complement operation) 2. 𝐴, 𝐵 ∈ 𝑨 ⟹ 𝐴 ∩ 𝐵 ∈ 𝑨 (closed under ∩) d The set of all algebra of sets on a set 𝑋 is denoted 𝒜(𝑋) such that e 𝒜(𝑋) ≜ {𝑨 ⊆ 𝟚𝑋 |𝑨 is an algebra of sets}. f An algebra of sets 𝑨 on 𝑋 is a σ-algebra on 𝑋 if 3. {𝐴𝑛 |𝑛∈ℤ} ⊆ 𝑨 ⟹ 𝐴 ∈ 𝑨 (closed under countable union operations). ⋃ 𝑛

and

𝑛∈ℤ

On every set 𝑋 with at least 2 elements, there are always two particular algebras of sets: the smallest algebra and the largest algebra, as demonstrated by Example 14.6 (next). Example 14.6. 12 Let 𝒜(𝑋) be the set of algebras of sets power set (Definition 14.3 page 201) on 𝑋. (smallest algebra) e {∅, 𝑋} ∈ 𝒜(𝑋) x 𝟚𝑋 ∈ 𝒜(𝑋) (largest algebra)


on a set 𝑋 and 𝟚𝑋 the

Isomorphically, all algebras of sets are boolean algebras (Definition 8.1 page 115) and all boolean algebras are algebras of sets (next theorem). Theorem 14.4 (Stone Representation Theorem). 13 Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. t 𝙇 is isomorphic to ( 𝑨, ∪, ∩, ∅, 𝑋 ; ⊆) h 𝙇 is Boolean ⟺ { for some algebra of sets (Definition 14.9 page 214) 𝑨 } m ✎PROOF: 1. Proof that algebra of sets ⟹ Boolean algebra: (a) Proof that 𝑺 is closed under ∪ and ∩: by hypothesis. (b) By item (1b) and by Theorem 14.6 (page 220), 𝙇 is a distributive lattice. (c) By item (1b) and properties of lattices (Theorem 4.3 page 62), 𝙇 is idempotent, commutative, associative, and absorptive. (d) Proof that 𝙇 has identity: 𝐴 ∪ ∅ = {𝑥 ∈ 𝑋|(𝑥 ∈ 𝐴) ∨ (𝑥 ∈ ∅)} = {𝑥 ∈ 𝑋|𝑥 ∈ 𝐴}

by definition of ∪ Definition 14.5 page 202 by definition of ∅ Definition 14.5 page 202

=𝐴 𝐴 ∩ 𝑋 = {𝑥 ∈ 𝑋|(𝑥 ∈ 𝐴) ∧ (𝑥 ∈ 𝑋)} = {𝑥 ∈ 𝑋|𝑥 ∈ 𝐴}

by definition of ∩ Definition 14.5 page 202 by definition of ∅ Definition 14.5 page 202

=𝐴 (e) Proof that 𝙇 is complemented: by hypothesis. 11

📘 Aliprantis and Burkinshaw (1998) page 95, 📘 Aliprantis and Burkinshaw (1998) page 151, 📘 Halmos (1950) page 21, 📘 Hausdorff (1937) page 91 12 📘 Stroock (1999) page 33, 📘 Aliprantis and Burkinshaw (1998) pages 95–96 13 📘 Levy (2002) page 257, 📘 Grätzer (2003) page 85, 📘 Joshi (1989) page 224, 📘 Saliǐ (1988) page 32 ⟨“Stone's Theorem”⟩, 📘 Stone (1936) version 0.30


Daniel J. Greenhoe


page 215


(f) Because 𝙇 is commutative (item (1c) page 214), distributive (item (1b) page 214), has identity (item (1d) page 214), and is complemented (item (1e) page 214), and by the definition of Boolean algebras (Definition 8.1 page 115), 𝙇 is a Boolean algebra. 2. Proof that Boolean algebra ⟹ algebra of sets: not included at this time.

✏

14.3.3 Rings of sets A ring of sets (next definition) is a family of subsets that is closed under an “addition-like” set union operator ∪ and “subtraction-like” set difference operator⧵. Using these two operations, it is not difficult to show that a ring of sets is also closed under a “multiplication-like” set intersection operator ∩. Because of this, a ring of sets behaves like an algebraic ring. Note however that a ring of sets is not necessarily a topology (Definition 14.8 page 211) because it does not necessarily include 𝑋 itself. Definition 14.10. 14 Let 𝑋 be a set with power set 𝟚𝑋 (Definition 14.3 page 201). 𝑹 ⊆ 𝟚𝑋 is a ring of sets on 𝑋 if 1. 𝐴, 𝐵 ∈ 𝑹 ⟹ 𝐴∪𝐵 (closed under ∪) 2. 𝐴, 𝐵 ∈ 𝑹 ⟹ 𝐴⧵𝐵 ∈ 𝑹 (closed under⧵) d The set of all rings of sets on a set 𝑋 is denoted ℛ(𝑋) such that e ℛ(𝑋) ≜ {𝑹 ⊆ 𝟚𝑋 |𝑹 is a ring of sets}. f A ring of sets 𝑹 on 𝑋 is a σ-ring on 𝑋 if ⟹ 𝐴 ∈ 𝑹 (closed under countable union operations). 3. {𝐴𝑛 |𝑛∈ℤ} ⊆ 𝑹 ⋃ 𝑛

and

𝑛∈ℤ

Example 14.7. The following table lists some rings of sets on a finite set 𝑋.

14

ℛ(∅)

rings ℛ(𝑋) on a set 𝑋 = { 𝑹1 = {∅} }

ℛ({𝑥})

=

ℛ({𝑥, 𝑦})

⎧ ⎪ ⎪ = ⎨ ⎪ ⎪ ⎩

𝑹1 = { ∅, } { 𝑹2 = { ∅, {𝑥} } } 𝑹1 𝑹2 𝑹3 𝑹4 𝑹5

= = = = =

{ { { { {

∅, ∅, {𝑥}, {𝑦}, ∅, {𝑥, 𝑦} ∅, ∅, {𝑥}, {𝑦}, {𝑥, 𝑦}

} } } } }

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

📘 Berezansky et al. (1996) page 4, 📘 Halmos (1950) page 19, 📘 Hausdorff (1937) page 90


Daniel J. Greenhoe


version 0.30

page 216


ℛ({𝑥, 𝑦, 𝑧})

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

𝑹1 𝑹2 𝑹3 𝑹4 𝑹5 𝑹6 𝑹7 𝑹8 𝑹9 𝑹10 𝑹11 𝑹12 𝑹13 𝑹14 𝑹15

={ ={ ={ ={ ={ ={ ={ ={ ={ ={ ={ ={ ={ ={ ={

∅, ∅, ∅, ∅, ∅, ∅, ∅, ∅, ∅, ∅, ∅, ∅, ∅, ∅, ∅,

{𝑥}, {𝑦}, {𝑧}, {𝑥, 𝑦}, {𝑥, 𝑧}, {𝑦, 𝑧}, {𝑥}, {𝑦}, {𝑥, 𝑦}, {𝑥}, {𝑧}, {𝑥, 𝑧}, {𝑦}, {𝑧}, {𝑦, 𝑧}, 𝑋 {𝑥}, {𝑦, 𝑧}, 𝑋 {𝑦}, {𝑥, 𝑧}, 𝑋 {𝑧}, {𝑥, 𝑦}, 𝑋 {𝑥}, {𝑦}, {𝑧}, {𝑥, 𝑦}, {𝑥, 𝑧}, {𝑦, 𝑧}, 𝑋

Example 14.8. Let 𝑋 ≜ {𝑥, 𝑦, 𝑧} be a set and 𝑹 be the family of sets 𝑹 ≜ {∅, 𝑋, {𝑥}, {𝑦}, {𝑧}, {𝑥, 𝑦}}. Note that (𝑹, ⊆, ∪, ∩) is a lattice as illustrated in the figure to the right. However, 𝑹 is not a ring of sets on 𝑋 because, for example, {𝑥, 𝑦, 𝑧}⧵{𝑥} = {𝑦, 𝑧} ∉ 𝑹. Example 14.9. Let 𝑋 ≜ {𝑥, 𝑦, 𝑧} be a set and 𝑹 be the family of sets 𝑹 ≜ {∅, 𝑋, {𝑥}, {𝑦}, {𝑧}}. Note that (𝑻 , ⊆) ∪∩ is a lattice as illustrated in the figure to the right. However, 𝑹 is not a ring of sets on 𝑋 because, for example, {𝑥, 𝑦, 𝑧}⧵{𝑥} = {𝑦, 𝑧} ∉ 𝑹.

} } } } } } } } } } } } } } }

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

{𝑥, 𝑦, 𝑧} {𝑥, 𝑦} {𝑥}

{𝑦}

{𝑧}

∅

{𝑥, 𝑦, 𝑧} {𝑥}

{𝑦} {𝑧} ∅

Proposition 14.7. 15 Let ℛ(𝑋) be the set of rings of sets (Definition 14.10 page 215) on a set 𝑋. p 𝑹1 and 𝑹2 (𝑹1 ∩ 𝑹2 ) r ⟹ { is a ring of sets } p { are rings of sets }

14.3.4

Partitions

Definition 14.11. 16 A set structure {𝑃𝑛 ∈ 𝟚𝑋 |𝑛=1,2,…,𝘕 } is a partition of the set 𝑋 if d e f

1. 2. 3.

𝑃𝑛 ≠ ∅ 𝑃𝑛 ∩ 𝑃 𝑚 = ∅ 𝑃 = 𝑋 ⋃ 𝑛

∀𝑛∈{1,2,…,𝘕 }

non-empty

and

∀𝑛≠𝑚

mutually exclusive

and

𝑛∈ℤ

Example 14.10. Let 𝐴, 𝐵 ⊆ 𝑋. Then the following are examples of partitions on 𝑋: 𝖼 1. {𝐴 ∪ 𝐵, (𝐴 ∪ 𝐵) } e 𝖼 x 2. {𝐴, 𝐵⧵𝐴, (𝐴 ∪ 𝐵) } 𝖼 3. {𝐴⧵𝐵, 𝐵⧵𝐴, 𝐴 ∩ 𝐵, (𝐴 ∪ 𝐵) } 15 16

📘 Kolmogorov and Fomin (1975) page 32, 📘 Bartle (2001) page 318 📘 Munkres (2000), page 23, 📘 Rota (1964), page 498, 📘 Halmos (1950) page 31

version 0.30


Daniel J. Greenhoe


page 217


Proposition 14.8. 17 Let 𝒫(𝑋) be the set of partitions on a set 𝑋. The relation ⊴∈ 𝟚ℙℙ defined as p def r 𝑷 ⊴𝑸 ⟺ ∀𝐵 ∈ 𝑸, ∃𝐴 ∈ 𝑷 such that 𝐵 ⊆ 𝐴 p is an ordering relation on 𝒫(𝑋). Example 14.11. The following table lists some partitions 𝑷 (𝑋) on a finite set 𝑋.

17

𝒫(∅)

partitions 𝒫(𝑋) on a set 𝑋 = { 𝑷1 = ∅ }

𝒫({𝑥})

= { 𝑷1 = {

𝒫({𝑥, 𝑦})

=

𝒫({𝑥, 𝑦, 𝑧})

⎧ ⎪ ⎪ = ⎨ ⎪ ⎪ ⎩

𝑷1 𝑷2 𝑷3 𝑷4 𝑷5

𝒫({𝑤, 𝑥, 𝑦, 𝑧})

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

𝑷1 𝑷2 𝑷3 𝑷4 𝑷5 𝑷6 𝑷7 𝑷8 𝑷9 𝑷10 𝑷11 𝑷12 𝑷13 𝑷14 𝑷15

𝑷1 = { { 𝑷2 = { = = = = = = = = = = = = = = = = = = = =

{ { { { { { { { { { { { { { { { { { { {

{𝑥} } } {𝑥}, {𝑦},

} {𝑥, 𝑦} } }

{𝑥, 𝑦, 𝑧} } {𝑦, 𝑧}, } {𝑦}, {𝑥, 𝑧}, } {𝑧}, {𝑥, 𝑦} } {𝑥}, {𝑦}, {𝑧} } {𝑥},

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

𝑋 } ⎫ {𝑤}, {𝑥, 𝑦, 𝑧} } ⎪ {𝑥}, {𝑤, 𝑦, 𝑧} } ⎪ ⎪ {𝑦}, {𝑤, 𝑥, 𝑧} } ⎪ {𝑧}, {𝑤, 𝑥, 𝑦} } ⎪ {𝑤, 𝑥}, {𝑦, 𝑧} } ⎪ {𝑤, 𝑦}, {𝑥, 𝑧} } ⎪ ⎪ {𝑤, 𝑧}, {𝑥, 𝑦}, } ⎬ {𝑤}, {𝑥} {𝑦, 𝑧} } ⎪ {𝑤}, {𝑦}, {𝑥, 𝑧} } ⎪ ⎪ {𝑤}, {𝑧}, {𝑥, 𝑦} } ⎪ {𝑥}, {𝑦}, {𝑤, 𝑧} } ⎪ {𝑥}, {𝑧}, {𝑤, 𝑦} } ⎪ {𝑦}, {𝑧}, {𝑤, 𝑥} } ⎪ ⎪ {𝑤}, {𝑥}, {𝑦}, {𝑧}, } ⎭

📘 Roman (2008) page 111, 📘 Comtet (1974) page 220, 📘 Grätzer (2007), page 697


Daniel J. Greenhoe


version 0.30

page 218

14.4

14.4. NUMBERS OF SET STRUCTURES

Numbers of set structures

Proposition 14.9. 18 The number of topologies 𝑡𝑛 on a finite set 𝑋𝑛 with 𝑛 elements is 𝑛 0 1 2 3 4 5 6 7 8 p 𝑡 1 1 4 29 355 6942 209, 527 9, 535, 241 642, 779, 354 r 𝑛 p 𝑛 9 10 𝑡𝑛 63, 260, 289, 423 8, 977, 053, 873, 043 Proposition 14.10. 19 Let 𝑡𝑛 be the number of topologies on a finite set with 𝑛 elements. 𝑡 lim 𝑛2 = ∞ (lower bound) 𝑛→∞ 𝑛 24 p 𝑡𝑛 r lim = 0 ∀𝜖>0 (upper bound) p 𝑛→∞ 1 +𝜖 𝑛2 ( ) 2 2 𝑡𝑛 > 𝑛𝑡𝑛−1 (rate of growth) Similar to the amazing relationship between 𝑒, 𝜋, 𝑖, 1, and 0 given by 𝑒𝑖𝜋 + 1 = 0, we find another relationship between 𝑒 and the number of partitions, rings of sets, and algebras of sets (Theorem 14.5 page 220). Definition 14.12. 20 The Bell numbers are the elements of the sequence ⦅𝐵𝑛 ⦆𝑛∈𝕎 defined as the solution to the following equation: d ∞ 𝐵𝑛 𝑛 e 𝑒𝑥 −1 𝑒 = 𝑥 . f ∑ 𝑛! 𝑛=0 The Bell numbers are also called the exponential numbers. Proposition 14.11. 21 Let ⦅𝐵𝑛 ⦆𝑛∈𝕎 be the sequence of Bell numbers. ⦅𝐵𝑛 ⦆ has the following values: p 𝑛 0 1 2 3 4 5 6 7 8 9 10 11 r p 𝐵𝑛 1 1 2 5 15 52 203 877 4140 21, 147 115, 975 678, 570 ✎PROOF:

By Definition 14.12 (page 218), the sequence ⦅𝐵𝑛 ⦆ is the solution to 𝑒𝑒

𝑥

∞

−1

=

𝐵𝑛 𝑛 𝑥 . ∑ 𝑛! 𝑛=0

Let 𝖿 (𝑛) (𝑥) be the 𝑛th derivative of a function 𝖿 ∶ ℝ → ℝ. The Maclaurin expansion of 𝖿(𝑥) is ∞

𝖿 (𝑥) =

𝖿 𝑛 (0) 𝑛 𝑥 ∑ 𝑛! 𝑛=0

18

💻 Sloane (2014) ⟨http://oeis.org/A000798⟩, 📘 Brown and Watson (1996), page 31, 📘 Comtet (1974) page 229, 📘 Comtet (1966), 📘 Chatterji (1967), page 7, 📘 Evans et al. (1967), 📘 Krishnamurthy (1966), page 157 19 📘 Chatterji (1967), pages 6–7, 📘 Kleitman and Rothschild (1970) 20 📘 Comtet (1974) pages 210–211, 📘 Rota (1964), page 499, 📘 Bell (1934) page 417, 📘 d'Ocagne (1887) page 371 21 💻 Sloane (2014) ⟨http://oeis.org/A000110⟩ version 0.30


Daniel J. Greenhoe


page 219

CHAPTER 14. SET STRUCTURES 𝑥

Let 𝖿(𝑥) ≜ 𝑒𝑒 . Then 𝖿 (0) (0) = 𝖿 (0) (𝑥)|𝑥=0 0

= 𝑒𝑒 =𝑒 (1)

𝖿 (0) = 𝖿 (1) (𝑥)|𝑥=0 d 𝑒𝑥 = 𝑒 d𝑥 |𝑥=0 𝑥

= 𝑒𝑒 𝑒𝑥 | 𝑥=0 =𝑒 d (1) 𝖿 (2) (0) = 𝖿 (𝑥)| d𝑥 𝑥=0 d 𝑒𝑥 𝑥 = 𝑒 𝑒 | d𝑥 𝑥=0 𝑥

𝑥

= (𝑒𝑒 𝑒𝑥 )𝑒𝑥 + 𝑒𝑒 𝑒𝑥 | 𝑥=0 𝑥

= 𝑒𝑒 (𝑒2𝑥 + 𝑒𝑥 )|

𝑥=0

= 2𝑒 d (2) 𝖿 (3) (0) = 𝖿 (𝑥)| d𝑥 𝑥=0 d 𝑒𝑥 2𝑥 = 𝑒 𝑒 + 𝑒𝑥 )| d𝑥 ( 𝑥=0 𝑥

𝑥

= 𝑒𝑒 𝑒𝑥 (𝑒2𝑥 + 𝑒𝑥 ) + 𝑒𝑒 (2𝑒2𝑥 + 𝑒𝑥 )| 𝑒𝑥

3𝑥

= 𝑒 (𝑒

+ 3𝑒

2𝑥

𝑥=0

𝑥

+ 𝑒 )|

𝑥=0

= 5𝑒 d (3) 𝖿 (𝑥)| 𝖿 (4) (0) = d𝑥 𝑥=0 d 𝑒𝑥 3𝑥 = 𝑒 𝑒 + 3𝑒2𝑥 + 𝑒𝑥 )| d𝑥 ( 𝑥=0 𝑥

𝑥

= (𝑒𝑒 𝑒𝑥 )(𝑒3𝑥 + 3𝑒2𝑥 + 𝑒𝑥 ) + 𝑒𝑒 (3𝑒3𝑥 + 6𝑒2𝑥 + 𝑒𝑥 )| 𝑥=0 𝑥

= 𝑒𝑒 (𝑒4𝑥 + 6𝑒3𝑥 + 7𝑒2𝑥 + 𝑒𝑥 )|

𝑥=0

= 15𝑒 d (4) 𝖿 (5) (0) = 𝖿 (𝑥)| d𝑥 𝑥=0 d 𝑒𝑥 4𝑥 = 𝑒 𝑒 + 6𝑒3𝑥 + 7𝑒2𝑥 + 𝑒𝑥 )| d𝑥 ( 𝑥=0 𝑥 d 𝑒𝑥 𝑥 𝑒 𝑒 )(𝑒4𝑥 + 6𝑒3𝑥 + 7𝑒2𝑥 + 𝑒𝑥 ) + 𝑒𝑒 (4𝑒4𝑥 + 18𝑒3𝑥 + 14𝑒2𝑥 + 𝑒𝑥 )| = ( d𝑥 𝑥=0 d 𝑒𝑥 5𝑥 4𝑥 3𝑥 2𝑥 𝑥 = 𝑒 𝑒 + 10𝑒 + 25𝑒 + 15𝑒 + 𝑒 )| d𝑥 ( 𝑥=0 = 52𝑒 d (5) 𝖿 (𝑥)| 𝖿 (6) (0) = d𝑥 𝑥=0 d 𝑒𝑥 5𝑥 𝑒 𝑒 + 10𝑒4𝑥 + 25𝑒3𝑥 + 15𝑒2𝑥 + 𝑒𝑥 )| = d𝑥 ( 𝑥=0 𝑥

𝑥

= (𝑒𝑒 𝑒𝑥 )(𝑒5𝑥 + 10𝑒4𝑥 + 25𝑒3𝑥 + 15𝑒2𝑥 + 𝑒𝑥 ) + 𝑒𝑒 (5𝑒5𝑥 + 40𝑒4𝑥 + 75𝑒3𝑥 + 30𝑒2𝑥 + 𝑒𝑥 )| 𝑥=0 𝑥

= 𝑒𝑒 (𝑒6𝑥 + 15𝑒5𝑥 + 65𝑒4𝑥 + 90𝑒3𝑥 + 31𝑒2𝑥 + 𝑒𝑥 )|

𝑥=0


Daniel J. Greenhoe


version 0.30

page 220

14.5. OPERATIONS ON SET STRUCTURES

= 203𝑒 𝑥

Thus, 𝑒𝑒 has Maclaurin expansion ∞

𝐵𝑛 𝑛 2 5 15 52 203 6 𝑒 = 𝑒(1 + 𝑥 + 𝑥2 + 𝑥3 + 𝑥4 + 𝑥5 + 𝑥 + ⋯) = 𝑒 𝑥 ∑ 𝑛! 2 3! 4! 5! 6! 𝑛=0 𝑒𝑥

✏ Theorem 14.5. 22 Let 𝑋𝑛 be a finite set with 𝑛 elements. Let ⦅𝐵𝑛 ⦆𝑛∈𝕎 be the sequence of Bell numbers. The number of partitions on 𝑋𝑛 is 𝐵𝑛 . t h The number of rings of sets on 𝑋𝑛 is 𝐵𝑛+1 . m The number of algebras of sets on 𝑋𝑛 is 𝐵𝑛 .

14.5

Operations on set structures

Proposition 14.12. closed under partition ∅ 𝑋 ✓ p 𝖼 r ∪ p ∩ △

⧵

ring of sets ✓

✓ ✓ ✓ ✓

algebra of sets ✓ ✓ ✓ ✓ ✓ ✓ ✓

topology ✓ ✓ ✓ ✓

✎PROOF: 1. Proof for closure in a topology: Definition 14.8 (page 211) 2. Proof for closure in a ring of sets: Definition 14.10 (page 215) and Theorem 14.14 (page 222) 3. Proof for closure in an algebra of sets: Definition 14.9 (page 214) and Theorem 14.13 (page 221)

✏ Theorem 14.6. Let 𝑻 be a set structure (Definition 14.1 page 201) on a set 𝑋. 𝑻 is a topology ⟹ ∀𝐴, 𝐵, 𝐶 ∈ 𝑻 𝐴∪𝐴 = 𝐴 𝐴∩𝐴 = 𝐴 𝐴∩𝐵 = 𝐵∩𝐴 𝐴∪𝐵 = 𝐵∪𝐴 t 𝐴 ∪ (𝐵 ∪ 𝐶) = (𝐴 ∪ 𝐵) ∪ 𝐶 𝐴 ∩ (𝐵 ∩ 𝐶) = (𝐴 ∩ 𝐵) ∩ 𝐶 h m 𝐴 ∩ (𝐴 ∪ 𝐵) = 𝐴 𝐴 ∪ (𝐴 ∩ 𝐵) = 𝐴 𝐴 ∪ (𝐵 ∩ 𝐶) = (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶) 𝐴 ∩ (𝐵 ∪ 𝐶) = (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶) property with emphasis on ∪

dual property with emphasis on ∩

(idempotent) (commutative) (associative) (absorptive) (distributive) property name

✎PROOF: 1. By Definition 14.8 (page 211), 𝑻 is a topology. 22

http://groups.google.com/group/sci.math/browse_thread/thread/70a73e734b69a6ec/

version 0.30


Daniel J. Greenhoe


page 221


2. By Theorem 14.4 (page 214), ( 𝑻 , ∪, ∩ ; ⊆) is a distributive lattice. 3. The properties listed are all properties of distributive lattices, as provided by Theorem 4.3 (page 62), Definition 6.2 (page 90), and Theorem 6.1 (page 90).

✏ Proposition 14.13. Let 𝑨 be a set structure (Definition 14.1 page 201) on a set 𝑋. ∈𝑨 (𝑨 includes the ∪ identity element) ⎧ 1. ∅ ⎪ 2. 𝑋 ∈𝑨 (𝑨 includes the ∩ identity element) ⎪ 3. 𝐴𝖼 ∈ 𝑨 ∀𝐴∈𝑨 (𝑨 is closed under 𝖼) p ⎪ 𝑨 is an r ∀𝐴,𝐵∈𝑨 (𝑨 is closed under ∪) ⟹ ⎨ 4. 𝐴 ∪ 𝐵 ∈ 𝑨 p { algebra of sets } ⎪ 5. 𝐴 ∩ 𝐵 ∈ 𝑨 ∀𝐴,𝐵∈𝑨 (𝑨 is closed under ∩) ⎪ 6. 𝐴⧵𝐵 ∈ 𝑨 ∀𝐴,𝐵∈𝑨 (𝑨 is closed under⧵) ⎪ ∀𝐴,𝐵∈𝑨 (𝑨 is closed under △) ⎩ 7. 𝐴△𝐵 ∈ 𝑨

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

✎PROOF: ∅ = 𝐴 ∩ 𝐴𝖼 𝑋 = 𝖼∅ 𝐴 ∪ 𝐵 = 𝖼(𝐴𝖼 ∩ 𝐵 𝖼 ) 𝐴⧵𝐵 = 𝐴 ∩ 𝐵

by de Morgan's Law (Theorem 14.8 page 221)

𝖼

𝐴△𝐵 = (𝐴⧵𝐵 𝖼 ) ∪ (𝐵⧵𝐴)

✏

(𝑨, ∪,⧵) is a ring of sets because ∪ and⧵are closed in 𝑨 (as shown above).

Theorem 14.7. 23 Let 𝑨 be a set structure (Definition 14.1 page 201) on a set 𝑋. 𝑨 is an algebra of sets ⟹ ∀𝐴, 𝐵, 𝐶 ∈ 𝑨

t h m

𝐴∪𝐴 𝐴∪𝐵 𝐴 ∪ (𝐵 ∪ 𝐶) 𝐴 ∪ (𝐴 ∩ 𝐵) 𝐴 ∪ (𝐵 ∩ 𝐶) 𝐴∪∅ 𝐴∪𝑋 𝐴 ∪ 𝐴𝖼 (𝐴𝖼 )𝖼 (𝐴 ∪ 𝐵)𝖼

= = = = = = = = = =

𝐴 𝐵∪𝐴 (𝐴 ∪ 𝐵) ∪ 𝐶 𝐴 (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶) 𝐴 𝑋 𝑋 𝐴 𝐴𝖼 ∩ 𝐵 𝖼

property emphasizing ∪

𝐴∩𝐴 𝐴∩𝐵 𝐴 ∩ (𝐵 ∩ 𝐶) 𝐴 ∩ (𝐴 ∪ 𝐵) 𝐴 ∩ (𝐵 ∪ 𝐶) 𝐴∩𝑋 𝐴∩∅ 𝐴 ∩ 𝐴𝖼 (𝐴 ∩ 𝐵)𝖼

= = = = = = = =

𝐴 𝐵∩𝐴 (𝐴 ∩ 𝐵) ∩ 𝐶 𝐴 (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶) 𝐴 ∅ ∅

= 𝐴𝖼 ∪ 𝐵 𝖼

dual property emphasizing ∩

(idempotent) (commutative) (associative) (absorptive) (distributive) (identity) (bounded) (complemented) (uniquely complemented) (de Morgan) property name

✎PROOF: 1. By Definition 14.9 (page 214), 𝑺 is an algebra of sets. 2. By the Stone Representation Theorem (Theorem 14.4 page 214), ( 𝑺, ∪, ∩, ∅, 𝑋 ; ⊆) is a Boolean algebra. 3. The properties listed are all properties of Boolean algebras (Theorem 8.2 page 120).

✏ Theorem 14.8.

24

Let 𝑨 be an algebra of sets (Definition 14.9 page 214) on a set 𝑋.

23

📘 Dieudonné (1969) pages 3–4, 📘 Copson (1968) page 9 📘 Michel and Herget (1993) page 12, 📘 Aliprantis and Burkinshaw (1998) page 4, 📘 Vaidyanathaswamy (1960) pages 3–4 24


Daniel J. Greenhoe


version 0.30

page 222

14.5. OPERATIONS ON SET STRUCTURES

𝑨 is an algebra of sets ⟹ 𝖼 𝘕 𝘕 ⎛ ⎞ ⎜ 𝐴𝑛 ⎟ = 𝐴𝖼 ⋂ 𝑛 ⎜⋃ ⎟ 𝑛=1 ⎝𝑛=1 ⎠ 𝘕 ⎛𝘕 ⎞ ⎜ 𝐴𝑛 ⎟ ∩ 𝐵 = 𝐴 ∩ 𝐵) ⋃( 𝑛 ⎟ ⎜⋃ 𝑛=1 ⎝𝑛=1 ⎠ 𝘕 𝘕 ⎛ ⎞ ⎜ 𝐴𝑛 ⎟ ⧵𝐵 = 𝐴 ⧵𝐵 ⋃( 𝑛 ) ⎜⋃ ⎟ 𝑛=1 ⎝𝑛=1 ⎠

t h m

property emphasizing ∪

∀𝐴1 , 𝐴2 , … , 𝐴𝘕 , 𝐵 ∈ 𝑨 and ∀𝘕 ∈ ℕ 𝖼 𝘕 ⎛𝘕 ⎞ ⎜ 𝐴𝑛 ⎟ = 𝐴𝖼 (de Morgan) ⋃ 𝑛 ⎜⋂ ⎟ 𝑛=1 𝑛=1 ⎠ ⎝ 𝘕 ⎛𝘕 ⎞ ⎛ distributive ⎜ 𝐴𝑛 ⎟ ∪ 𝐵 = 𝐴𝑛 ∪ 𝐵 ) ⎜⎜ with respect to ( ⋂ ⎟ ⎜⋂ ⎝ ∪ and ∩ 𝑛=1 ⎝𝑛=1 ⎠ 𝘕 ⎛𝘕 ⎞ ⎛ distributive ⎜ 𝐴𝑛 ⎟ ⧵𝐵 = 𝐴𝑛 ⧵𝐵 ) ⎜⎜ with respect to ( ⋂ ⎜⋂ ⎟ ⎝ ⧵and ∩ 𝑛=1 ⎝𝑛=1 ⎠ dual property emphasizing ∩

⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠

property name

✎PROOF: 1. By Theorem 14.4 (page 214), the lattice ( 𝑋, ∪, ∩ ; ⊆) is Boolean. 2. The first four properties are true any Boolean system Theorem 8.4 (page 121). 3. Proof for the remaining two:

⎛𝘕 ⎞ ⎛𝘕 ⎞ ⎜ 𝐴𝑛 ⎟ ⧵𝐵 = ⎜ 𝐴𝑛 ⎟ ∩ 𝐵 𝖼 ⎜⋂ ⎟ ⎜⋂ ⎟ ⎝𝑛=1 ⎠ ⎝𝑛=1 ⎠


𝘕

=

(𝐴 ∩ 𝐵 𝖼 ) ⋂ 𝑛

by previous result

𝑛=1 𝘕

=

(𝐴 ⧵ 𝐵) ⋂ 𝑛


𝑛=1

⎛𝘕 ⎞ ⎛𝘕 ⎞ ⎜ 𝐴𝑛 ⎟ ⧵𝐵 = ⎜ 𝐴𝑛 ⎟ ∩ 𝐵 𝖼 ⎜⋃ ⎟ ⎜⋃ ⎟ ⎝𝑛=1 ⎠ ⎝𝑛=1 ⎠


𝘕

=

(𝐴 ∩ 𝐵 𝖼 ) ⋃ 𝑛

by previous result

𝑛=1 𝘕

=


(𝐴 ⧵ 𝐵) ⋃ 𝑛 𝑛=1

✏ Proposition 14.14. p r p

25

𝑹 is a ⎧ ⎪ ⎨ ring of sets ⎪ ⎩ on 𝑋

25

⎫ ⎪ ⎬ ⎪ ⎭

Let 𝑹 be a set structure (Definition 14.1 page 201) on a set 𝑋. ∈𝑹 (𝑹 includes the ∪ identity element) ⎧ 1. ∅ ⎪ 2. 𝐴 ∪ 𝐵 ∈ 𝑹 ∀𝐴,𝐵∈𝑹 (𝑹 is closed under ∪) ⎪ ⟹ ⎨ 3. 𝐴 ∩ 𝐵 ∈ 𝑹 ∀𝐴,𝐵∈𝑹 (𝑹 is closed under ∩) ⎪ 4. 𝐴⧵𝐵 ∈ 𝑹 ∀𝐴,𝐵∈𝑹 (𝑹 is closed under⧵) ⎪ 5. 𝐴△𝐵 ∈ 𝑹 ∀𝐴,𝐵∈𝑹 (𝑹 is closed under △) ⎩

and and and and

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

📘 Berezansky et al. (1996) page 4, 📘 Halmos (1950) pages 19–20

version 0.30


Daniel J. Greenhoe


page 223


✎PROOF: 𝐴△𝐵 = (𝐴⧵𝐵) ∪ (𝐵⧵𝐴) 𝐴 ∩ 𝐵 = (𝐴 ∪ 𝐵)⧵(𝐴△𝐵) 𝐴⧵𝐴 = ∅

✏ Theorem 14.9. 26 Let 𝑹 be a set structure (Definition 14.1 page 201) on a set 𝑋. If 𝑹 is an ring of sets on 𝑋, then (𝑹, △, ∩) is an algebraic ring; in particular, 𝐴△∅ = 𝐴 ∀𝐴∈𝑹 𝐴 ∩ ∅ = ∅ ∀𝐴∈𝑹 t 𝐴△𝑋 = 𝐴𝖼 ∀𝐴∈𝑹 𝐴 ∩ 𝑋 = 𝐴 ∀𝐴∈𝑹 h 𝐴△∅ = 𝐴 ∀𝐴∈𝑹 𝐴 ∩ 𝐴 = 𝐴 ∀𝐴∈𝑹 m 𝐴 ∩ (𝐵 △𝐶) = (𝐴 ∩ 𝐵)△(𝐴 ∩ 𝐶) ∀𝐴,𝐵,𝐶∈𝑹 properties emphasizing △

properties emphasizing ∩

✎PROOF: 1. Proof that (𝑹, ∪,⧵) is an algebraic ring: by Theorem 14.9 (page 223) 2. Proof that a ring of sets is equivalent to (𝑹, ∪,⧵): This is proven simply by noting that ∪ and⧵(the two operations in a ring of sets (𝑹, ∪,⧵)) can be expressed in terms of △ and ∩ (the two operations in the algebraic ring (𝑹, △, ∩)) and vice-versa. And this is demonstrated by Theorem 14.1 (page 203).

1. Proof that (𝑺, △) is a group: see Proposition 14.1 (page 203). 2. Proof that 𝐴 ∩ (𝐵 ∩ 𝐶) = (𝐴 ∩ 𝐵) ∩ 𝐶: 𝐴 ∩ (𝐵 ∩ 𝐶) = {𝑥 ∈ 𝑋|(𝑥 ∈ 𝐴) ∧ [(𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶)]}

by definition of ∩ page 202

= {𝑥 ∈ 𝑋|[(𝑥 ∈ 𝐴) ∧ (𝑥 ∈ 𝐵)] ∧ (𝑥 ∈ 𝐶)} = (𝐴 ∩ 𝐵) ∩ 𝐶

by definition of ∩ page 202

3. Proof that 𝐴 ∩ (𝐵 △𝐶) = (𝐴 ∩ 𝐵)△(𝐴 ∩ 𝐶): 𝐴 ∩ (𝐵 △𝐶) = {𝑥 ∈ 𝑋|(𝑥 ∈ 𝐴) ∧ [(𝑥 ∈ 𝐵) ⊕ (𝑥 ∈ 𝐶)]}

by definition of ∩, △ page 202

= {𝑥 ∈ 𝑋|[(𝑥 ∈ 𝐴) ∧ (𝑥 ∈ 𝐵)] ⊕ [(𝑥 ∈ 𝐴) ∧ (𝑥 ∈ 𝐶)]} = (𝐴 ∩ 𝐵)△(𝐴 ∩ 𝐶)


4. Proof that (𝐴△𝐵) ∩ 𝐶 = (𝐴 ∩ 𝐶)△(𝐵 ∩ 𝐶): (𝐴△𝐵) ∩ 𝐶 = {𝑥 ∈ 𝑋|[(𝑥 ∈ 𝐴) ⊕ (𝑥 ∈ 𝐵)] ∧ (𝑥 ∈ 𝐶)}


= {𝑥 ∈ 𝑋|[(𝑥 ∈ 𝐴) ∧ (𝑥 ∈ 𝐶)] ⊕ [(𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶)]} = (𝐴 ∩ 𝐶)△(𝐵 ∩ 𝐶)


✏ 26

📘 Vaidyanathaswamy (1960) pages 17–18, 📘 Kelley and Srinivasan (1988) page 22, 📘 Wilker (1982), page 211, 📘 Vaidyanathaswamy (1960) page 19 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 224

14.6. LATTICES OF SET STRUCTURES

14.6

Lattices of set structures

14.6.1

Ordering relations

The set inclusion relation ⊆ (Definition 14.13 page 224) is an order relation (Definition 3.2 page 46) on set structures, as demonstrated by Proposition 14.15 (next proposition). Definition 14.13. Let 𝑺 be a set structure (Definition 14.1 page 201) on a set 𝑋. d The relation ⊆∈ 𝟚𝑺𝑺 is defined as e 𝐴⊆𝐵 if 𝑥∈𝐴 ⟹ 𝑥∈𝐵 ∀𝑥 ∈ 𝑋 f Proposition 14.15 (order properties). Let 𝑺 be a set structure (Definition 14.1 page 201) on a set 𝑋. The pair (𝑺, ⊆) is an ordered set. In particular, p 𝐴 ⊆ 𝐴 ∀𝐴∈𝑺 (reflexive) and r 𝐴 ⊆ 𝐵 and 𝐵 ⊆ 𝐶 ⟹ 𝐴 ⊆ 𝐶 ∀𝐴,𝐵,𝐶∈𝑺 (transitive) and p 𝐴 ⊆ 𝐵 and 𝐵 ⊆ 𝐴 ⟹ 𝐴 = 𝐵 ∀𝐴,𝐵∈𝑺 (anti-symmetric). ✎PROOF:

By Definition 3.2 (page 46), a relation is an order relation if it is reflexive, transitive, and anti-

symmetric. 1. Proof that ⊆ is reflexive on 𝟚𝑋 : 𝑥∈𝐴 ⟹ 𝑥∈𝐴 ⟹ 𝐴⊆𝐴 2. Proof that ⊆ is transitive on 𝟚𝑋 : 𝑥∈𝐴 ⟹ 𝑥∈𝐵

by first left hypothesis

⟹ 𝑥∈𝐶

by second left hypothesis

⟹ 𝐴⊆𝐶


3. Proof that ⊆ is anti-symmetric on 𝟚𝑋 : 𝐴 ⊆ 𝐵 ⟹ (𝑥 ∈ 𝐴 ⟹ 𝑥 ∈ 𝐵) 𝐵 ⊆ 𝐴 ⟹ (𝑥 ∈ 𝐵 ⟹ 𝑥 ∈ 𝐴) 𝐴 ⊆ 𝐵 and 𝐵 ⊆ 𝐴 ⟹ (𝑥 ∈ 𝐴 ⟺ 𝑥 ∈ 𝐵) ⟹ 𝐴=𝐵

✏ In a set structure that is closed under the union operation ∪ and intersection operation ∩, the greatest lower bound of any two elements 𝐴 and 𝐵 is simply 𝐴 ∩ 𝐵 and least upper bound is simply 𝐴 ∪ 𝐵 (Proposition 14.16 page 224). However, this may not be true for a set structure that is not closed under these operations (Example 14.12 page 225). Proposition 14.16. Let 𝑺 be a set structure (Definition 14.1 page 201) on a set 𝑋. p If 𝑺 is closed under ∪ and ∩ then r 𝐴 ∪ 𝐵 is the least upper bound of 𝐴 and 𝐵 in (𝑺, ⊆) (∪ = ∨) p 𝐴 ∩ 𝐵 is the greatest lower bound of 𝐴 and 𝐵 in (𝑺, ⊆) (∩ = ∧). version 0.30


Daniel J. Greenhoe

and


page 225


✎PROOF: 1. Proof that 𝐴 ∪ 𝐵 is the least upper bound: 𝐴 = {𝑥 ∈ 𝑋|𝑥 ∈ 𝐴} ⊆ {𝑥 ∈ 𝑋|𝑥 ∈ 𝐴 or 𝑥 ∈ 𝐵} =𝐴∪𝐵


𝐵 = {𝑥 ∈ 𝑋|𝑥 ∈ 𝐵} ⊆ {𝑥 ∈ 𝑋|𝑥 ∈ 𝐴 or 𝑥 ∈ 𝐵} by Definition 14.5 page 202

=𝐴∪𝐵 𝐴 ⊆ 𝐶 and 𝐵 ⊆ 𝐶 ⟹ {𝑥 ∈ 𝐴 and 𝑦 ∈ 𝐵 ⟹ 𝑥, 𝑦 ∈ 𝐶 } ⟹ {𝑥 ∈ 𝐴 or 𝑥 ∈ 𝐵 ⟹ 𝑥 ∈ 𝐶} ⟹ {𝑥 ∈ 𝐴 ∪ 𝐵

⟹

𝑥 ∈ 𝐶}

⟹ 𝐴∪𝐵 ⊆𝐶 2. Proof that 𝐴 ∩ 𝐵 is the greatest lower bound: 𝐴 ∩ 𝐵 = {𝑥 ∈ 𝑋|𝑥 ∈ 𝐴 and 𝑥 ∈ 𝐵 }


⊆ {𝑥 ∈ 𝑋|𝑥 ∈ 𝐴} =𝐴 𝐴 ∩ 𝐵 = {𝑥 ∈ 𝑋|𝑥 ∈ 𝐴 and 𝑥 ∈ 𝐵 }


⊆ {𝑥 ∈ 𝑋|𝑥 ∈ 𝐵} =𝐵 𝐶 ⊆ 𝐴 and 𝐶 ⊆ 𝐵 ⟹ {𝑥 ∈ 𝐶 ⟹ 𝑥 ∈ 𝐴 and 𝑥 ∈ 𝐶 ⟹ 𝑥 ∈ 𝐵 } ⟹ {𝑥 ∈ 𝐶 ⟹ 𝑥 ∈ 𝐴 or 𝑥 ∈ 𝐵} ⟹ {𝑥 ∈ 𝐶

⟹

𝑥 ∈ 𝐴 ∩ 𝐵}

⟹ 𝐶 ⊆𝐴∩𝐵

✏ Example 14.12. The set structure 𝑺 ≜ {∅, {𝑥}, {𝑦}, {𝑧}, {𝑥, 𝑦}, {𝑥, 𝑦, 𝑧}} ordered by the set inclusion relation ⊆ is illustrated by the Hasse diagram to the right. Note that {𝑥} ∨ {𝑧} = {𝑥, 𝑦, 𝑧} ≠ {𝑥, 𝑧} = {𝑥} ∪ {𝑧}. That is, the set union operation ∪ is not equivalent to the order join operation ∨.

{𝑥, 𝑦, 𝑧} {𝑥, 𝑦} {𝑥}

{𝑦}

{𝑧}

∅

14.6.2 Lattices of topologies Example 14.13. 27 Example 14.3 (page 211) lists the four topologies on the set 𝑋 ≜ {𝑥, 𝑦}. The lattice of these topologies ( {𝑻1 , 𝑻2 , 𝑻3 , 𝑻4 }, ∪, ∩ ; ⊆) is illustrated by the Hasse diagram to the right. 27

𝟚{𝑥,𝑦} ≜ { ∅, {𝑥}, {𝑦}, {𝑥, 𝑦}}

{ ∅, {𝑥}, {𝑥, 𝑦}}

finer coarser

{ ∅, {𝑦}, {𝑥, 𝑦}}

{ ∅, {𝑥, 𝑦}}

📘 Isham (1999), page 44, 📘 Isham (1989), page 1515


Daniel J. Greenhoe


version 0.30

page 226


𝑻77 𝑻33

𝑻35

𝑻65

𝑻66

𝑻53

𝑻56 𝑻25

𝑻13

𝑻31

𝑻64

𝑻46

𝑻52 𝑻12

𝑻22

𝑻42

𝑻41

𝑻14

𝑻11

𝑻21

𝑻44

𝑻24

𝑻40 𝑻10 𝑻01

𝑻20 𝑻02

𝑻04 𝑻00

Figure 14.4: Lattice of topologies on 𝑋 ≜ {𝑥, 𝑦, 𝑧} (see Example 14.14 page 226) Example 14.14. 28 Let a given topology in 𝒯 ({𝑥, 𝑦, 𝑧}) be represented by a Hasse diagram as illustrated to the right, where a circle present means the indicated set is in the topology, and a circle absent means the indicated set is not in the topology. Example 14.4 (page 212) lists the 29 topologies 𝒯 ({𝑥, 𝑦, 𝑧}). The lattice of these 29 topologies ( 𝒯 ({𝑥, 𝑦, 𝑧}), ∪, ∩ ; ⊆) is illustrated in Figure 14.4 (page 226). The five topologies 𝑻1 , 𝑻41 , 𝑻22 , 𝑻14 , and 𝑻77 are also algebras of sets (Definition 14.9 page 214); these five sets are shaded in Figure 14.4.

{𝑥, 𝑦, 𝑧} {𝑥, 𝑦}

{𝑥, 𝑧} {𝑦, 𝑧}

{𝑥}

{𝑧} ∅

{𝑦}

Theorem 14.10. 29 Let 𝒯 (𝑋) be the lattice of topologies on a set 𝑋 with | 𝑋 | elements. t | 𝑋 | ≤ 2 ⟹ 𝒯 (𝑋) is distributive h m | 𝑋 | ≥ 3 ⟹ 𝒯 (𝑋) is not modular (and not distributive) Theorem 14.11. 30 Let 𝒯 (𝑋) be the lattice of topologies on a set 𝑋. t h 𝒯 (𝑋) is self-dual ⟺ |𝑋 | ≤ 3 m Theorem 14.12. 31 t h Every lattice of topologies is complemented. m 28

📘 Isham (1999), page 44, 📘 Isham (1989), page 1516, 📘 Steiner (1966), page 386 📘 Steiner (1966), page 384 30 📘 Steiner (1966), page 385 31 📘 van Rooij (1968), 📘 Steiner (1966), page 397, 📘 Gaifman (1961), 📘 Hartmanis (1958)

29

version 0.30


Daniel J. Greenhoe


page 227


Theorem 14.13. 32 t Every topology (Definition 14.8 page 211) except the discrete topology and indiscrete topology (Example h m 14.2 page 211) in the lattice of topologies on a set 𝑋 has at least | 𝑋 | − 1 complements. Example 14.15. Example 14.4 (page 212) lists the 29 topologies on a set 𝑋 ≜ {𝑥, 𝑦, 𝑧}. By Theorem 14.13 (page 227), with the exception of 𝑻00 (the indiscrete topology) and 𝑻77 (the discrete topology), each of those topologies has exactly | 𝑋 | − 1 = 3 − 1 = 2 complements. Listed below are the 29 topologies on {𝑥, 𝑦, 𝑧} along with their respective complements. topologies on {𝑥, 𝑦, 𝑧} 𝑻00 𝑻01 𝑻02 𝑻04 𝑻10 𝑻20 𝑻40 𝑻11 𝑻21 𝑻41 𝑻12 𝑻22 𝑻42 𝑻14 𝑻24 𝑻44 𝑻31 𝑻52 𝑻64 𝑻13 𝑻25 𝑻46 𝑻33 𝑻53 𝑻35 𝑻65 𝑻56 𝑻66 𝑻77

={∅, 𝑋 ={∅, {𝑥}, 𝑋 {𝑦}, ={∅, 𝑋 {𝑧}, ={∅, 𝑋 {𝑥, 𝑦}, ={∅, 𝑋 {𝑥, 𝑧}, ={∅, 𝑋 {𝑦, 𝑧}, 𝑋 ={∅, {𝑥, 𝑦}, ={∅, {𝑥}, 𝑋 {𝑥, 𝑧}, ={∅, {𝑥}, 𝑋 {𝑦, 𝑧}, 𝑋 ={∅, {𝑥}, {𝑦}, {𝑥, 𝑦}, ={∅, 𝑋 {𝑦}, {𝑥, 𝑧}, ={∅, 𝑋 {𝑦}, {𝑦, 𝑧}, 𝑋 ={∅, {𝑧}, {𝑥, 𝑦}, ={∅, 𝑋 {𝑧}, {𝑥, 𝑧}, ={∅, 𝑋 {𝑧}, {𝑦, 𝑧}, 𝑋 ={∅, {𝑥, 𝑦}, {𝑥, 𝑧}, ={∅, {𝑥}, 𝑋 {𝑦}, {𝑥, 𝑦}, {𝑥, 𝑧}, ={∅, 𝑋 {𝑧}, {𝑥, 𝑧}, {𝑦, 𝑧}, 𝑋 ={∅, {𝑥, 𝑦}, ={∅, {𝑥}, {𝑦}, 𝑋 {𝑧}, {𝑥, 𝑧}, ={∅, {𝑥}, 𝑋 {𝑦}, {𝑧}, {𝑦, 𝑧}, 𝑋 ={∅, {𝑥, 𝑦}, {𝑥, 𝑧}, ={∅, {𝑥}, {𝑦}, 𝑋 {𝑥, 𝑦}, {𝑦, 𝑧}, 𝑋 ={∅, {𝑥}, {𝑦}, {𝑧}, {𝑥, 𝑦}, {𝑥, 𝑧}, ={∅, {𝑥}, 𝑋 {𝑧}, {𝑥, 𝑧}, {𝑦, 𝑧}, 𝑋 ={∅, {𝑥}, {𝑦}, {𝑧}, {𝑥, 𝑦}, {𝑦, 𝑧}, 𝑋 ={∅, {𝑦}, {𝑧}, {𝑥, 𝑧}, {𝑦, 𝑧}, 𝑋 ={∅, ={∅, {𝑥}, {𝑦}, {𝑧}, {𝑥, 𝑦}, {𝑥, 𝑧}, {𝑦, 𝑧}, 𝑋

Theorem 14.14. 33 t h 𝒯 (𝑋) is a topology of sets m Theorem 14.15.

34

⟹

} } } } } } } } } } } } } } } } } } } } } } } } } } } } }

1st complement 𝑻77 𝑻56 𝑻65 𝑻53 𝑻65 𝑻53 𝑻33 𝑻64 𝑻52 𝑻22 𝑻64 𝑻41 𝑻31 𝑻41 𝑻52 𝑻31 𝑻42 𝑻21 𝑻11 𝑻24 𝑻12 𝑻11 𝑻04 𝑻04 𝑻02 𝑻02 𝑻01 𝑻01 𝑻00

2nd compl. 𝑻66 𝑻35 𝑻33 𝑻66 𝑻56 𝑻35 𝑻46 𝑻46 𝑻14 𝑻25 𝑻14 𝑻25 𝑻22 𝑻13 𝑻13 𝑻44 𝑻24 𝑻12 𝑻44 𝑻42 𝑻21 𝑻40 𝑻20 𝑻40 𝑻10 𝑻20 𝑻10

𝒯 (𝑋) is atomic. { 𝒯 (𝑋) is anti-atomic.

Let 𝒯 (𝑋) be the lattice of topologies on a set 𝑋 and let 𝑛 ≜ | 𝑋 |.

32

📘 Hartmanis (1958), 📘 Schnare (1968), page 56, 📘 Watson (1994), 📘 Brown and Watson (1996), page 32 📘 Larson and Andima (1975), page 179, 📘 Frölich (1964), 📘 Vaidyanathaswamy (1960), 📘 Vaidyanathaswamy (1947) 34 📘 Larson and Andima (1975), page 179, 📘 Frölich (1964) 33


Daniel J. Greenhoe


version 0.30

page 228

t h m


𝒯 (𝑋) contains 2𝑛 − 2 atoms 𝒯 (𝑋) contains 2| 𝑋 | atoms 𝒯 (𝑋) contains 𝑛(𝑛 − 1) anti-atoms |𝑋| 𝒯 (𝑋) contains 22 anti-atoms

14.6.3

for finite 𝑋. for infinite 𝑋. for finite 𝑋. for infinite 𝑋.

Lattices of algebra of sets

Example 14.16. The following table lists some algebras of sets on a finite set 𝑋. Lattices of algebras of sets are illustrated in Figure 14.7 (page 230) and Figure 14.5 (page 229). algebra of sets 𝒜(𝑋) on a set 𝑋 = {∅} }

𝒜(∅)

=

{ 𝑨1

𝒜({𝑥})

=

{ 𝑨1 = {∅, {𝑥}} }

𝒜({𝑥, 𝑦})

=

𝑨1 = {∅, 𝑋} { 𝑨2 = {∅, {𝑥}, {𝑦}, 𝑋} }

𝒜({𝑥, 𝑦, 𝑧})

=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

𝒜({𝑤, 𝑥, 𝑦, 𝑧}) ⎧ 𝑨1 = ⎪ 𝑨2 = ⎪ 𝑨3 = ⎪ ⎪ 𝑨4 = ⎪ 𝑨5 = ⎪ 𝑨6 = ⎪ 𝑨 = ⎪ 7 ⎨ 𝑨8 = ⎪ 𝑨9 = ⎪ 𝑨 = ⎪ 10 ⎪ 𝑨11 = ⎪ 𝑨12 = ⎪ 𝑨13 = ⎪ 𝑨 = ⎪ 14 𝑨 ⎩ 15 =

= { ∅, { ∅, { ∅, { ∅, { ∅, { ∅, { ∅, { ∅, { ∅, { ∅, { ∅, { ∅, { ∅, { ∅, 𝟚𝑋

14.6.4

𝑨1 𝑨2 𝑨3 𝑨4 𝑨5

= = = = =

{ { { { {

∅, {𝑦, 𝑧}, ∅, {𝑥}, {𝑦}, {𝑥, 𝑧}, ∅, {𝑧}, {𝑥, 𝑦}, ∅, ∅, {𝑥}, {𝑦}, {𝑧}, {𝑥, 𝑦}, {𝑥, 𝑧}, {𝑦, 𝑧},

{𝑤}, {𝑥}, {𝑦}, {𝑧}, {𝑤, 𝑥}, {𝑦, 𝑧}, {𝑤, 𝑦}, {𝑥, 𝑧}, {𝑤, 𝑧}, {𝑥, 𝑦}, {𝑤}, {𝑥}, {𝑤, 𝑥}, {𝑦, 𝑧}, {𝑤}, {𝑦}, {𝑤, 𝑦}, {𝑥, 𝑧}, {𝑤}, {𝑧}, {𝑤, 𝑧}, {𝑥, 𝑦}, {𝑥}, {𝑦}, {𝑤, 𝑧}, {𝑥, 𝑦}, {𝑥}, {𝑧}, {𝑤, 𝑦}, {𝑥, 𝑧}, {𝑦}, {𝑧}, {𝑤, 𝑥}, {𝑦, 𝑧},

𝑋 𝑋 𝑋 𝑋 𝑋 𝑋 𝑋 𝑋 {𝑤, 𝑦, 𝑧}, {𝑥, 𝑦, 𝑧}, 𝑋 {𝑤, 𝑥, 𝑧}, {𝑥, 𝑦, 𝑧}, 𝑋 {𝑤, 𝑥, 𝑦}, {𝑥, 𝑦, 𝑧}, 𝑋 {𝑤, 𝑥, 𝑧}, {𝑤, 𝑦, 𝑧}, 𝑋 {𝑤, 𝑥, 𝑦}, {𝑤, 𝑦, 𝑧}, 𝑋 {𝑤, 𝑥, 𝑦}, {𝑤, 𝑥, 𝑧}, 𝑋 {𝑥, 𝑦, 𝑧}, {𝑤, 𝑦, 𝑧}, {𝑤, 𝑥, 𝑧}, {𝑤, 𝑥, 𝑦},

} } } } } } } } } } } } } }

𝑋 𝑋 𝑋 𝑋 𝑋

} } } } }

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

Lattices of rings of sets

Example 14.17. There are a total of 15 rings of sets on the set 𝑋 ≜ {𝑥, 𝑦, 𝑧}. These rings of sets are listed in Example 14.7 (page 215) and illustrated in Figure 14.6 (page 229). The five rings containing 𝑋 (𝑹11 –𝑹15 ) are also algebras of sets (Proposition 14.18 page 232), and thus also Boolean algebras (Theorem 14.4 version 0.30


Daniel J. Greenhoe


page 229


𝑨15

𝟚𝑋

∅, 𝑤, 𝑥, 𝑤𝑥, ∅, 𝑤, 𝑦, 𝑤𝑦, ∅, 𝑦, 𝑧, 𝑤𝑥, 𝑦𝑧, 𝑤𝑦𝑧, 𝑥𝑧, 𝑤𝑥𝑧, 𝑦𝑧, 𝑤𝑥𝑦, 𝑥𝑦𝑧, 𝑋 𝑥𝑦𝑧, 𝑋 𝑥𝑤𝑧, 𝑋 𝑨09

𝑨10

∅, 𝑤, 𝑥𝑦𝑧, 𝑋

∅, 𝑤𝑥, 𝑦𝑧, 𝑋

𝑨02

𝑨06

𝑨12 ∅, 𝑥, 𝑦, 𝑤𝑧, ∅, 𝑥, 𝑧, 𝑤𝑦, ∅, 𝑤, 𝑧, 𝑤𝑧, 𝑥𝑦, 𝑤𝑥𝑧, 𝑥𝑧, 𝑤𝑥𝑦, 𝑥𝑦, 𝑤𝑥𝑦, 𝑤𝑦𝑧, 𝑋 𝑤𝑦𝑧, 𝑋 𝑥𝑦𝑧, 𝑋

𝑨14

∅, 𝑥, 𝑤𝑦𝑧, 𝑋 𝑨03

𝑨11

𝑨13

∅, 𝑤𝑦, 𝑥𝑧, 𝑋

∅, 𝑦, 𝑤𝑥𝑧, 𝑋 𝑨04

𝑨07 ∅, 𝑋

∅, 𝑤𝑧, 𝑥𝑦, 𝑋 𝑨08

∅, 𝑧, 𝑤𝑥𝑦, 𝑋 𝑨05

𝑨01

Figure 14.5: lattice of algebras of sets on {𝑤, 𝑥, 𝑦, 𝑧} (Example 14.16 page 228)

𝑹15

𝑹08

𝑹09

𝑹12

𝑹13

𝑹14

𝑹10

𝑹02

𝑹03

𝑹04

𝑹11

𝑹05

𝑹06 𝑹07

𝑹01

Figure 14.6: Lattice of rings of sets on 𝑋 ≜ {𝑥, 𝑦, 𝑧} (Example 14.17 page 228)


Daniel J. Greenhoe


version 0.30

page 230


partitions on {𝑥, 𝑦, 𝑧} (𝒫({𝑥, 𝑦, 𝑧}), ⊴)

rings on {𝑥, 𝑦} (ℛ({𝑥, 𝑦}), ⊆)

algebra of sets on {𝑥, 𝑦, 𝑧} (𝒜({𝑥, 𝑦, 𝑧}), ⊆) 𝑨77

𝑨22

𝑨41

𝑨14

𝑨00

Figure 14.7: Lattices of set structures (see Example 14.18 (page 230), Example 14.7 (page 215), and Example 14.16 (page 228)) page 214).

14.6.5

The five algebras of sets are shaded Figure 14.6.

Lattices of partitions of sets

Example 14.18. There are a total of 5 partitions of sets on the set 𝑋 ≜ {𝑥, 𝑦, 𝑧}. These sets are listed in Example 14.11 (page 217) and illustrated in Figure 14.7 (page 230). Example 14.19. There are a total of 15 partitions of sets on the set 𝑋 ≜ {𝑤, 𝑥, 𝑦, 𝑧}. These sets are listed in Example 14.11 (page 217) and illustrated in Figure 14.8 (page 232). In 1946, Philip Whitman proposed an amazing conjecture—that all finite lattices are isomorphic to a lattice of partitions. A proof for this was published some 30 years later by Pavel Pudlák and Jiří Tůma (next theorem). Theorem 14.16. t h 𝙇 is finite m

35

Let 𝙇 be a lattice. ⟹

𝙇 is isomorphic to a lattice of partitions

Example 14.20. There are five unlabeled lattices on a five element set as stated in Proposition 4.2 (page 67) and illustrated in Example 4.11 (page 67). All of these lattices are isomorphic to a lattice of partitions (Theorem 14.16 page 230), as illustrated next. lattices on 5 element set as lattices of partitions {{𝑥}, {𝑦}, {𝑧}} {{𝑤}, {𝑥}, {𝑦}, {𝑧}} {{𝑥}, {𝑦, 𝑧}}

{{𝑤}, {𝑥}, {𝑦, 𝑧}} {{𝑧}, {𝑥, 𝑦}} {{𝑦}, {𝑥, 𝑧}} {{𝑤, 𝑥}, {𝑦, 𝑧}}

{{𝑤}, {𝑥}, {𝑦, 𝑧}}

{{𝑥, 𝑦, 𝑧}}

𝑦, 𝑧}} ⟨proof⟩, 📘 Whitman (1946) 📘 Pudlák and Tůma (1980) ⟨improved proof⟩, 📘 Pudlák and {{𝑤, Tůma𝑥,(1977) ⟨conjecture⟩, 📘 Saliǐ (1988) page vii ⟨list of lattice theory breakthroughs⟩ 35

version 0.30


Daniel J. Greenhoe


page 231


{{𝑤}, {𝑥}, {𝑦}, {𝑧}} {{𝑤}, {𝑥, 𝑦}, {𝑧}} {{𝑤}, {𝑥, 𝑦, 𝑧}}

{{𝑤}, {𝑥}, {𝑦}, {𝑧}} {{𝑤, 𝑥}, {𝑦, 𝑧}}

{{𝑤}, {𝑥}, {𝑦, 𝑧}}

{{𝑤, 𝑥, 𝑦}, {𝑧}} {{𝑤, 𝑥, 𝑦, 𝑧}}

{{𝑤, 𝑥, 𝑦, 𝑧}}

{{𝑤, 𝑥, 𝑦, 𝑧}} {{𝑥, 𝑦, 𝑧}} {{𝑥, 𝑦}} {{𝑥}} {∅}

14.7 Relationships between set structures Proposition 14.17. 36 p 𝑹 is a ring of sets r } p { on a set 𝑋

𝑹 ∪ 𝑋 is an algebra of sets { on 𝑋 }

⟹

Theorem 14.17. Let 𝑋 be a set. t 𝑨 is an algebra of sets h { } on 𝑋 m

⟺

1.

{

2.

𝑨 is a topology on 𝑋 𝑨 is a ring of sets on 𝑋

and

}

✎PROOF: 𝑨 is an algebra of sets on 𝑋 ⟹ 𝑨 is closed under ∪, ∩, 𝖼,⧵, ∅, 𝑋 1. 𝑨 is a topology on 𝑋 ⎧ ⎪ AND ⟹ ⎨ ⎪ ⎩ 2. 𝑨 is a ring of sets on 𝑋 1. 𝑨 is a topology on 𝑋 ⎧ ⎪ AND ⎨ ⎪ 2. 𝑨 is a ring of sets on 𝑋 ⎩

⎫ ⎪ ⎬ ⟹ 𝑨 is closed under 𝖼 and ∩ ⎪ ⎭ ⟹ 𝑨 is a ring of sets


⎫ ⎪ ⎬ ⎪ ⎭


✏ Corollary 14.1. Let 𝑋 be a set and 𝟚𝑋 the power set of 𝑋. c 𝑨 ⊆ 𝟚𝑋 |𝑨 is an algebra of sets on 𝑋 } o { = {𝑻 ⊆ 𝟚𝑋 |𝑻 is a topology on 𝑋 } ∩ {𝑹 ⊆ 𝟚𝑋 |𝑹 is a ring of sets on 𝑋 } r ✎PROOF: {𝑻 |𝑻 is a topology} ∩ {𝑹|𝑹 is a ring of sets} = {𝒀 |𝒀 is a topology AND a ring of sets}


= {𝒀 |𝒀 is an algebra of sets}


= {𝑨|𝑨 is an algebra of sets}

by change of variable

✏ 36

📘 Berezansky et al. (1996) page 4, 📘 Halmos (1950) page 21


Daniel J. Greenhoe


version 0.30

page 232

14.7. RELATIONSHIPS BETWEEN SET STRUCTURES

{𝑤}, {𝑥} {𝑦}, {𝑧}

𝑷09

𝑷10

{𝑤}, {𝑥} {𝑦, 𝑧}

{𝑤}, {𝑥, 𝑦, 𝑧} 𝑷02

𝑷14

{𝑤}, {𝑦} {𝑥, 𝑧}

{𝑤, 𝑥}, {𝑦, 𝑧}

𝑷15

{𝑦}, {𝑧} {𝑤, 𝑥}

{𝑥}, {𝑤, 𝑦, 𝑧} 𝑷06

𝑷03

𝑷12 {𝑥}, {𝑦} {𝑤, 𝑧}

{𝑤, 𝑦}, {𝑥, 𝑧}

{𝑥}, {𝑧} {𝑤, 𝑦}

{𝑦}, {𝑤, 𝑥, 𝑧} 𝑷07

𝑷11

𝑷13

𝑷04

{𝑤}, {𝑧} {𝑥, 𝑦}

{𝑤, 𝑧}, {𝑥, 𝑦}

{𝑧}, {𝑤, 𝑥, 𝑦}

𝑷08

𝑷05

{𝑤, 𝑥, 𝑦, 𝑧} 𝑷01

Figure 14.8: Lattice of partitions of sets on 𝑋 ≜ {𝑤, 𝑥, 𝑦, 𝑧} (Example 14.19 page 230) Example 14.21. Note that the intersection of the lattice of topologies on {𝑥, 𝑦, 𝑧} (Figure 14.4 page 226) and the lattice of rings of sets on {𝑥, 𝑦, 𝑧} (Figure 14.6 page 229) is equal to the lattice of algebras of sets on {𝑥, 𝑦, 𝑧} (Figure 14.7 page 230). Proposition 14.18. Let ℛ(𝑋) be the set of rings of sets (Definition 14.10 page 215) and 𝒜(𝑋) the set of algebras of sets (Definition 14.9 page 214) on a set 𝑋. p 1. 𝑹 is a ring of sets and r ⟺ { 𝑹 is an algebra of sets } { } 2. 𝑋 ∈ 𝑹 p ✎PROOF: 𝐴𝖼 = 𝑋 ⧵𝐴


𝐴 ∩ 𝐵 = 𝐴⧵(𝐴⧵𝐵)


Therefore, (𝑹 ∪ 𝑋) is closed under 𝖼 and ∩, and thus by the definition of algebras of sets (Definition 14.9 page 214), (𝑹 ∪ 𝑋) is an algebra of sets. ✏

Definition 14.14. 37 d The Borel set 𝐁(𝑋, 𝑻 ) generated by the topological space (𝑋, 𝑻 ) e f is the 𝜎-algebra generated by the topology 𝑻 . Example 14.22. Suppose we have a dice with the standard six possible outcomes 𝑋. Suppose also we construct the following topology 𝑻 on 𝑋, and this in turn generates the following Borel set (𝜎algebra) 𝐵 on 𝑋: 37

📘 Aliprantis and Burkinshaw (1998) page 97

version 0.30


Daniel J. Greenhoe


page 233

CHAPTER 14. SET STRUCTURES ⦁ @ ⦁ ⦁@ @⦁@ @ ⦁@ @ @⦁

⦁ @⦁ ⦁ @ @⦁@ @ ⦁@ @ @⦁

⦁ @ ⦁ ⦁@ @@ @⦁ ⦁@ @ @⦁

⦁ @⦁ ⦁@ @ @⦁ ⦁@@ @ @⦁

⦁ @⦁ ⦁ @ @ @⦁@ @⦁ @ @⦁

⦁ @⦁ ⦁@ @ @⦁@ @⦁ @ @⦁

⦁ @ ⦁ ⦁@ @ @ @ @ ⦁ @ @⦁

⦁ @⦁ ⦁ @ @ @⦁@ @ @ @⦁

⦁ @⦁ ⦁@ @ @@ @⦁ @ @⦁

⦁ @ ⦁ @ @ @ @ ⦁ ⦁@ @ @⦁

⦁ @ ⦁@ @ @ @ ⦁ @⦁ @ @⦁

⦁ @ @⦁ @ @⦁@ @⦁ @ @⦁

⦁ @ ⦁ @ @ @ @ @ ⦁ @ @⦁

⦁ @ ⦁@ @ @ @ ⦁ @ @ @⦁

⦁ @ ⦁ @ @ @⦁@ @ @ @⦁

⦁ @ @⦁ @ @⦁@ @ @ @⦁

⦁ @ ⦁@ @ @@ @⦁ @ @⦁

⦁ @ @⦁ @ @@ @⦁ @ @⦁

⦁ @ @ @ @ ⦁@ @ @ @⦁

⦁ @ @⦁ @ @@ @ @ @⦁

⦁ @ @ @ @⦁@ @ @ @⦁

⦁ @ ⦁@ @ @@ @ @ @⦁

⦁ @ @ @ @@ @⦁ @ @⦁

⦁ @ @ @ @@ @⦁ @ @⦁

? ⦁ @ @⦁ @ @ @ ⦁ @ @ @⦁

? ⦁ @ ⦁@ @ @⦁@ @ @ @⦁

⦁ @ @ - ⦁@⦁@⦁ @ ⦁ ⦁@⦁ @ @⦁

? ⦁ @ ⦁ @ @ @@ @⦁ @ @⦁

⦁ @ @ @ @@ @ @ @⦁

Figure 14.9: Algebras of sets generated by topologies on the set 𝑋 ≜ {𝑥, 𝑦, 𝑧} (see Example 14.23 page 233)

e x

𝑋 = {⚀,⚁,⚂,⚃,⚄,⚅} ⎧ ⎪ 𝑻 = ⎨{⏟}, {⚀,⚁,⚂,⚃,⚄,⚅} ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟, ⎪ ∅ 𝛺 ⎩ ⎧ ⎪ 𝐵 = ⎨{⏟}, {⚀,⚁,⚂,⚃,⚄,⚅} ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟, ⎪ ∅ 𝛺 ⎩

{⚀,⚁,⚂,⚃} {⚃,⚄,⚅}, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟, ⏟⏟⏟⏟⏟⏟⏟ first four

last three

{⚀,⚁,⚂,⚃} ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟, {⚃,⚄,⚅} ⏟⏟⏟⏟⏟⏟⏟, first four

last three

⎫ ⎪ , ⎬ {1234} ∩ {456} ⎪ ⎭ {⚃} ⏟

{⚃} ⏟

,

{1234} ∩ {456}

⎫ ⎪ {⚀,⚁,⚂,⚄,⚅} {⚀,⚁,⚂}, ⎬ ⏟ , ⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟, {⚄,⚅} ⎪ ({4}) ∩ {456} {4} {1234} ∩ {4} ⎭

Example 14.23. There are a total of 29 topologies on the set 𝑋 ≜ {𝑥, 𝑦, 𝑧} (Proposition 14.9 page 218); and of these, 5 are also algebras of sets, 24 are not. Figure 14.9 (page 233) illustrates the 24 topologies on the set {𝑥, 𝑦, 𝑧} that are not algebras of sets and the 5 algebras of sets that they generate.

14.8 Literature 📚 Literature survey: 1. Origin of the symbols ∪ and ∩: 📘 Peano (1888a) 📘 Peano (1888b) 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 234

14.8. LITERATURE

2. There is some difference in the definition of ring of sets: (a) ring of sets defined as closed under △, ∩: 📘 Stone (1936), page 38 📘 Kolmogorov and Fomin (1975) page 31 📘 Kolmogorov and Fomin (1999) page 20 📘 Constantinescu (1984) page 155 (b) ring of sets defined as closed under ∪,⧵(compatible definition): 📘 Wilker (1982), page 211 📘 Kelley and Srinivasan (1988) page 21 📘 Aliprantis and Burkinshaw (1998) page 96 📘 Haaser and Sullivan (1991) page 2 📘 Hewitt and Ross (1994) page 118 (c) ring of sets defined as closed under ∪,⧵, ∅ (compatible definition): 📘 Rao (2004) page 15 (d) ring of sets defined as closed under ∪, ∩ (incompatible definition): 📘 Hausdorff (1927) ⟨???,p.77?⟩ 📘 Hausdorff (1937) page 90 📘 Birkhoff (1937), page 443 📘 Erdös and Tarski (1943), page 315 📘 MacLane and Birkhoff (1999) page 485 3. Relationship to lattices (order theory): 📘 Stone (1936) 4. More references dealing with set structures … 📘 Vaidyanathaswamy (1947) 📘 Bagley (1955) 📘 Hartmanis (1958) 📘 Vaidyanathaswamy (1960) 📘 Gaifman (1961) 📘 Gaifman (1966) 📘 Steiner (1966) 📘 van Rooij (1968) 📘 Schnare (1968) 📘 Rayburn (1969) 📘 Larson and Andima (1975) 📘 Pudlák and Tůma (1980) 📘 Brown and Watson (1991) 📘 Watson (1994) 📘 Brown and Watson (1996) 5. Partitions 📘 Deza and Deza (2006) page 142 📘 Day (1981) 📘 Rota (1964) 6. Distributive and modular properties in lattice of topologies (a) Remark that “It can be shewn easily that the lattice of topologies is not distributive.” 📘 Vaidyanathaswamy (1947) 📘 Vaidyanathaswamy (1960) page 134 (b) Proof that the lattice of 𝑇 1 topologies is not modular: 📘 Bagley (1955) (c) Proof that the lattice of topologies on any set with 3 or more elements is not modular (and thus also not distributive): 📘 Steiner (1966), page 384 7. Complements in lattice of topologies: (a) Proof that every lattice of topologies over a finite set is complemented: 📘 Hartmanis (1958) version 0.30


Daniel J. Greenhoe


page 235

CHAPTER 14. SET STRUCTURES (b) Proof that every lattice of topologies over a countably infinite set is complemented: 📘 Gaifman (1961) (c) Proof that every lattice of topologies over a any arbitrary set is complemented: 📘 Steiner (1966), page 397 (d) 📘 van Rooij (1968) ̂ (e) Every topology in 𝛴(𝑋) has at least 2 complements for | 𝑋 | ≥ 3: 📘 Hartmanis (1958) ̂ (f) Every topology in 𝛴(𝑋) has at least | 𝑋 | − 1 complements for | 𝑋 | ≥ 2: 📘 Schnare (1968) ̂ (g) A large number of topologies in 𝛴(𝑋) have at least 2| 𝑋 | complements for | 𝑋 | ≥ 4: 📘 Brown and Watson (1996)

📚


Daniel J. Greenhoe


version 0.30

page 236

version 0.30

14.8. LITERATURE


Daniel J. Greenhoe


CHAPTER 15 RELATIONS AND FUNCTIONS

15.1 Relations ❝A

dual relative term, such as “lover,” “benefactor,” “servant,” is a common name signifying a pair of objects. Of the two members of the pair, a determinate one is generally the first, and the other the second; so that if the order is reversed, the pair is not considered as remaining the same.❞ Charles Sanders Peirce (1839–1914), American mathematician and logician 1

15.1.1 Definition and examples A relation on the sets 𝑋 and 𝑌 is any subset of the Cartesian product 𝑋 × 𝑌 (next definition). Alternatively, a relation is a generalization of a function (Definition 15.8 page 249) in the sense that both are subsets of a Cartesian product, but the relation allows mapping from a single element in its domain to two different elements in its range, whereas functions do not— a single element in a function's domain may map to one and only one element in its range. The set of all relations in 𝑋 × 𝑌 is denoted 𝟚𝑋𝑌 , which is suitable since the number of relations in 𝑋 × 𝑌 when 𝑋 and 𝑌 are finite is 2| 𝑋 |⋅| 𝑌 | (Proposition 15.1 page 238). Examples include the following: Example 15.2 page 238 Relations in the Cartesian product {𝑥1 , 𝑥2 , 𝑥3 } × {𝑦1 , 𝑦2 } Example 15.20 page 251 Functions in the Cartesian product {𝑥1 , 𝑥2 , 𝑥3 } × {𝑦1 , 𝑦2 } Example 15.21 page 251 Functions in the Cartesian product {𝑥, 𝑦, 𝑧} × {𝑥, 𝑦, 𝑧} Example 15.18 page 250 discrete examples Example 15.19 page 250 continuous examples Definition 15.1. 1

2

Let 𝑋 and 𝑌 be sets.

quote: 📘 Peirce (1883a), page 187 image: http://www-history.mcs.st-andrews.ac.uk/history/PictDisplay/Peirce_Benjamin.html 2 📘 Maddux (2006) page 4, 📘 Halmos (1960) page 26

page 238

d e f

15.1. RELATIONS

A relation Ⓡ ∶ 𝑋 → 𝑌 is any subset of 𝑋 × 𝑌 . That is, Ⓡ⊆𝑋×𝑌 A pair (𝑥, 𝑦) ∈ Ⓡ is alternatively denoted 𝑥Ⓡ𝑦. The set of all relations that are subsets of 𝑋 × 𝑌 is denoted 𝟚𝑋𝑌 ; that is, 𝟚𝑋𝑌 ≜ {Ⓡ|Ⓡ ⊆ (𝑋 × 𝑌 )}.

Example 15.1. Let

𝑋

𝑋 ≜ {1, 2, 3}

Ⓡ

𝑌

(1, 𝐴)

𝑌 ≜ {𝐴, 𝐵}

1 b

Ⓡ ≜ {(1, 𝐴) , (2, 𝐴) , (3, 𝐵)}

2 b

The sets 𝑋 and 𝑌 and the relation Ⓡ are illustrated to the right.

3 b

b

𝐴

b

𝐵

(2, 𝐴)

(3, 𝐵)

Proposition 15.1. Let 𝟚𝑋𝑌 be the set of all relations from a set 𝑋 to a set 𝑌 . Let | ⋅ | be the counting measure for sets. p r 𝟚𝑋𝑌 | = 2| 𝑋×𝑌 | = 2| 𝑋 |⋅| 𝑌 | |⏟ p number of possible relations in 𝑋 × 𝑌

✎PROOF: 1. Let 𝑋 be a finite set with 𝑚 elements. 2. Let 𝑌 be a finite set with 𝑛 elements. 3. Then the number of elements in 𝑋 × 𝑌 is 𝑚𝑛. 4. A relation is any subset of 𝑋 × 𝑌 , which may (represent this with a 1) or may not (represent this with a 0) contain a given element of 𝑋 × 𝑌 . 5. Therefore, the number of possible relations is 2𝑚𝑛 = 2| 𝑋 |⋅| 𝑌 | .

✏ Example 15.2 (next) lists all of the 64 possible relations in the Cartesian product {𝑥1 , 𝑥2 , 𝑥3 } × {𝑦1 , 𝑦2 }. Eight of these 64 relations are also functions. These eight functions are listed in Example 15.20 (page 251). Of these eight functions, six are surjective. These six surjective functions are listed in Example 15.27 (page 254). Example 15.2. Let 𝑋 ≜ {𝑥1 , 𝑥2 , 𝑥3 } and 𝑌 ≜ {𝑦1 , 𝑦2 }. Let 𝟚𝑋𝑌 be the set of all relations in 𝑋 × 𝑌 . There are a total of | 𝟚𝑋𝑌 | = 2| 𝑋 |⋅| 𝑌 | = 23×2 = 64 possible relations. These are listed below. Of these 64 relations, only 8 are functions, as listed in Example 15.20 (page 251). relations in {𝑥1 , 𝑥2 , 𝑥3 } × {𝑦1 , 𝑦2 } Ⓡ1 Ⓡ2 Ⓡ3 Ⓡ4 version 0.30

= = = =

∅ { (𝑥1 , 𝑦1 ) , { { (𝑥1 , 𝑦1 ) ,

} } }

(𝑥1 , 𝑦2 ) (𝑥1 , 𝑦2 )


Daniel J. Greenhoe


page 239

CHAPTER 15. RELATIONS AND FUNCTIONS

Ⓡ5 Ⓡ6 Ⓡ7 Ⓡ8 Ⓡ9 Ⓡ10 Ⓡ11 Ⓡ12 Ⓡ13 Ⓡ14 Ⓡ15 Ⓡ16 Ⓡ17 Ⓡ18 Ⓡ19 Ⓡ20 Ⓡ21 Ⓡ22 Ⓡ23 Ⓡ24 Ⓡ25 Ⓡ26 Ⓡ27 Ⓡ28 Ⓡ29 Ⓡ30 Ⓡ31 Ⓡ32 Ⓡ33 Ⓡ34 Ⓡ35 Ⓡ36 Ⓡ37 Ⓡ38 Ⓡ39 Ⓡ40 Ⓡ41 Ⓡ42 Ⓡ43 Ⓡ44 Ⓡ45 Ⓡ46 Ⓡ47 Ⓡ48 Ⓡ49 Ⓡ50 Ⓡ51 Ⓡ52 Ⓡ53 Ⓡ54

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

{ { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { {


(𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) ,

(𝑥2 , 𝑦1 ) (𝑥2 , 𝑦1 ) (𝑥1 , 𝑦2 ) , (𝑥2 , 𝑦1 ) (𝑥1 , 𝑦2 ) , (𝑥2 , 𝑦1 ) , (𝑥2 , 𝑦2 ) (𝑥2 , 𝑦2 ) (𝑥1 , 𝑦2 ) (𝑥2 , 𝑦2 ) (𝑥1 , 𝑦2 ) (𝑥2 , 𝑦2 ) 𝑥 , 𝑦 ( 2 1 ) ( 𝑥2 , 𝑦 2 ) (𝑥2 , 𝑦1 ) (𝑥2 , 𝑦2 ) (𝑥1 , 𝑦2 ) , (𝑥2 , 𝑦1 ) (𝑥2 , 𝑦2 ) (𝑥1 , 𝑦2 ) , (𝑥2 , 𝑦1 ) , (𝑥2 , 𝑦2 )

(𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) ,

(𝑥1 , 𝑦2 ) (𝑥1 , 𝑦2 ) (𝑥2 , 𝑦1 ) (𝑥2 , 𝑦1 ) (𝑥1 , 𝑦2 ) , (𝑥2 , 𝑦1 ) (𝑥1 , 𝑦2 ) , (𝑥2 , 𝑦1 ) , (𝑥2 , 𝑦2 ) (𝑥2 , 𝑦2 ) (𝑥2 , 𝑦2 ) (𝑥1 , 𝑦2 ) 𝑥 , 𝑦 (𝑥2 , 𝑦2 ) ( 1 2) (𝑥2 , 𝑦1 ) (𝑥2 , 𝑦2 ) (𝑥2 , 𝑦1 ) (𝑥2 , 𝑦2 ) , 𝑥 , 𝑦 ( 1 2 ) (𝑥2 , 𝑦1 ) (𝑥2 , 𝑦2 ) (𝑥1 , 𝑦2 ) , (𝑥2 , 𝑦1 ) , (𝑥2 , 𝑦2 )

(𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 )

(𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) ,

(𝑥1 , 𝑦2 ) (𝑥1 , 𝑦2 ) (𝑥2 , 𝑦1 ) (𝑥2 , 𝑦1 ) (𝑥1 , 𝑦2 ) , (𝑥2 , 𝑦1 ) (𝑥1 , 𝑦2 ) , (𝑥2 , 𝑦1 ) , (𝑥2 , 𝑦2 ) (𝑥2 , 𝑦2 ) (𝑥1 , 𝑦2 ) (𝑥2 , 𝑦2 ) 𝑥 , 𝑦 ( 1 2) (𝑥2 , 𝑦2 ) (𝑥2 , 𝑦1 ) (𝑥2 , 𝑦2 ) (𝑥2 , 𝑦1 ) (𝑥2 , 𝑦2 ) 𝑥 , 𝑦 , ( 1 2 ) (𝑥2 , 𝑦1 ) (𝑥2 , 𝑦2 ) (𝑥1 , 𝑦2 ) , (𝑥2 , 𝑦1 ) , (𝑥2 , 𝑦2 )

(𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) ,

(𝑥1 , 𝑦2 ) (𝑥1 , 𝑦2 )

(𝑥1 , 𝑦1 ) , Daniel J. Greenhoe

(𝑥2 , 𝑦1 ) (𝑥2 , 𝑦1 )

(𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 )

(𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 )


} } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } version 0.30

page 240

15.1. RELATIONS

Ⓡ55 Ⓡ56 Ⓡ57 Ⓡ58 Ⓡ59 Ⓡ60 Ⓡ61 Ⓡ62 Ⓡ63 Ⓡ64

= = = = = = = = = =

{ { { { { { { { { {

(𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) , (𝑥1 , 𝑦1 ) ,

(𝑥1 , 𝑦2 ) , (𝑥2 , 𝑦1 ) (𝑥1 , 𝑦2 ) , (𝑥2 , 𝑦1 ) , (𝑥2 , 𝑦2 ) (𝑥2 , 𝑦2 ) (𝑥1 , 𝑦2 ) (𝑥2 , 𝑦2 ) 𝑥 , 𝑦 ( 1 2) (𝑥2 , 𝑦2 ) (𝑥2 , 𝑦1 ) (𝑥2 , 𝑦2 ) (𝑥2 , 𝑦1 ) (𝑥2 , 𝑦2 ) 𝑥 , 𝑦 , ( 1 2 ) (𝑥2 , 𝑦1 ) (𝑥2 , 𝑦2 ) (𝑥1 , 𝑦2 ) , (𝑥2 , 𝑦1 ) , (𝑥2 , 𝑦2 )

(𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 ) (𝑥3 , 𝑦1 )

Example 15.3. Let 𝑋 ≜ {1, 2, 3}, 𝑌 ≜ {1, 2}, and 𝟚𝑋𝑌 the set of all of the 23×2 = 64 relations in 𝑋 × 𝑌 . Furthermore, let 𝑥1 ≜ 1, 𝑥2 ≜ 2, 𝑥3 ≜ 3, 𝑦1 ≜ 1, and 𝑦2 ≜ 2. Then the following common relations are the relations of Example 15.2 (page 238): ≤ ≥ < > =

15.1.2

≡ ≡ ≡ ≡ ≡

{(1, 1) , (1, 2) , (2, 2)} {(1, 1) , (2, 1) , (2, 2) , (3, 1) , (3, 2)} {(1, 2)} {(2, 1) , (3, 1) , (3, 2)} {(1, 1) , (2, 2)}

≡ ≡ ≡ ≡ ≡

Ⓡ12 Ⓡ62 Ⓡ3 Ⓡ53 Ⓡ10

𝑋

(𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 ) (𝑥3 , 𝑦2 )

≤

} } } } } } } } } }

𝑌

(1, 1)

1

b

2

b

(1, 2)

3

b

(2, 2)

b

1 b

2

Calculus of Relations

Proposition 15.2. 3 Let 𝟚𝑋𝑌 be the set of all relations in 𝑋 × 𝑌 . ∅ ∈ 𝟚𝑋𝑌 (∅ is a relation) ✎PROOF: ∅⊆𝑋×𝑌 ⟹ ∅ is a relation.

by definition of relation Definition 15.1 page 237

✏ Proposition 15.3. 4 Let 𝟚𝑋𝑌 be the set of all relations from the sets 𝑋 to the set 𝑌 . p Ⓡ ∈ 𝟚𝑋𝑌 (Ⓡ is a relation) and r ⟹ Ⓢ ∈ 𝟚𝑋𝑌 (Ⓢ is a relation) } Ⓢ ⊆ Ⓡ (Ⓢ is a subset of Ⓡ) p ✎PROOF: Ⓢ⊆Ⓡ

by right hypothesis

⊆𝑋×𝑌


⟹ ∅ is a relation.


✏ 3 4

📘 Suppes (1972) page 58 📘 Suppes (1972) page 58

version 0.30


Daniel J. Greenhoe


page 241


A function does not always have an inverse that is also a function. But unlike functions, every relation has an inverse that is also a relation. Note that since all functions are relations, every function does have an inverse that is at least a relation, and in some cases this inverse is also a function. Definition 15.2. 5 Let Ⓡ be a relation in 𝟚𝑋𝑌 . −1 the inverse of relation Ⓡ if d Ⓡ is −1 e Ⓡ ≜ {(𝑦, 𝑥) ∈ 𝑌 × 𝑋| (𝑥, 𝑦) ∈ Ⓡ} f The inverse relation Ⓡ−1 is also called the converse of Ⓡ. Example 15.4.

Let 𝑋 ≜ {1, 2, 3} and 𝑌 ≜ {𝐴, 𝐵} and Ⓡ ≜ {(1, 𝐴) , (2, 𝐴) , (2, 𝐵) , (3, 𝐵)}. −1 Then Ⓡ = {(𝐴, 1) , (𝐴, 2) , (𝐵, 2) , (𝐵, 3)}. The sets 𝑋 and 𝑌 and the relations Ⓡ and Ⓡ−1 are illustrated below. Ⓡ Ⓡ−1 𝑋 𝑌 𝑋 (1, 𝐴)

1 b

2 b

3 b

(2, 𝐴) (2, 𝐵)

(𝐴, 1) b

𝐴

1 b

2 b

3

(𝐴, 2) (𝐵, 2)

b

𝐵b

(3, 𝐵)

(𝐵, 3)

Example 15.5.

Example 15.6. Let Ⓡ be the ellipse relation in 𝟚ℝℝ such that 2 2 Ⓡ ≜ {(𝑥, 𝑦) ∈ ℝ2 | 𝑥2 + 𝑦2 = 1}. 2

𝑌

1 b

b

1

2 b

b

2

3 b

b

3

𝑦

𝑦

2

2

1

1

Then the inverse relation Ⓡ−1 is 2 2 Ⓡ−1 = {(𝑥, 𝑦) ∈ ℝ2 | 𝑥2 + 22 = 1}. 2 2 Both of these relations are illustrated to the right.

≤

𝑋

Let 𝑋 ≜ {1, 2, 3}. Then the “less than or equal to” relation ≤ in 𝟚𝑋𝑋 is ≤≡ {(1, 1) , (1, 2) , (1, 3) , (2, 2) , (2, 3) , (3, 3)} and it's inverse ≤−1 is equivalent to the “greater than or equal to” relation ≥: ≤−1 ≡ {(1, 1) , (2, 1) , (3, 1) , (2, 2) , (3, 2) , (3, 3)} ≡≥.

1 𝑥

−2 −1 −1

1

−2

2

𝑥 −2 −1 −1

1

2

−2

Ⓡ

Ⓡ−1

Example 15.7. Let 𝐈 ∈ 𝑋 𝑋 be an identity function, and 𝖿, 𝖿 −1 ∈ 𝑋 𝑋 be functions. 𝖿 −1 is the inverse of 𝖿 if 𝖿𝖿 −1 = 𝖿 −1 𝖿 = 𝐈. Theorem 15.1. 6 Let Ⓡ be a relation with inverse Ⓡ−1 . t h (Ⓡ−1 )−1 = Ⓡ m 5

📘 Suppes (1972) page 61 ⟨Defintion 6, inverse=“converse”⟩, 📘 Kelley (1955) page 7, 📘 Peirce (1883a) page 188 ⟨inverse=“converse”⟩ 6 📘 Kelley (1955) page 8, 📘 Peirce (1883a) page 188 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 242

15.1. RELATIONS

✎PROOF: −1 −1

(Ⓡ )

{(𝑥, 𝑦) | (𝑦, 𝑥) ∈ Ⓡ}−1 = ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

by definition of Ⓡ−1 (Definition 15.2 page 241)

Ⓡ−1

by definition of Ⓡ−1 (Definition 15.2 page 241)

= {(𝑥, 𝑦) | (𝑦, 𝑥) ∈ {(𝑦, 𝑥) | (𝑥, 𝑦) ∈ Ⓡ}} = {(𝑥, 𝑦) | (𝑥, 𝑦) ∈ Ⓡ} =Ⓡ

✏ Example 15.8.

Let and and Then and

𝑋 𝑌 Ⓡ Ⓡ−1 −1 −1 (Ⓡ )

≜ ≜ ≜ = =

{1, 2, 3} {𝐴, 𝐵} {(1, 𝐴) , (2, 𝐴) , (2, 𝐵) , (3, 𝐵)}. {(𝐴, 1) , (𝐴, 2) , (𝐵, 2) , (𝐵, 3)} {(1, 𝐴) , (2, 𝐴) , (2, 𝐵) , (3, 𝐵)} = Ⓡ. −1

The sets 𝑋 and 𝑌 and the relations Ⓡ, Ⓡ−1 , and (Ⓡ−1 ) are illustrated below. −1 −1 (Ⓡ ) Ⓡ−1 Ⓡ 𝑌 𝑋 𝑌 𝑋 (1, 𝐴)

1

(𝐴, 1)

b

b

(2, 𝐴)

2 b

3 b

(2, 𝐵)

𝐴 𝐵b

(1, 𝐴)

1b

(𝐴, 2)

2b

(2, 𝐴)

(𝐵, 2)

3b

(2, 𝐵)

(3, 𝐵)

(𝐵, 3)

b

𝐴

b

𝐵

(3, 𝐵)

Example 15.9. Let 𝑋 ≜ {1, 2, 3}. Let ≤∈ 𝟚𝑋𝑋 be the “less than or equal to” relation in 𝟚𝑋𝑋 . −1

−1 (≤ )

−1

≜ ({(1, 1) , (1, 2) , (1, 3) , (2, 2) , (2, 3) , (3, 3)}−1 ) = ({(1, 1) , (2, 1) , (3, 1) , (2, 2) , (3, 2) , (3, 3)})−1 = ({(1, 1) , (1, 2) , (1, 3) , (2, 2) , (2, 3) , (3, 3)}) ≜≤

𝑋

≤

𝑋

≤−1

𝑋

−1

−1 (≤ )

𝑋

b

1b

1b b

1

2 b

2b

2b b

2

3 b

3b

3b b

3

1

Definition 15.3. 7 Let Ⓡ ∈ 𝟚𝑋𝑌 and Ⓢ ∈ 𝟚𝑌 𝑍 be relations. Let ∧ be the logical and function. d The composition function ∘ on relations Ⓡ and Ⓢ is defined as e Ⓢ ∘ Ⓡ ≜ {(𝑥, 𝑧) |∃𝑦 such that (𝑥, 𝑦) ∈ Ⓡ ∧ (𝑦, 𝑧) ∈ Ⓢ} f Theorem 15.2. 8 Let 𝑋, 𝑌 , and 𝑍 be sets. t (Ⓡ ∘ Ⓢ)−1 = (Ⓢ−1 ) ∘ (Ⓡ−1 ) ∀Ⓡ ∈ 𝟚𝑊 𝑋 , Ⓢ ∈ 𝟚𝑋𝑌 h ∀Ⓡ ∈ 𝟚𝑊 𝑋 , Ⓢ ∈ 𝟚𝑋𝑌 , Ⓠ ∈ 𝟚𝑌 𝑍 m Ⓠ ∘ (Ⓢ ∘ Ⓡ) = (Ⓠ ∘ Ⓢ) ∘ Ⓡ 7 8

(idempotent) (associative)

📘 Kelley (1955) pages 7–8, 📘 Fuhrmann (2012) page 2 📘 Kelley (1955) page 8

version 0.30


Daniel J. Greenhoe


page 243


✎PROOF: (Ⓡ ∘ Ⓢ)−1 = {(𝑥, 𝑧) |∃𝑦 such that (𝑥, 𝑦) ∈ Ⓡ and (𝑦, 𝑧) ∈ Ⓢ}−1 by definition of ∘ (page 242) = {(𝑧, 𝑥) | (𝑥, 𝑧) ∈ {(𝑥, 𝑧) |∃𝑦 such that (𝑥, 𝑦) ∈ Ⓡ and (𝑦, 𝑧) ∈ Ⓢ}} by definition of Ⓡ−1 (page 241) = {(𝑧, 𝑥) |∃𝑦 such that

(𝑥, 𝑦) ∈ Ⓡ and (𝑦, 𝑧) ∈ Ⓢ}

= {(𝑧, 𝑥) |∃𝑦 such that

(𝑦, 𝑥) ∈ Ⓡ−1 and (𝑧, 𝑦) ∈ Ⓢ−1 }

by definition of Ⓡ−1 (page 241)

= (Ⓢ−1 ) ∘ (Ⓡ−1 )

by definition of ∘ (page 242)

Ⓠ ∘ (Ⓢ ∘ Ⓡ) = {(𝑤, 𝑧) |∃𝑦 such that

(𝑤, 𝑦) ∈ (Ⓢ ∘ Ⓡ) and (𝑦, 𝑧) ∈ Ⓠ}

by definition of ∘ (page 242) (𝑤, 𝑦) ∈ {(𝑤, 𝑦) |∃𝑥 such that

= {(𝑤, 𝑧) |∃𝑦 such that

(𝑤, 𝑥) ∈ Ⓡ and (𝑥, 𝑦) ∈ Ⓢ} and (𝑦, 𝑧) ∈ Ⓠ}

by definition of ∘ (page 242) = {(𝑤, 𝑧) |∃𝑥, 𝑦

such that

(𝑤, 𝑥) ∈ Ⓡ and (𝑥, 𝑦) ∈ Ⓢ and (𝑦, 𝑧) ∈ Ⓠ}

= {(𝑤, 𝑧) |∃𝑥 such that

(𝑤, 𝑥) ∈ Ⓡ and (𝑥, 𝑧) ∈ {(𝑥, 𝑧) |∃𝑦 such that

= {(𝑤, 𝑧) |∃𝑥 such that

(𝑤, 𝑥) ∈ Ⓡ and (𝑥, 𝑧) ∈ (Ⓢ ∘ Ⓠ)}

(𝑥, 𝑦) ∈ Ⓢ and (𝑦, 𝑧) ∈ Ⓠ}}

by definition of ∘ (page 242) = (Ⓠ ∘ Ⓢ) ∘ Ⓡ by definition of ∘ (page 242)

✏ Example 15.10. ≜ {1, 2, 3} ≜ {𝐴, 𝐵} ≜ {red, green, blue} ≜ {(1, 𝐴) , (2, 𝐴) , (2, 𝐵) , (3, 𝐵)}. ≜ {(𝐴, red) , (𝐴, green) , (𝐵, blue)}. = {(1, red) , (1, green) , (2, green) , (2, blue) , = {(red, 1) , (green, 1) , (green, 2) , (blue, 2) , = Ⓢ−1 ∘ Ⓡ−1 The quanitities are illustrated below. Ⓡ Ⓢ 𝑋 𝑌 𝑍 Let and and and and Then and

𝑋 𝑌 𝑍 Ⓡ Ⓢ Ⓡ∘Ⓢ (Ⓡ ∘ Ⓢ)−1

(1, 𝐴)

1

𝐴b

b

(2, 𝐴)

2 b

3 b

(𝐴, red) b

(3, blue)}. (blue, 3)}.

red

(𝐴, green)

green b

(2, 𝐵) b

(3, 𝐵)

𝐵

b

(𝐵, blue)

blue

Definition 15.4. 9 Let Ⓡ ∈ 𝟚𝑋𝑌 be a relation. The domain of Ⓡ is 𝓓(Ⓡ) ≜ {𝑥 ∈ 𝑋|∃𝑦 such that (𝑥, 𝑦) ∈ Ⓡ} . d The image set of Ⓡ is 𝓘(Ⓡ) ≜ {𝑦 ∈ 𝑌 |∃𝑥 such that (𝑥, 𝑦) ∈ Ⓡ} . e f The null space of Ⓡ is 𝓝 (Ⓡ) ≜ {𝑥 ∈ 𝑋| (𝑥, 0) ∈ Ⓡ} . The range of Ⓡ is any set 𝓡(Ⓡ) such that 𝓘(Ⓡ) ⊆ 𝓡(Ⓡ) 9

📘 Munkres (2000), page 16, 📘 Kelley (1955) page 7


Daniel J. Greenhoe


version 0.30

page 244

15.1. RELATIONS

Example 15.11. Let Ⓡ ≜ sin 𝑥. Then … 𝓓(Ⓡ) 𝓘(Ⓡ) 𝓝 (Ⓡ) 𝓡(Ⓡ)

= = = =

ℝ −1 ≤ 𝑦 ≤ 1 {𝑛𝜋|𝑛 ∈ ℤ} . ℝ

1 −4𝜋 −3𝜋 −2𝜋 −1𝜋 0𝜋 −1

1𝜋

2𝜋

3𝜋

4𝜋

1𝜋

2𝜋

3𝜋

4𝜋

𝑥

Example 15.12. Let Ⓡ ≜ cos 𝑥. Then … 𝓓(Ⓡ) 𝓘(Ⓡ) 𝓝 (Ⓡ) 𝓡(Ⓡ)

= = = =

ℝ −1 ≤ 𝑦 ≤ 1 1 {(𝑛 + 2 )𝜋|𝑛 ∈ ℤ} . ℝ

1 −4𝜋 −3𝜋 −2𝜋 −1𝜋 0𝜋 −1

𝑥

Example 15.13. (Rudin, 1991)99 Let 𝙓 and 𝙔 be linear functions and 𝑌 𝑋 be the set of all functions from 𝙓 to 𝙔 . Let 𝖿 be an function in 𝑌 𝑋 . The domain of 𝖿 is 𝓓(𝖿) ≜ 𝙓 The range of 𝖿 is 𝓘(𝖿) ≜ {𝒚 ∈ 𝙔 |∃𝒙 ∈ 𝙓 such that 𝒚 = 𝖿𝒙} The null space of 𝖿 is 𝓝 (𝖿) ≜ {𝒙 ∈ 𝙓 |𝖿𝒙 = 0} Theorem 15.3. 𝓓 t h m

𝓓

(⋃ 𝑖∈𝐼

Ⓡ𝑖

10

)

Let 𝓓(Ⓡ) be the domain of a relation Ⓡ and 𝓘(Ⓡ) the image of Ⓡ. =

⋃

𝓘

𝓓 (Ⓡ𝑖 )

𝑖∈𝐼

Ⓡ𝑖 ⊆ 𝓓 Ⓡ ⋂ ( 𝑖) (⋂ ) 𝑖∈𝐼 𝑖∈𝐼 𝓓(Ⓡ⧵Ⓢ) ⊇ 𝓓(Ⓡ)⧵𝓓(Ⓢ)

𝓘

(⋃ 𝑖∈𝐼

Ⓡ𝑖

)

=

𝓘 Ⓡ ⋃ ( 𝑖) 𝑖∈𝐼

Ⓡ𝑖 ⊆ 𝓘 Ⓡ ⋂ ( 𝑖) (⋂ ) 𝑖∈𝐼 𝑖∈𝐼 𝓘(Ⓡ⧵Ⓢ) ⊇ 𝓘(Ⓡ)⧵𝓘(Ⓢ)

✎PROOF: 𝓓

(⋃ 𝑖∈𝐼

Ⓡ𝑖

)

= = = = =

{ { { {

𝑥|∃𝑦 such that

(𝑥, 𝑦) ∈

⋃

Ⓡ𝑖

𝑖∈𝐼

𝑥|∃𝑦 such that 𝑥|∃𝑦 such that 𝑥|

(𝑥, 𝑦) ∈

⋁

{

(𝑥, 𝑦) |

(𝑥, 𝑦) ∈ Ⓡ𝑖

𝑖

}

∃𝑦 such that ⋁[

(𝑥, 𝑦) ∈ Ⓡ𝑖 ]


⋁

(𝑥, 𝑦) ∈ Ⓡ𝑖

𝑖

𝑖

⋃{


} }}


}

(𝑥, 𝑦) ∈ Ⓡ𝑖 }


𝑖

=

⋃

𝓓 (Ⓡ𝑖 )


𝑖

𝓓

(⋂ 𝑖∈𝐼

Ⓡ𝑖

)

= =

10

{ {


(𝑥, 𝑦) ∈

⋂ 𝑖∈𝐼


(𝑥, 𝑦) ∈

{

Ⓡ𝑖


}

(𝑥, 𝑦) |

⋀ 𝑖

(𝑥, 𝑦) ∈ Ⓡ𝑖

}}


📘 Suppes (1972) pages 60–61

version 0.30


Daniel J. Greenhoe


page 245


=

𝑥|∃𝑦 such that {

⋀

(𝑥, 𝑦) ∈ Ⓡ𝑖

𝑖

=

𝑥| [∃𝑦 such that { ⋀ 𝑖

=

𝑥|∃𝑦 such that ⋂{

}

(𝑥, 𝑦) ∈ Ⓡ𝑖 ] } (𝑥, 𝑦) ∈ Ⓡ𝑖 }


𝑖

=

⋂

𝓓 (Ⓡ𝑖 )


𝑖

✏ Example 15.14. Let and and and

𝑋 𝑌 Ⓡ Ⓢ

≜ ≜ ≜ ≜

{1, 2, 3} {𝐴, 𝐵, 𝐶} {(1, 𝐴) , (2, 𝐵) , (3, 𝐶)} {(1, 𝐴) , (1, 𝐵) , (2, 𝐶) , (3, 𝐶)}.

𝑋 1 2 3

Ⓡ

𝑌

𝑋

b b

b b

b

𝐴 𝐵 𝐶

b

1 2 3

Ⓢ

𝑌 b

b

b b

b

𝐴 𝐵 𝐶

b

𝓓(Ⓡ ∪ Ⓢ) = 𝓓({(1, 𝐴) , (2, 𝐵) , (3, 𝐶)} ∪ {(1, 𝐴) , (1, 𝐵) , (2, 𝐶) , (3, 𝐶)}). = 𝓓{(1, 𝐴) , (1, 𝐵) , (2, 𝐵) , (2, 𝐶) , (3, 𝐶)} = {1, 2, 3} = {1, 2, 3} ∪ {1, 2 3} = 𝓓Ⓡ ∪ 𝓓Ⓢ 𝓓(Ⓡ ∩ Ⓢ) = {(1, 𝐴) , (3, 𝐶)} = {1, 3} ⊆ {1, 2, 3} ∩ {1, 2 3} = 𝓓Ⓡ ∩ 𝓓Ⓢ 𝓘(Ⓡ ∪ Ⓢ) = 𝓘({(1, 𝐴) , (2, 𝐵) , (3, 𝐶)} ∪ {(1, 𝐴) , (1, 𝐵) , (2, 𝐶) , (3, 𝐶)}). = 𝓘{(1, 𝐴) , (1, 𝐵) , (2, 𝐵) , (2, 𝐶) , (3, 𝐶)} = {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐶} ∪ {𝐴, 𝐵 𝐶} = 𝓘Ⓡ ∪ 𝓘Ⓢ 𝓘(Ⓡ ∩ Ⓢ) = {(1, 𝐴) , (3, 𝐶)} = {𝐴, 𝐶} ⊆ {𝐴, 𝐵, 𝐶} ∩ {𝐴, 𝐵 𝐶} = 𝓘Ⓡ ∩ 𝓘Ⓢ Example 15.15. Let and and and

𝑋 𝑌 Ⓡ Ⓢ

≜ ≜ ≜ ≜

{−1, 0, 1} {−1, 0, 1} {(−1, −1) , (0, 0) , (1, 1)} {(−1, 1) , (0, 0) , (1, −1)}.

𝑋 1 0 −1

Ⓡ

𝑌

b

b b

b b

b

𝑋 1 0 −1

1 0 −1

Ⓢ

𝑌

b

b b

b b

b

1 0 −1

𝓓(Ⓡ ∪ Ⓢ) = 𝓓({(−1, −1) , (0, 0) , (1, 1)} ∪ {(−1, 1) , (0, 0) , (1, −1)}). = 𝓓{(−1, −1) , (0, 0) , (1, 1) , (−1, 1) , (1, −1)} 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 246

15.1. RELATIONS

= {−1, 0, 1} = {−1, 0, 1} ∪ {−1, 0 1} = 𝓓Ⓡ ∪ 𝓓Ⓢ 𝓓(Ⓡ ∩ Ⓢ) = 𝓓({(−1, −1) , (0, 0) , (1, 1)} ∩ {(−1, 1) , (0, 0) , (1, −1)}). = 𝓓{(0, 0)} = {0} ⊆ {−1, 0, 1} ∩ {−1, 0 1} = 𝓓Ⓡ ∩ 𝓓Ⓢ 𝓘(Ⓡ ∪ Ⓢ) = 𝓘({(−1, −1) , (0, 0) , (1, 1)} ∪ {(−1, 1) , (0, 0) , (1, −1)}). = 𝓘{(−1, −1) , (0, 0) , (1, 1) , (−1, 1) , (1, −1)} = {−1, 0, 1} = {−1, 0, 1} ∪ {−1, 0 1} = 𝓘Ⓡ ∪ 𝓓Ⓢ 𝓘(Ⓡ ∩ Ⓢ) = 𝓘({(−1, −1) , (0, 0) , (1, 1)} ∩ {(−1, 1) , (0, 0) , (1, −1)}). = 𝓘{(0, 0)} = {0} ⊆ {−1, 0, 1} ∩ {−1, 0 1} = 𝓘Ⓡ ∩ 𝓘Ⓢ Example 15.16. 𝑦

𝖿

1 Let 𝖿 (𝑥) and 𝗀(𝑥)

≜ ≜

𝑥 −𝑥.

𝑥 −1 −1

1 𝗀

𝓓(𝖿 ∪ 𝗀) = 𝓓 ({(𝑥, 𝑦) ∈ ℝ2 |𝑦 = 𝑥} ∪ {(𝑥, 𝑦) ∈ ℝ2 |𝑦 = −𝑥}) = 𝓓 {(𝑥, 𝑦) ∈ ℝ2 |𝑦 = 𝑥 or 𝑦 = −𝑥} =ℝ =ℝ∪ℝ = (𝓓 {(𝑥, 𝑦) ∈ ℝ2 |𝑦 = 𝑥}) ∪ (𝓓 {(𝑥, 𝑦) ∈ ℝ2 |𝑦 = −𝑥}) 𝓓(𝖿 ∩ 𝗀) = 𝓓 ({(𝑥, 𝑦) ∈ ℝ2 |𝑦 = 𝑥} ∩ {(𝑥, 𝑦) ∈ ℝ2 |𝑦 = −𝑥}) = 𝓓 {(𝑥, 𝑦) ∈ ℝ2 |𝑦 = 𝑥 and 𝑦 = −𝑥} = 𝓓{(0, 0)} = {0} ⊆ℝ =ℝ∩ℝ = (𝓓 {(𝑥, 𝑦) ∈ ℝ2 |𝑦 = 𝑥}) ∩ (𝓓 {(𝑥, 𝑦) ∈ ℝ2 |𝑦 = −𝑥}) 𝓘(𝖿 ∪ 𝗀) = 𝓘 ({(𝑥, 𝑦) ∈ ℝ2 |𝑦 = 𝑥} ∪ {(𝑥, 𝑦) ∈ ℝ2 |𝑦 = −𝑥}) = 𝓘 {(𝑥, 𝑦) ∈ ℝ2 |𝑦 = 𝑥 or 𝑦 = −𝑥} =ℝ =ℝ∪ℝ version 0.30


Daniel J. Greenhoe


page 247


= (𝓘 {(𝑥, 𝑦) ∈ ℝ2 |𝑦 = 𝑥}) ∪ (𝓘 {(𝑥, 𝑦) ∈ ℝ2 |𝑦 = −𝑥}) 𝓘(𝖿 ∩ 𝗀) = 𝓘 ({(𝑥, 𝑦) ∈ ℝ2 |𝑦 = 𝑥} ∩ {(𝑥, 𝑦) ∈ ℝ2 |𝑦 = −𝑥}) = 𝓘 {(𝑥, 𝑦) ∈ ℝ2 |𝑦 = 𝑥 and 𝑦 = −𝑥} = 𝓘{(0, 0)} = {0} ⊆ℝ =ℝ∩ℝ = (𝓘 {(𝑥, 𝑦) ∈ ℝ2 |𝑦 = 𝑥}) ∩ (𝓘 {(𝑥, 𝑦) ∈ ℝ2 |𝑦 = −𝑥}) Definition 15.5. 11 Let Ⓡ be a relation in 𝟚𝑋𝑌 . d Ⓡ(𝐴) ≜ {𝑦 ∈ 𝑌 |∃𝑥 ∈ 𝐴 such that (𝑥, 𝑦) ∈ Ⓡ} e −1 Ⓡ (𝐵) ≜ {𝑥 ∈ 𝑋|∃𝑦 ∈ 𝐵 such that (𝑥, 𝑦) ∈ Ⓡ} f

∀𝐴∈𝟚𝑋

(image of 𝐴 under Ⓡ)

∀𝐵∈𝟚𝑌

(image of 𝐵 under Ⓡ−1 )

Theorem 15.4. 12 Ⓡ(∅) = ∅ t h m

𝐴𝑖 Ⓡ = Ⓡ 𝐴 ⋃ ( 𝑖) (⋃ ) 𝑖∈𝐼 𝑖∈𝐼 Ⓡ ⊆ Ⓡ 𝐴 𝐴𝑖 ⋂ ( 𝑖) (⋂ ) 𝑖∈𝐼 𝑖∈𝐼

✎PROOF: Ⓡ(∅) = {𝑦 ∈ 𝑌 |∃𝑥 ∈ ∅

such that

(𝑥, 𝑦) ∈ Ⓡ}


=∅ Ⓡ 𝐴 𝐴𝑖 = 𝑦 ∈ 𝑌 |∃𝑥 ∈ ⋃ 𝑖 (⋃ ) { 𝑖∈𝐼 𝑖∈𝐼

(𝑥, 𝑦) ∈ Ⓡ

such that

=

𝑦 ∈ 𝑌 |∃𝑥 ∈ 𝑥 ∈ 𝑋| 𝑥 ∈ 𝐴𝑖 ⋁ } { { 𝑖∈𝐼

=

𝑦 ∈ 𝑌 |∃𝑥 ∈ 𝑋 {

such that

=

𝑦 ∈ 𝑌 |∃𝑥 ∈ 𝑋 {

such that

[⋁ 𝑖∈𝐼


}

such that

(𝑥, 𝑦) ∈ Ⓡ

}


𝑥 ∈ 𝐴𝑖 ∧ (𝑥, 𝑦) ∈ Ⓡ ] }

𝑥 ∈ 𝐴𝑖 ∧ (𝑥, 𝑦) ∈ Ⓡ] ⋁[ } 𝑖∈𝐼

=

𝑦 ∈ 𝑌| ∃𝑥 ∈ 𝑋 ⋁[ { 𝑖∈𝐼

=

𝑦 ∈ 𝑌 |∃𝑥 ∈ 𝑋 ⋃{

such that such that

𝑥 ∈ 𝐴𝑖 ∧ (𝑥, 𝑦) ∈ Ⓡ] }

𝑥 ∈ 𝐴𝑖 ∧ (𝑥, 𝑦) ∈ Ⓡ}


𝑖∈𝐼

=


Ⓡ 𝐴 ⋃ ( 𝑖) 𝑖∈𝐼

Ⓡ 𝐴 𝐴𝑖 = 𝑦 ∈ 𝑌 |∃𝑥 ∈ ⋂ 𝑖 (⋂ ) { 𝑖∈𝐼 𝑖∈𝐼 = 11 12

such that

(𝑥, 𝑦) ∈ Ⓡ

𝑦 ∈ 𝑌 |∃𝑥 ∈ 𝑥 ∈ 𝑋| 𝑥 ∈ 𝐴𝑖 ⋀ } { { 𝑖∈𝐼


}

such that

(𝑥, 𝑦) ∈ Ⓡ

}


📘 Kelley (1955) page 8 📘 Kelley (1955) page 8


Daniel J. Greenhoe


version 0.30

page 248

15.1. RELATIONS

= = ⊆ =

{ { {

𝑦 ∈ 𝑌 |∃𝑥 ∈ 𝑋

such that

𝑦 ∈ 𝑌 |∃𝑥 ∈ 𝑋

such that

[⋀ 𝑖∈𝐼

𝑥 ∈ 𝐴𝑖 ∧ (𝑥, 𝑦) ∈ Ⓡ ] }

𝑥 ∈ 𝐴𝑖 ∧ (𝑥, 𝑦) ∈ Ⓡ] ⋀[ } 𝑖∈𝐼

𝑦 ∈ 𝑌|

∃𝑥 ∈ 𝑋 ⋀[

such that

𝑖∈𝐼

𝑦 ∈ 𝑌 |∃𝑥 ∈ 𝑋 ⋂{

such that

𝑥 ∈ 𝐴𝑖 ∧ (𝑥, 𝑦) ∈ Ⓡ] }

𝑥 ∈ 𝐴𝑖 ∧ (𝑥, 𝑦) ∈ Ⓡ}


𝑖∈𝐼

=


Ⓡ 𝐴 ⋂ ( 𝑖) 𝑖∈𝐼

✏ Definition 15.6 (next) provides some properties associated with special types of relations. Relations can be defined based on their properties. For example, equivalence relations (Definition 1.9 page 7) are reflexive, symmetric, and transitive; whereas order relations are (Definition 3.2 page 46) are reflexive, anti-symmetric, and transitive. Definition 15.6. 13 Let 𝑋 be a set and Ⓡ a relation in 𝟚𝑋𝑋 . Ⓡ is reflexive if 𝑥Ⓡ𝑥 Ⓡ is irreflexive if (𝑥, 𝑥) ∉ Ⓡ Ⓡ is symmetric if 𝑥Ⓡ𝑦 ⟹ 𝑦Ⓡ𝑥 d Ⓡ is asymmetric if 𝑥Ⓡ𝑦 ⟹ (𝑦, 𝑥) ∉ Ⓡ e if 𝑥Ⓡ𝑦 and 𝑦Ⓡ𝑥 ⟹ 𝑥 = 𝑦 f Ⓡ is anti-symmetric Ⓡ is transitive if 𝑥Ⓡ𝑦 and 𝑦Ⓡ𝑧 ⟹ 𝑥Ⓡ𝑧 Ⓡ is connected if 𝑥 ≠ 𝑦 ⟹ 𝑥Ⓡ𝑦 or 𝑦Ⓡ𝑥 Ⓡ is strongly connected if 𝑥Ⓡ𝑦 or 𝑦Ⓡ𝑥

∀𝑥∈𝑋 ∀𝑥∈𝑋 ∀𝑥,𝑦∈𝑋 ∀𝑥,𝑦∈𝑋 ∀𝑥,𝑦∈𝑋 ∀𝑥,𝑦,𝑧∈𝑋 ∀𝑥,𝑦,𝑧∈𝑋 ∀𝑥,𝑦,𝑧∈𝑋

Definition 15.7. 14 � The identity element ○(𝑋) with respect to Ⓡ ∈ 𝟚𝑋𝑋 is defined as d � e ○(𝑋) ≜ {(𝑥, 𝑥) | (𝑥, 𝑥) ∈ Ⓡ} . f The identity element ○(𝑋) � � may also be denoted as simply ○. � be the identity element in 𝟚𝑋𝑋 with respect to the composition function ∘. Proposition 15.4. Let ○ p � ∘Ⓡ=Ⓡ∘○ � =Ⓡ r ○ ∀Ⓡ ∈ 𝟚𝑋𝑋 p

Example 15.17. (Michel and Herget, 1993)411 Let 𝙓 be a linear space and 𝑋 𝑋 the set of all functions from 𝙓 to 𝙓 (Definition 15.8 page 249). Let 𝐈 be an function in 𝑋 𝑋 . 𝐈 is an identity function in 𝑋 𝑋 if 𝐈𝒙 = 𝒙 ∀𝒙 ∈ 𝙓 . � be the identity element in 𝟚𝑋𝑋 with respect to Theorem 15.5. 15 Let Ⓡ be a relation in 𝟚𝑋𝑋 . Let ○ composition. � Ⓡ is reflexive ⟺ ○ ⊆ Ⓡ ⟺ Ⓡ = Ⓡ−1 t Ⓡ is symmetric h Ⓡ is anti-symmetric ⟺ Ⓡ ∩ Ⓡ−1 = ∅ m Ⓡ is transitive ⟺ Ⓡ∘Ⓡ ⊆ Ⓡ Ⓡ is transitive and reflexive ⟹ Ⓡ∘Ⓡ = Ⓡ 13

📘 Suppes (1972) page 69 ⟨Defintion 10–Definition 17⟩, 📘 Kelley (1955) page 9 📘 Kelley (1955) page 9 15 📘 Kelley (1955) page 9 14

version 0.30


Daniel J. Greenhoe


page 249


✎PROOF: Ⓡ is reflexive ⟺ (𝑥, 𝑥) ∈ Ⓡ

∀𝑥 ∈ 𝑋


� ⊆Ⓡ ⟺ ○


Ⓡ is symmetric ⟺ [(𝑥, 𝑦) ∈ Ⓡ ⟹ (𝑦, 𝑥) ∈ Ⓡ] ⟺ Ⓡ = Ⓡ−1


Ⓡ is anti-symmetric ⟺ [(𝑥, 𝑦) ∈ Ⓡ ⟹ (𝑦, 𝑥) ∉ Ⓡ] Ⓡ−1 = ∅ ⋂ Ⓡ is transitive ⟺ [(𝑥, 𝑦) , (𝑦, 𝑧) ∈ Ⓡ ⟹ (𝑥, 𝑧) ∈ Ⓡ] ⟺ Ⓡ

⟺ Ⓡ∘Ⓡ⊆Ⓡ

by Definition 15.6 page 248 by Definition 15.2 page 241 by Definition 15.6 page 248 by Definition 15.3 page 242

� ⊆Ⓡ Ⓡ is transitive and reflexive ⟺ [Ⓡ ∘ Ⓡ ⊆ Ⓡ and ○ ] � ⟹ [Ⓡ ∘ Ⓡ ⊆ Ⓡ and Ⓡ = ○ ∘ Ⓡ ⊆ Ⓡ ∘ Ⓡ]

by previous results � page 248 by definition of ○

⟺ [Ⓡ ∘ Ⓡ ⊆ Ⓡ and Ⓡ ⊆ Ⓡ ∘ Ⓡ] ⟹ Ⓡ∘Ⓡ=Ⓡ

✏

15.2 Functions The function is a special case of the relation in that while both are subsets of a Cartesian product, an element in the domain of a function can only map to one element in the range (Definition 15.8— next definition). The set of all functions in the Cartesian product 𝑋×𝑌 is denoted 𝑌 𝑋 ; this is suitable because the number of functions in 𝑋 × 𝑌 for finite 𝑋 and 𝑌 is | 𝑌 || 𝑋 | (Proposition 15.5 page 250). The fact that not all functions are relations is demonstrated in Example 15.18 (page 250) (discrete cases) and Example 15.19 (page 250) (continuous cases).

15.2.1 Definition and examples Definition 15.8. 16 Let 𝑋 and 𝑌 be sets. Let ∧ be the “logical and” operation (Definition 12.1 page 185). A relation 𝖿 ∈ 𝟚𝑋𝑌 is a function if (for each 𝑥, there is only one 𝖿(𝑥)) (𝑥, 𝑦1 ) ∈ 𝖿 ∧ (𝑥, 𝑦2 ) ∈ 𝖿 ⟹ 𝑦1 = 𝑦2 d 𝑋𝑌 e The set of all functions in 𝟚 is denoted f 𝑌 𝑋 ≜ {𝖿 ∈ 𝟚𝑋𝑌 |𝖿 is a function} . A function may also be referred to as a correspondence, transformation, or map. As indicated in Definition 15.8 (previous definition), functions customarily come disguised in different names depending on the context in which they are found. This is particularly true with respect to vector spaces, as illustrated next: ➀ function: maps from a field to a field ➁ functional: maps from a vector space to a field ➂ function: maps from a vector space to a vector space However, no matter what name is used, a function is still a function as long as it satisfies Definition 15.8. 16

📘 Suppes (1972) page 86, 📘 Kelley (1955) page 10, 📘 Bourbaki (1939), 📘 Bottazzini (1986), page 7


Daniel J. Greenhoe


version 0.30

page 250

15.2. FUNCTIONS

Definition 15.9. 17 𝑛 A function 𝖿 ∈ 𝑌 𝑋 3 A function 𝖿 ∈ 𝑌 𝑋 d 2 e A function 𝖿 ∈ 𝑌 𝑋 f 1 A function 𝖿 ∈ 𝑌 𝑋 ≜ 𝑌 𝑋 0 A function 𝖿 ∈ 𝑌 𝑋 ≜ 𝑌

is said to have arity 𝑛. is said to be ternary. is said to be binary. is said to be unary. is said to be nullary.

Example 15.18. The figure below illustrates two discrete examples of relations that are functions and two that are not. 𝑋

𝑌

𝑋

𝑌

𝑋

𝑌

𝑋

𝑌

𝑥4

𝑦4

𝑥4

𝑦4

𝑥4

𝑦4

𝑥4

𝑦4

𝑥3

𝑦3

𝑥3

𝑦3

𝑥3

𝑦3

𝑥3

𝑦3

𝑥2

𝑦2

𝑥2

𝑦2

𝑥2

𝑦2

𝑥2

𝑦2

𝑥1

𝑦1

𝑥1

𝑦1

𝑥1

𝑦1

𝑥1

𝑦1

two relations in 𝟚𝑋𝑌 that are functions

two relations in 𝟚𝑋𝑌 that are not functions

Example 15.19. 18 The figures below illustrates one example of a continuous relation that is not a function and one that is.

𝑦

𝑦 𝑥

−1

1

2 2 {(𝑥, 𝑦) ∈ 𝑋 × 𝑌 |𝑥 + 𝑦 = 1}

(a relation that is not a function) Proposition 15.5. measure for sets. p r p

19

𝑥 −1

1

2 √ (𝑥, 𝑦) ∈ 𝑋 × 𝑌 | 𝑦 = 1 − 𝑥 for − 1 < 𝑥 < 1 { } 𝑦=0 otherwise (a relation that is a function)

Let 𝑌 𝑋 be the set of all functions from a set 𝑋 to a set 𝑌 . Let | ⋅ | be the counting 𝑌 𝑋 | = | 𝑌 || 𝑋 | |⏟

number of possible functions in 𝑋 × 𝑌

✎PROOF:

Let 𝑋 ≜ {𝑥1 , 𝑥2 , … , 𝑥𝑚 }. Let 𝑌 ≜ {𝑦1 , 𝑦2 , … , 𝑦𝑛 }. Then 𝑥1 can map to exactly one of the 𝑛 elements in set 𝑌 : 𝑦1 , 𝑦2 , …, or 𝑦𝑛 . Likewise, 𝑥2 can also map to one of the 𝑛 elements in set 𝑌 . So, the total number of possible functions in 𝑌 𝑋 is 𝑛 𝑚 = | 𝑌 || 𝑋 | 17

📘 Burris and Sankappanavar (2000), pages 25–26 📘 Apostol (1975) page 34 19 📘 Comtet (1974) page 4 18

version 0.30


Daniel J. Greenhoe


page 251


✏ | 𝑋 |⋅| 𝑌 |

3×2

Example 15.20. Let 𝑋 ≜ {𝑥1 , 𝑥2 , 𝑥3 } and 𝑌 ≜ {𝑦1 , 𝑦2 }. There are a total of | ℝ | = 2 =2 = 64 possible relations on 𝑋 × 𝑌 , as listed in Example 15.2 (page 238). Let 𝔽 ≜ ⦅𝑭1 , 𝑭2 , 𝑭3 , …⦆ be the set of all functions from 𝑋 to 𝑌 . There are a total of | 𝔽 | = | 𝑌 || 𝑋 | = 23 = 8 possible functions. These 8 functions are listed below. Of these 8 functions, 6 are surjective, as listed in Example 15.27 (page 254). functions on {𝑥1 , 𝑥2 , 𝑥3 } × {𝑦1 , 𝑦2 } 𝑭1 = { (𝑥1 , 𝑦1 ) , (𝑥2 , 𝑦1 ) , (𝑥3 , 𝑦1 ) } 𝑭5 = { (𝑥1 , 𝑦1 ) , (𝑥2 , 𝑦1 ) , (𝑥3 , 𝑦2 ) } 𝑭2 = { (𝑥1 , 𝑦2 ) , (𝑥2 , 𝑦1 ) , (𝑥3 , 𝑦1 ) } 𝑭6 = { (𝑥1 , 𝑦2 ) , (𝑥2 , 𝑦1 ) , (𝑥3 , 𝑦2 ) } 𝑭3 = { (𝑥1 , 𝑦1 ) , (𝑥2 , 𝑦2 ) , (𝑥3 , 𝑦1 ) } 𝑭7 = { (𝑥1 , 𝑦1 ) , (𝑥2 , 𝑦2 ) , (𝑥3 , 𝑦2 ) } 𝑭4 = { (𝑥1 , 𝑦2 ) , (𝑥2 , 𝑦2 ) , (𝑥3 , 𝑦1 ) } 𝑭8 = { (𝑥1 , 𝑦2 ) , (𝑥2 , 𝑦2 ) , (𝑥3 , 𝑦2 ) } Example 15.21. Let 𝑋 ≜ {𝑥, 𝑦, 𝑧} There are a total of | ℝ | = 2| 𝑋×𝑋 | = 2| 𝑋 |⋅| 𝑋 | = 23×3 = 29 = 512 possible relations on 𝑋 2 . Of these 512 relations, only 27 are functions. These 27 functions are listed below. Of these 27 functions, only 7 are surjective functions, as listed in Example 15.28 (page 255).

𝑭1 𝑭2 𝑭3 𝑭4 𝑭5 𝑭6 𝑭7 𝑭8 𝑭9 𝑭10 𝑭11 𝑭12 𝑭13 𝑭14

= = = = = = = = = = = = = =

{ { { { { { { { { { { { { {

(𝑥, 𝑥) , (𝑥, 𝑦) , (𝑥, 𝑧) , (𝑥, 𝑥) , (𝑥, 𝑦) , (𝑥, 𝑧) , (𝑥, 𝑥) , (𝑥, 𝑦) , (𝑥, 𝑧) , (𝑥, 𝑥) , (𝑥, 𝑦) , (𝑥, 𝑧) , (𝑥, 𝑥) , (𝑥, 𝑦) ,

functions on {𝑥, (𝑦, 𝑥) , (𝑧, 𝑥) } (𝑦, 𝑥) , (𝑧, 𝑥) } (𝑦, 𝑥) , (𝑧, 𝑥) } (𝑦, 𝑦) , (𝑧, 𝑥) } (𝑦, 𝑦) , (𝑧, 𝑥) } (𝑦, 𝑦) , (𝑧, 𝑥) } (𝑦, 𝑧) , (𝑧, 𝑥) } (𝑦, 𝑧) , (𝑧, 𝑥) } (𝑦, 𝑧) , (𝑧, 𝑥) } (𝑦, 𝑥) , (𝑧, 𝑦) } (𝑦, 𝑥) , (𝑧, 𝑦) } (𝑦, 𝑥) , (𝑧, 𝑦) } (𝑦, 𝑦) , (𝑧, 𝑦) } (𝑦, 𝑦) , (𝑧, 𝑦) }

𝑦, 𝑧} × {𝑥, 𝑦, 𝑧} 𝑭15 = { (𝑥, 𝑧) , 𝑭16 = { (𝑥, 𝑥) , 𝑭17 = { (𝑥, 𝑦) , 𝑭18 = { (𝑥, 𝑧) , 𝑭19 = { (𝑥, 𝑥) , 𝑭20 = { (𝑥, 𝑦) , 𝑭21 = { (𝑥, 𝑧) , 𝑭22 = { (𝑥, 𝑥) , 𝑭23 = { (𝑥, 𝑦) , 𝑭24 = { (𝑥, 𝑧) , 𝑭25 = { (𝑥, 𝑥) , 𝑭26 = { (𝑥, 𝑦) , 𝑭27 = { (𝑥, 𝑧) ,

(𝑦, 𝑦) , (𝑦, 𝑧) , (𝑦, 𝑧) , (𝑦, 𝑧) , (𝑦, 𝑥) , (𝑦, 𝑥) , (𝑦, 𝑥) , (𝑦, 𝑦) , (𝑦, 𝑦) , (𝑦, 𝑦) , (𝑦, 𝑧) , (𝑦, 𝑧) , (𝑦, 𝑧) ,

(𝑧, 𝑦) (𝑧, 𝑦) (𝑧, 𝑦) (𝑧, 𝑦) (𝑧, 𝑧) (𝑧, 𝑧) (𝑧, 𝑧) (𝑧, 𝑧) (𝑧, 𝑧) (𝑧, 𝑧) (𝑧, 𝑧) (𝑧, 𝑧) (𝑧, 𝑧)

} } } } } } } } } } } } }

Definition 15.10. 20 Let 𝑌 𝑋 be the set of functions from a set 𝑋 to a set 𝑌 . 𝑋 𝑋 d Functions 𝖿 ∈ 𝑌 and 𝗀 ∈ 𝑌 are equal if e 𝖿(𝑥) = 𝗀(𝑥) ∀𝑥 ∈ 𝑋 f This is denoted as 𝖿 ≗ 𝗀.

15.2.2 Properties of functions Let 𝖿 be a function (Definition 15.8 page 249) in 𝑌 𝑋 with inverse relation 𝖿 −1 in 𝟚𝑋𝑌 . 𝖿(∅) = ∅ ∀𝖿∈𝑌 𝑋 𝖿 −1 (∅) = ∅ ∀𝖿∈𝑌 𝑋 𝐴⊆𝐵 ⟹ 𝖿 (𝐴) ⊆ 𝖿(𝐵) ∀𝖿∈𝑌 𝑋 , 𝐴,𝐵∈𝟚𝑋 (isotone) −1 −1 𝑋 𝑌 𝐴 ⊆ 𝐵 ⟹ 𝖿 (𝐴) ⊆ 𝖿 (𝐵) ∀𝖿∈𝑌 , 𝐴,𝐵∈𝟚 (isotone)

Theorem 15.6. 1.

t h m

2. 3. 4.

20 21

21

📘 Berberian (1961) page 73 📘 Davis (2005) pages 6–8, 📘 Vaidyanathaswamy (1960) page 10


Daniel J. Greenhoe


version 0.30

page 252

15.2. FUNCTIONS

✎PROOF:

1. Proof that 𝖿(∅) = ∅: 𝖿(∅) = {𝑦 ∈ 𝑌 |∃𝑥 ∈ ∅

such that

(𝑥, 𝑦) ∈ 𝖿 }


=∅

by definition of ∅ page 5

2. Proof that 𝐴 ⊆ 𝐵 ⟹ 𝖿(𝐴) ⊆ 𝖿(𝐵): 𝖿 (𝐴) = {𝑦 ∈ 𝑌 |∃𝑥 ∈ 𝐴

such that

(𝑥, 𝑦) ∈ 𝖿 }


⊆ {𝑦 ∈ 𝑌 |∃𝑥 ∈ 𝐵

such that

(𝑥, 𝑦) ∈ 𝖿 }

by left hypothesis

= 𝖿 (𝐵)


3. Proof that 𝖿 −1 (∅) = ∅: 𝖿 −1 (∅) = {𝑥 ∈ 𝑋|∃𝑦 ∈ ∅

such that

(𝑥, 𝑦) ∈ 𝖿 }


=∅

by definition of ∅ page 5

4. Proof that 𝐴 ⊆ 𝐵 ⟹ 𝖿 −1 (𝐴) ⊆ 𝖿 −1 (𝐵): 𝖿 −1 (𝐴) = {𝑥 ∈ 𝑋|∃𝑦 ∈ 𝐴

such that

(𝑥, 𝑦) ∈ 𝖿 −1 }


⊆ {𝑥 ∈ 𝑋|∃𝑦 ∈ 𝐵

such that

(𝑥, 𝑦) ∈ 𝖿 }

by left hypothesis

= 𝖿 −1 (𝐵)


✏

15.2.3

Types of functions

In general, a function 𝖿 ∈ 𝑌 𝑋 can be described as “into” because 𝖿 maps each element of 𝑋 into 𝑌 such that 𝖿(𝑋) ⊆ 𝑌 . However there are some common more restrictive special types of functions. These are defined in Definition 15.11 (next defintion). Definition 15.11. 22 Let 𝖿 ∈ 𝑌 𝑋 . 𝖿 is surjective (also called onto) if 𝖿(𝑋) = 𝑌 𝖿 is injective (also called one-to-one) if 𝖿(𝑥𝑛 ) = 𝖿(𝑥𝑚 ) ⟹ 𝑥𝑛 = 𝑥𝑚 𝖿 is bijective (also called one-to-one if 𝖿 is both surjective and injective. d and onto) e We also define the following sets of functions: f 𝒮j (𝑋, 𝑌 ) ≜ {𝖿 ∈ 𝑌 𝑋 |𝖿 is surjective} (the set of all surjective functions in 𝑌 𝑋 ) 𝑋 ℐj (𝑋, 𝑌 ) ≜ {𝖿 ∈ 𝑌 |𝖿 is injective} (the set of all injective functions in 𝑌 𝑋 ) 𝑋 ℬj (𝑋, 𝑌 ) ≜ {𝖿 ∈ 𝑌 |𝖿 is bijective} (the set of all bijective functions in 𝑌 𝑋 ) The types described in Definition 15.11 are illustrated below: 22

📘 Michel and Herget (1993), pages 14–15, 📘 Fuhrmann (2012) page 2, 📘 Comtet (1974) page 5

version 0.30


Daniel J. Greenhoe


page 253


𝑋

𝑌

𝑋

𝑌

𝑋

𝑌

𝑋

𝑌

𝑥4 b

b

𝑦4

𝑥4 b

b

𝑦3

𝑥3 b

b

𝑦4

𝑥4 b

b

𝑦4

𝑥3 b

b

𝑦3

𝑥3 b

b

𝑦2

𝑥2 b

b

𝑦3

𝑥3 b

b

𝑦3

𝑥2 b

b

𝑦2

𝑥2 b

b

𝑦1

𝑥1 b

b

𝑦2

𝑥2 b

b

𝑦2

𝑥1 b

b

𝑦1

𝑥1 b

b

𝑦1

𝑥1 b

b

𝑦1

“into” (arbitrary function in 𝑌 𝑋 )

“onto” surjective

“one-to-one” injective

“one-to-one and onto” bijective

Example 15.22. 1 ℝ

In the set ℝ , the function sin 𝑥 is not injective, not surjective, and not bijective.

−4𝜋 −3𝜋 −2𝜋 −1𝜋 0𝜋 −1

1𝜋

2𝜋

3𝜋

4𝜋

1𝜋

2𝜋

3𝜋

4𝜋

1

2

3

4

𝑥

Example 15.23.

3 2 1 ℝ

In the set ℝ , the function 𝑥 sin 𝑥 is surjective, but not injective and not bijective.

−4𝜋 −3𝜋 −2𝜋 −1𝜋 0𝜋 −1

𝑥

−2 −3

Example 15.24. 1 ℝ

In the set ℝ , the function 𝑦 = surjective, and bijective.

1𝑥 4

is injective,

𝑥 −4 −3 −2 −1 −1

3

𝑒𝑥

2

Example 15.25. In the set ℝℝ , the function 𝑒𝑥 is injective, but not surjective and not bijective.

1 𝑥 −3 −2 −1

0

1

Example 15.26. In the set ℝℝ , the function tan 𝑥 is not injective, not surjective (it's range does not include 𝜋2 , 3𝜋 , etc.) and not bijective. 2 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 254

15.2. FUNCTIONS

1

−2𝜋 − 3𝜋 − 5𝜋 2 2

−𝜋

− 𝜋2− 𝜋4

𝜋 4

𝜋 2

𝜋

3𝜋 2

2𝜋

5𝜋 2

𝑥

−1

Theorem 15.7. 23 𝖿 and 𝗀 are t 𝗀∘𝖿 is h 𝖿 and 𝗀 are m 𝗀∘𝖿 is

surjective surjective injective injective

⟹ ⟹ ⟹ ⟹

𝗀∘𝖿 𝗀 𝗀∘𝖿 𝖿

is surjective is surjective is injective is injective

✎PROOF: 𝖿, 𝗀 are surjective ⟹ 𝖿 (𝑋) = 𝑌 , and 𝗀(𝑌 ) = 𝑍

by definition of surjective page 252

⟹ 𝗀 ∘ 𝖿(𝑋) = 𝗀(𝑌 ) = 𝑍 ⟹ 𝗀 ∘ 𝖿 is surjective 𝗀 ∘ 𝖿 is surjective ⟹ 𝗀 ∘ 𝖿(𝑋) = 𝑍

by definition of surjective page 252 by definition of surjective page 252

⟹ 𝗀(𝖿 (𝑋)) = 𝑍 ⟹ 𝗀(𝑌 ) = 𝑍

because 𝖿(𝑋) ⊆ 𝑌 and by isotone property page 251

⟹ 𝗀 is surjective

by definition of surjective page 252

𝗀 ∘ 𝖿 (𝑥1 ) = 𝗀 ∘ 𝖿 (𝑥2 ) ⟹ 𝗀(𝖿(𝑥1 )) = 𝗀(𝖿 (𝑥2 )) ⟹ 𝖿 (𝑥1 ) = 𝖿(𝑥2 )

because 𝗀 is injective

⟹ 𝑥1 = 𝑥 2

because 𝖿 is injective

⟹ 𝗀 ∘ 𝖿 is injective 𝖿 (𝑥1 ) = 𝖿 (𝑥2 ) ⟹ 𝗀(𝖿 (𝑥1 )) = 𝗀(𝖿 (𝑥2 )) ⟹ 𝗀 ∘ 𝖿 (𝑥1 ) = 𝗀 ∘ 𝖿 (𝑥2 ) ⟹ 𝑥1 = 𝑥 2

because 𝗀 ∘ 𝖿 is injective

⟹ 𝖿 is injective

✏ Theorem 15.8 (Bernstein-Cantor-Schröder Theorem). 24 ⟹ ∃𝗁 ∈ ℬj (𝑋, 𝑌 ) (∃𝖿 ∈ ℐj (𝑋, 𝑌 )) and (∃𝗀 ∈ ℐj (𝑌 , 𝑋)) Example 15.27. Let 𝑋 ≜ {𝑥1 , 𝑥2 , 𝑥3 } and 𝑌 ≜ {𝑦1 , 𝑦2 }. There are a total of | ℝ | = 23×2 = 64 possible relations, as listed in Example 15.2 (page 238). There are a total of | 𝔽 | = 23 = 8 possible functions, as listed in Example 15.20 (page 251). Let 𝕊 ≜ ⦅𝑺1 , 𝑺2 , 𝑺3 , …⦆ be the set of all surjective functions 23

📘 Durbin (2000), pages 16–17 📘 Schröder (2003), page 116, 📘 Nievergelt (2002), page 213, 📘 Suppes (1972) page 95, 📘 Fraenkel (1953), pages 102–103 ⟨???⟩ 24

version 0.30


Daniel J. Greenhoe


page 255


from 𝑋 to 𝑌 . There are a total of | 𝕊 | = 6 possible surjective functions, as listed next: surjective functions on {𝑥1 , 𝑥2 , 𝑥3 } × {𝑦1 , 𝑦2 } 𝑺1 = { (𝑥1 , 𝑦2 ) , (𝑥2 , 𝑦1 ) , (𝑥3 , 𝑦1 ) } 𝑺4 = { (𝑥1 , 𝑦1 ) , (𝑥2 , 𝑦1 ) , (𝑥3 , 𝑦2 ) } 𝑺2 = { (𝑥1 , 𝑦1 ) , (𝑥2 , 𝑦2 ) , (𝑥3 , 𝑦1 ) } 𝑺5 = { (𝑥1 , 𝑦2 ) , (𝑥2 , 𝑦1 ) , (𝑥3 , 𝑦2 ) } 𝑺3 = { (𝑥1 , 𝑦2 ) , (𝑥2 , 𝑦2 ) , (𝑥3 , 𝑦1 ) } 𝑺6 = { (𝑥1 , 𝑦1 ) , (𝑥2 , 𝑦2 ) , (𝑥3 , 𝑦2 ) } Example 15.28. Let 𝑋 ≜ {𝑥, 𝑦, 𝑧} There are a total of | ℝ | = 2| 𝑋×𝑋 | = 2| 𝑋 |⋅| 𝑋 | = 23×3 = 29 = 512 possible relations on 𝑋 × 𝑋. Of these 512 relations, only 27 are functions. These 27 functions are listed in Example 15.21 (page 251). Of these 27 functions, only 7 are surjective functions, as listed below. Actually, in the case of a function mapping from a finite set onto the same finite set, The set 𝕊 of surjective functions is equal to the set of injective functions and the set of bijective functions. surjective functions on {𝑥, 𝑦, 𝑧} × {𝑥, 𝑦, 𝑧} 𝑺1 = { (𝑥, 𝑧) , (𝑦, 𝑥) , (𝑧, 𝑥) } 𝑺5 = { (𝑥, 𝑥) , (𝑦, 𝑧) , (𝑧, 𝑦) } 𝑺2 = { (𝑥, 𝑧) , (𝑦, 𝑦) , (𝑧, 𝑥) } 𝑺6 = { (𝑥, 𝑦) , (𝑦, 𝑥) , (𝑧, 𝑧) } 𝑺3 = { (𝑥, 𝑦) , (𝑦, 𝑧) , (𝑧, 𝑥) } 𝑺7 = { (𝑥, 𝑥) , (𝑦, 𝑦) , (𝑧, 𝑧) } 𝑺4 = { (𝑥, 𝑧) , (𝑦, 𝑥) , (𝑧, 𝑦) }

15.2.4 Image relations Consider two subsets 𝐴 and 𝐵 of the domain of a function 𝖿 . What is the relationship between the image under 𝖿 of their union and the union of their images under 𝖿? Are they equal? Is one a subset of the other? What is the relationship between the image of their intersection under 𝖿 and the intersection of their images 𝖿? Theorem 15.9 (next theorem) answers these questions. 𝖿

𝑋

𝑌 𝖿(𝐴) 𝖿 (𝐵)

𝐵

𝐴

Theorem 15.9. t h m

𝖿

(⋃

𝐴𝑖

𝑖∈𝐼

𝖿

(⋂ 𝑖∈𝐼

𝐴𝑖

) )

25

Let 𝖿 be a function in 𝑌 𝑋 . =

𝖿 𝐴 ⋃ ( 𝑖)

∀𝖿∈𝑌 𝑋 , 𝐴𝑖 ∈𝟚𝑋

𝖿 𝐴 ⋂ ( 𝑖)

∀𝖿∈𝑌 𝑋 , 𝐴𝑖 ∈𝟚𝑋

(additive)

𝑖∈𝐼

⊆

𝑖∈𝐼

✎PROOF:

25

📘 Davis (2005) pages 6–7, 📘 Vaidyanathaswamy (1960) page 10


Daniel J. Greenhoe


version 0.30

page 256

15.2. FUNCTIONS

1. Proof that 𝖿 (⋃𝑖∈𝐼 𝐴𝑖 ) = ⋃𝑖∈𝐼 𝖿 (𝐴𝑖 ):

𝖿

(⋃ 𝑖∈𝐼

𝐴𝑖

)

=

𝑦 ∈ 𝑌 |∃𝑥 ∈ 𝐴 ⋃ 𝑖 { 𝑖∈𝐼

=

𝑦 ∈ 𝑌 |∃𝑥 ∈ 𝐴𝑖 ⋃{

such that such that

(𝑥, 𝑦) ∈ 𝖿


}

(𝑥, 𝑦) ∈ 𝖿 }

𝑖∈𝐼

=

𝖿 𝐴 ⋃ ( 𝑖)


𝑖∈𝐼

2. Proof that 𝖿 (⋂𝑖∈𝐼 𝐴𝑖 ) ⊆ ⋂𝑖∈𝐼 𝖿 (𝐴𝑖 ):

𝖿

(⋂ 𝑖∈𝐼

𝐴𝑖

)

=

𝑦 ∈ 𝑌 |∃𝑥 ∈ 𝐴 ⋂ 𝑖 { 𝑖∈𝐼

=

𝑦 ∈ 𝑌 |∃𝑥 {

such that

(𝑥, 𝑦) ∈ 𝖿


}

[𝑥 ∈ 𝐴𝑖 ] and (𝑥, 𝑦) ∈ 𝖿 ] ⋀ }

such that


𝑖∈𝐼

⊆

𝑦 ∈ 𝑌 | [∃𝑥 ∈ 𝐴𝑖 ⋀ { 𝑖∈𝐼

=

𝑦 ∈ 𝑌 |∃𝑥 ∈ 𝐴𝑖 ⋂{

such that such that

(𝑥, 𝑦) ∈ 𝖿 ]

}

(𝑥, 𝑦) ∈ 𝖿 }


𝑖∈𝐼

=


𝖿 𝐴 ⋂ ( 𝑖) 𝑖∈𝐼

✏ Theorem 15.10. 26 Let 𝖿 −1 ∈ 𝑋 𝑌 be the inverse of a function 𝖿 ∈ 𝑌 𝑋 . 𝖿 −1 (𝑌 ) = 𝑋 ∀𝖿 ∈𝑌 𝑋 𝖿 −1 (𝐴𝖼 ) = 𝖼 [𝖿 −1 (𝐴)] ∀𝖿 ∈𝑌 𝑋 , 𝐴∈𝟚𝑌 t −1 𝐴𝑖 = 𝖿 −1 (𝐴𝑖 ) ∀𝖿 ∈𝑌 𝑋 , 𝐴𝑖 ∈𝟚𝑌 h 𝖿 ⋃ ⋃ ( 𝑖∈𝐼 ) m 𝑖∈𝐼 𝖿 −1

(⋂ 𝑖∈𝐼

𝐴𝑖

)

=

⋂

𝖿 −1 (𝐴𝑖 )

∀𝖿 ∈𝑌 𝑋 , 𝐴𝑖 ∈𝟚𝑌

𝑖∈𝐼

✎PROOF:

1. Proof that 𝖿 −1 (𝐴𝖼 ) = 𝖼 [𝖿 −1 (𝐴)]: 𝖼 [𝖿 −1 (𝑌 )] = 𝖼 {𝑥 ∈ 𝑋|∃𝑦 ∈ 𝐴 = {𝑥 ∈ 𝑋|¬{∃𝑦 ∈ 𝐴 = {𝑥 ∈ 𝑋|∄𝑦 ∈ 𝐴 = {𝑥 ∈ 𝑋|∃𝑦 ∈ 𝐴

such that such that such that

𝖼

such that

(𝑥, 𝑦) ∈ 𝖿 }


(𝑥, 𝑦) ∈ 𝖿 }}


(𝑥, 𝑦) ∈ 𝖿 }


(𝑥, 𝑦) ∈ 𝖿 }

= 𝖿 −1 (𝐴𝖼 ) 26


📘 Davis (2005) pages 7–8, 📘 Vaidyanathaswamy (1960) page 10

version 0.30


Daniel J. Greenhoe


page 257


2. Proof that 𝖿 −1 (⋃𝑖∈𝐼 𝐴𝑖 ) = ⋃𝑖∈𝐼 𝖿 −1 (𝐴𝑖 ): 𝖿 −1

(⋃ 𝑖∈𝐼

𝐴𝑖

)

= = =

{ {

𝑥 ∈ 𝑋|∃𝑦 ∈

⋃

𝐴𝑖

(𝑥, 𝑦) ∈ 𝖿

such that

𝑖∈𝐼

𝑥 ∈ 𝑋|

∃𝑦 ∈ 𝐴𝑖 ⋁{

such that

𝑖∈𝐼

∃𝑥 ∈ 𝑋|𝑦 ∈ 𝐴𝑖 ⋃{

such that

}


(𝑥, 𝑦) ∈ 𝖿 } }

(𝑥, 𝑦) ∈ 𝖿 }


𝑖∈𝐼

=

⋃

𝖿 −1 (𝐴𝑖 )


𝑖∈𝐼

3. Proof that 𝖿 −1 (𝑌 ) = 𝑋: 𝖿 −1 (𝑌 ) = 𝖿 −1 (𝓘𝑋 ∪ 𝑌 ⧵𝓘𝑋) = 𝖿 −1 (𝓘𝑋) ∪ 𝖿 −1 (𝑌 ⧵𝓘𝑋)

by item 4

=𝑋∪∅


=𝑋 4. Proof that 𝖿 −1 (⋂𝑖∈𝐼 𝐴𝑖 ) = ⋂𝑖∈𝐼 𝖿 −1 (𝐴𝑖 ): 𝖿 −1

(⋂ 𝑖∈𝐼

𝐴𝑖

)

= = = =

{ { {

𝑥 ∈ 𝑋|∃𝑦 ∈

⋂

𝐴𝑖

such that

𝑖∈𝐼

𝑥 ∈ 𝑋|∃𝑦 𝑥 ∈ 𝑋|

such that

[∃𝑦 ∈ 𝐴𝑖 ⋀

(𝑥, 𝑦) ∈ 𝖿

}

𝐴 and (𝑥, 𝑦) ∈ 𝖿 by Definition 14.5 page 202 𝑦∈ ⋀ 𝑖 }} { 𝑖∈𝐼 such that

(𝑥, 𝑦) ∈ 𝖿]

𝑖∈𝐼

𝑥 ∈ 𝑋|∃𝑦 ∈ 𝐴𝑖 ⋂{


such that

(𝑥, 𝑦) ∈ 𝖿 }

}

by definition of function page 249 by Definition 14.5 page 202

𝑖∈𝐼

=

⋂

𝖿 −1 (𝐴𝑖 )


𝑖∈𝐼

5. Proof that 𝖿 −1 (𝑌 ⧵𝐴) = 𝑋 ⧵ 𝖿 −1 (𝐴): 𝖿 −1 (𝑌 ⧵ 𝐴) = 𝖿 −1 (𝑌 ∩ 𝐴𝖼 ) = 𝖿 −1 (𝑌 ) ∩ 𝖿 −1 (𝐴𝖼 )

by 6.

= 𝑋 ∩ 𝖿 −1 (𝐴𝖼 )

by 5.

= 𝑋 ∩ 𝖼 [𝖿

−1

(𝐴)]

by 3.

= 𝑋 ⧵𝖿 −1 (𝐴)


✏

15.2.5 Indicator functions By the axiom of extension, a set is uniquely defined by the elements that are in that set. Thus, we are often interested in the Boolean result of whether an element is in a set 𝐴, or is not in 𝐴, but exclude the possibility of both being true. That a statement is either true or false but definitely not both is called the law of the excluded middle and is a fundamental property of all Boolean algebras 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 258

15.2. FUNCTIONS

({1, 0}, ∨, ∧). 27 The indicator function (next definition) is a convenient “indicator” of whether or not a particular element is in a set, and has several interesting properties (Theorem 15.11 page 258). Definition 15.12. 28 Let 𝑋 be a set. 𝑋 The indicator function 𝟙 ∈ {0, 1}𝟚 is defined as d 1 for 𝑥 ∈ 𝐴 ∀𝑥∈𝑋, 𝐴∈𝟚𝑋 e 𝟙𝐴 (𝑥) = { 0 for 𝑥 ∉ 𝐴 ∀𝑥∈𝑋, 𝐴∈𝟚𝑋 f The indicator function 𝟙 is also called the characteristic function. Theorem 15.11. 29 Let 𝟙 be the indicator function (Definition 15.12 page 258). Let 𝑥 ∨ 𝑦 represent the maximum of {𝑥, 𝑦}. 𝟙∅ = 0 𝟙𝑋 = 1 t 𝟙𝐴∪𝐵 = 𝟙𝐴 ∨ 𝟙𝐵 𝟙𝐴∩𝐵 = 𝟙𝐴 𝟙𝐵 h 𝟙𝐴⧵𝐵 = 𝟙𝐴 (1 − 𝟙𝐵 ) m 𝟙𝐴△𝐵 = 𝟙𝐴 𝟙𝐵 𝟙𝐴𝖼 = 1 − 𝟙𝐴 ✎PROOF: 𝟙𝐴∪𝐵 (𝑥) ≜

1 { 0

for 𝑥 ∈ 𝐴 ∪ 𝐵 for 𝑥 ∉ 𝐴 ∪ 𝐵

=

1 { 0

for 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 otherwise

=

1 { 0

for 𝑥 ∈ 𝐴 1 ∨ otherwise } { 0

∀𝑥∈𝑋

by Definition 15.12

∀𝑥∈𝑋 ∀𝑥∈𝑋


for 𝑥 ∈ 𝐵 otherwise }

= 𝟙𝐴 (𝑥) ∨ 𝟙𝐵 (𝑥)

by Definition 15.12

𝟙𝐴∩𝐵 (𝑥) ≜

1 { 0

for 𝑥 ∈ 𝐴 ∩ 𝐵 for 𝑥 ∉ 𝐴 ∩ 𝐵

=

1 { 0

for 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 otherwise

=

1 { 0

1 for 𝑥 ∈ 𝐴 ∧ otherwise } { 0

∀𝑥∈𝑋

by Definition 15.12

∀𝑥∈𝑋 ∀𝑥∈𝑋


for 𝑥 ∈ 𝐵 otherwise }

= 𝟙𝐴 (𝑥) ∧ 𝟙𝐵 (𝑥) = 𝟙 𝐴 𝟙𝐵

by Definition 15.12

𝟙𝐴𝖼 (𝑥) =

1 { 0

for 𝑥 ∈ 𝐴𝖼 for 𝑥 ∉ 𝐴𝖼

=

1 { 0

for 𝑥 ∉ 𝐴 for 𝑥 ∈ 𝐴

∀𝑥∈𝑋 ∀𝑥∈𝑋 ∀𝑥∈𝑋 ∀𝑥∈𝑋

by Definition 15.12 by Definition 14.5 page 202

= 1 − 𝟙𝐴 𝟙𝐴⧵𝐵 = 𝟙𝐴∩𝐵𝖼 = 𝟙 𝐴 𝟙𝐵 𝖼 = 𝟙𝐴 (1 − 𝟙𝐵 ) 27

excluded middle: Theorem 12.2 page 185 📘 Feller (1971), page 104, 📘 Aliprantis and Burkinshaw (1998), page 126, 📘 Hausdorff (1937), page 22, 📘 de la Vallée-Poussin (1915) page 440 29 📘 Aliprantis and Burkinshaw (1998), page 126, 📘 Hausdorff (1937), pages 22–23 28

version 0.30


Daniel J. Greenhoe


page 259


𝟙𝐴△𝐵 = 𝟙(𝐴⧵𝐵𝖼 )∪(𝐵⧵𝐴𝖼 ) = (𝟙𝐴⧵𝐵𝖼 ) ∨ (𝟙𝐵⧵𝐴𝖼 ) = [𝟙𝐴 (1 − 𝟙𝐵𝖼 )] ∨ [𝟙𝐵 (1 − 𝟙𝐴𝖼 )] = [𝟙𝐴 (1 − 1 + 𝟙𝐵 )] ∨ [𝟙𝐵 (1 − 1 + 𝟙𝐴 )] = [𝟙𝐴 𝟙𝐵 ] ∨ [𝟙𝐵 𝟙𝐴 ] = 𝟙 𝐴 𝟙𝐵 𝟙∅ = 𝟙𝐴⧵𝐴 = 𝟙 𝐴 (1 − 𝟙 𝐴 ) = 𝟙 𝐴 − 𝟙 𝐴 𝟙𝐴 = 𝟙 𝐴 − 𝟙𝐴 =0 𝟙𝑋 = 𝟙𝐴∪𝐴𝖼 = 𝟙 𝐴 ∨ 𝟙 𝐴𝖼 = 𝟙 𝐴 ∨ (1 − 𝟙 𝐴 ) =1

✏ Definition 15.13. d e The Kronecker delta function 𝛿𝑛̄ is defined as f

𝛿𝑛̄ ≜

1 for 𝑛 = 0 { 0 for 𝑛 ≠ 0.̇

and

∀𝑛∈ℤ

Proposition 15.6. Let 𝟙 be the indicator function (Definition 15.12 page 258) and 𝛿 ̄ be the Kronecker delta function (Definition 15.13 page 259). p r 𝛿𝑛̄ = 𝟙{0} (𝑛) ∀𝑛∈ℤ p

15.2.6 Calculus of functions Definition 15.14. 30 Let 𝑌 𝑋 be the set of all functions from a set 𝑋 to a set 𝑌 . ∀𝑥∈𝑋, 𝖿 ∈𝑌 𝑋 (negation) [−𝖿](𝑥) ≜ −[𝖿(𝑥)] 𝑋 ∀𝑥∈𝑋 𝖿 ,𝗀∈𝑌 (function addition) d [𝖿 ⨢ 𝗀](𝑥) ≜ 𝖿 (𝑥) + 𝗀(𝑥) e [𝖿 − 𝗀](𝑥) ≜ 𝖿 (𝑥) + [−𝗀](𝑥) ∀𝑥∈𝑋 𝖿,𝗀∈𝑌 𝑋 (function subtraction) f 𝑋 ∀𝑥∈𝑋 𝖿,𝗀∈𝑌 (function multiplication) [𝗀𝖿 ](𝑥) ≜ 𝗀[𝖿 (𝑥)] 𝑋 ∀𝑥∈𝑋,𝛼∈𝙔 𝖿 ∈𝑌 (scalar multiplication) [𝛼𝖿](𝑥) ≜ 𝛼[𝖿 (𝑥)] Definition 15.15. in 𝑋 𝑋 . 𝐈 d 𝑛 ⎧ ⎪ 𝑛 e 𝖿 ≜ ⎨ ∏1 𝖿 f −1 𝑛 ⎪ ⎩ (𝖿 ) 30

Let 𝖿 be a function in 𝑋 𝑋 with inverse relation 𝖿 −1 and let 𝐈 be the identity function for 𝑛 = 0 for 𝑛 ∈ ℕ for 𝑛 ∈ ℤ−

📘 Michel and Herget (1993) page 409, 📘 Cayley (1858), 📘 Riesz (1913), 📘 Hilbert et al. (1927) page 6


Daniel J. Greenhoe


version 0.30

page 260

15.2. FUNCTIONS

Theorem 15.12. (𝖿𝗀)−1 1. t 2. 𝗁(𝗀𝖿) h m 3. (𝖿 ⨢ 𝗀)𝗁 4. 𝛼(𝖿 𝗀)

31

Let 𝑋, 𝑌 , and 𝑍 be sets. = (𝗀−1 )(𝖿 −1 ) ∀𝖿 ∈𝑌 𝑋 , 𝗀∈𝑍 𝑌 = (𝗁𝗀)𝖿 ∀𝖿 ∈𝑋 𝑊 , 𝗀∈𝑌 𝑋 , 𝗁∈𝑍 𝑌 = (𝖿𝗁) ⨢ (𝗀𝗁) ∀𝖿 ,𝗀∈𝑌 𝑋 , 𝗁∈𝑍 𝑌 = (𝛼𝖿)𝗀 ∀𝖿 ∈𝑌 𝑋 , 𝗀∈𝑍 𝑌

(idempotent) (associative) (right distributive) (homogenous)

✎PROOF: 1. Proof of the idempotent property: (a) Note that 𝖿𝗀 = 𝖿 ∘ 𝗀, where ∘ is the composition function (Definition 15.3 page 242). (b) The result follows from Theorem 15.2 (page 242), where it is demonstrated to be true for the more general case of relations. 2. Proof of the associative property: This result follows from Theorem 15.2 (page 242), where it is demonstrated to be true for the more general case of relations. 3. Proof of the right distributive property: [(𝖿 ⨢ 𝗀)𝗁]𝑥 = (𝖿 ⨢ 𝗀)(𝗁𝑥) = [𝖿(𝗁𝑥)] ⨢ [𝗀(𝗁𝑥)]


= [(𝖿 𝗁)𝑥] ⨢ [(𝗀𝗁)𝑥]



4. Proof of the homogeneous property: [𝛼 [𝖿𝗀]](𝑥) = 𝛼 [[𝖿 𝗀](𝑥)] = 𝛼 [𝖿 [𝗀(𝑥)]]


= [𝛼𝖿][𝗀(𝑥)] = [[𝛼𝖿]𝗀](𝑥)



✏ Theorem 15.13. 1. (𝒜, ⨢) t 2. (𝒜, ⨢, ⋅) h m 3. (𝒜, ⨢) 4. (𝒜, ⨢, ⋅)

Let 𝒜 ≜ 𝑋 𝑋 be the set of functions on 𝑋 𝑋 . is an additive group. is a ring. is a linear space. is an algebra.

✎PROOF: 1. additive group: 1. 𝖿 ⨢𝟘 = 𝟘+𝖿 =𝖿 2. 𝖿 ⨢ (−𝖿 ) = (−𝖿 ) + 𝖿 = 𝟘 3. (𝖿 ⨢ 𝗀) + 𝗁 = 𝖿 ⨢ (𝗀 + 𝗁)

∀𝖿 ∈𝒜

(𝟘 ∈ 𝒜 is the identity element)

∀𝖿∈𝒜

((−𝖿 ) is the inverse of 𝖿 )

∀𝖿 ,𝗀,𝗁∈𝒜

((𝒜, ⋅) is associative)

2. ring: 1. 2. 3. 4. 31

(𝒜, +, ∗) is a group with respect to (𝒜, +) 𝖿 (𝗀𝗁) = (𝖿 𝗀)𝗁 𝖿 (𝗀 + 𝗁) = (𝖿 𝗀) + (𝖿 𝗁) (𝖿 ⨢ 𝗀)𝗁 = (𝖿 𝗁) + (𝗀𝗁)

(additive group) ∀𝖿,𝗀,𝗁∈𝒜

(associative with respect to ∗)

∀𝖿,𝗀,𝗁∈𝒜

(∗ is left distributive over +)


(∗ is right distributive over +).

📘 Kelley (1955) page 8, 📘 Berberian (1961) page 88 ⟨Theorem IV.5.1⟩

version 0.30


Daniel J. Greenhoe


page 261


3. linear space: 1. 2. 3. ∃𝟘 ∈ 𝑋 4. ∃𝗀 ∈ 𝑋 5. 6. 7. 8.

(𝖿 ⨣ 𝗀) ⨣ 𝗁 𝖿 ⨣𝗀 such that 𝖿 ⨢ 𝟘 such that 𝖿 ⨣ 𝗀 𝛼 ⊗ (𝖿 ⨣ 𝗀) (𝛼 + 𝛽) ⊗ 𝖿 𝛼(𝛽 ⊗ 𝖿 ) 1⊗𝖿

= = = = = = = =

𝖿 ⨣ (𝗀 ⨣ 𝗁) 𝗀⨣𝖿 𝖿 𝟘 (𝛼 ⊗ 𝖿) ⨣ (𝛼 ⊗ 𝗀) (𝛼 ⊗ 𝖿) ⨣ (𝛽 ⊗ 𝖿) (𝛼 ⋅ 𝛽) ⊗ 𝖿 𝖿

1. (𝖿 𝗀)𝗁 2. 𝖿 (𝗀 ⨢ 𝗁) 3. (𝖿 ⨢ 𝗀)𝗁 4. 𝛼(𝗀𝗁)

𝖿 (𝗀𝗁) (𝖿 𝗀) + (𝖿 𝗁) (𝖿 𝗁) + (𝗀𝗁) (𝛼𝗀)𝗁 = 𝗀(𝛼𝗁)


(⨣ is associative)

∀𝖿,𝗀∈𝒜

(⨣ is commutative)

∀𝖿∈𝑋𝒜

(⨣ identity)

∀𝖿∈𝒜

(⨣ inverse)

∀𝛼∈𝑆 and 𝖿,𝗀∈𝒜

(⊗ distributes over ⨣)

∀𝛼,𝛽∈𝑆 and 𝖿∈𝒜

(⊗ pseudo-distributes over +)

∀𝛼,𝛽∈𝑆 and 𝖿∈𝒜

(⋅ associates with ⊗)

∀𝖿∈𝒜

(⊗ identity)

4. algebra: = = = =


(associative)


(left distributive)

∀𝖿 ,𝗀,𝗁∈𝒜

(right distributive)

∀𝗀,𝗁∈𝒜 and 𝛼∈𝔽

(scalar commutative)

✏ Theorem 15.14. Let 𝒜 ≜ {𝖿 ∈ 𝑋 𝑋 |∃𝖿 −1 on 𝑋 𝑋 . t h (𝒜, ⋅) is a (multiplicative) group. m

such that

𝖿 −1 𝖿 ≗ 𝖿𝖿 −1 ≗ 𝐈} be the set of invertible functions

✎PROOF: 1. multiplicative group: 1. 𝖿 𝐈 = 𝐈𝖿 = 𝖿 ∀𝖿 ∈ 𝒜 (𝐈 ∈ 𝒜 is the identity element) −1 −1 2. 𝖿 𝖿 = 𝖿 𝖿 = 𝐈 ∀𝖿 ∈ 𝒜 (𝖿 −1 is the inverse of 𝖿 ) 3. (𝖿𝗀)𝗁 = 𝖿 (𝗀𝗁) ∀𝖿 , 𝗀, 𝗁 ∈ 𝒜 ((𝒜, ⋅) is associative) 2. field:

1. (𝑋, +, ∗) is a ring 2. 𝒙𝒚 = 𝒚𝒙 3. (𝑋 ⧵{0}, ∗) is a group

(ring) ∀𝒙,𝒚∈𝑋

(commutative with respect to ∗) (group with respect to ∗).

✏ Theorem 15.15. Let 𝓓(𝖿 ) be the domain of an function 𝖿 and 𝓘(𝖿 ) the image of 𝖿.

t h m

𝓓

𝖿𝑖 = 𝓓 𝖿 ⋃ ( 𝑖) (⋃ ) 𝑖∈𝐼 𝑖∈𝐼

𝓘

𝖿𝑖 = 𝓘 𝖿 ⋃ ( 𝑖) (⋃ ) 𝑖∈𝐼 𝑖∈𝐼

𝓓

𝖿𝑖 ⊆ 𝓓 𝖿 ⋂ ( 𝑖) (⋂ 𝑖∈𝐼 ) 𝑖∈𝐼 𝓓(𝖿 ⧵𝗀) ⊇ 𝓓(𝖿 )⧵𝓓(𝗀)

𝓘

𝖿𝑖 ⊆ 𝓘 𝖿 ⋂ ( 𝑖) (⋂ 𝑖∈𝐼 ) 𝑖∈𝐼 𝓘(𝖿 ⧵𝗀) ⊇ 𝓘(𝖿 )⧵𝓘(𝗀)

✎PROOF:

These results follow from Theorem 15.3 (page 244).

Definition 15.16.

32

✏

Let 𝙓 and 𝙔 be linear spaces over a field 𝔽 and with dual spaces

𝙓 ∗ ≜ {𝖿(𝒙; 𝒙∗ ) ∈ 𝔽 𝙓 |𝒙∗ ∈ 𝙓 ∗ } (set of functionals with parameter 𝒙∗ from 𝙓 to 𝔽 ) 𝙔 ∗ ≜ {𝗀(𝒚; 𝒚∗ ) ∈ 𝔽 𝙔 |𝒚∗ ∈ 𝙔 ∗ } . (set of functionals with parameter 𝒚∗ from 𝙔 to 𝔽 ) 32

📘 Michel and Herget (1993) page 420, 📘 Giles (2000), page 171


Daniel J. Greenhoe


version 0.30

page 262

15.3. TEMPERED DISTRIBUTIONS

Let 𝖿 ∈ 𝑌 𝑋 be a function. d A function 𝖿 ∗ in 𝙓 ∗𝙔 ∗ is the conjugate of the function 𝖿 if e 𝗀(𝖿 𝒙; 𝒚∗ ) = 𝖿(𝒙; 𝖿 ∗ 𝒚∗ ) ∀𝒙∈𝙓 , 𝖿 ∈𝙓 ∗ , 𝗀∈𝙔 ∗ f

15.3

Tempered Distributions ❝I am sure that something must be found.

There must exist a notion of generalized functions which are to functions what the real numbers are to the rationals.❞ Giuseppe Peano (1858–1932), Italian mathematician33

Definition 15.17. 34 A test function is any function 𝜙 that satisfies 1. 𝜙 ∈ ℂℝ d 2. 𝜙 is infinitely differentiable. e The set of all test functions is denoted ℂ∞ (ℝ). A test function 𝜙 belongs to the Schwartz class f 𝑆 if, for some set of constants {𝐶𝑛,𝑘 |𝑛, 𝑘 ∈ 𝕎 }, (1 + |𝑥|)𝑛 |𝜙(𝑘) | ≤ 𝐶𝑛,𝑘 ∀𝑛, 𝑘 ∈ 𝕎, ∀𝑥 ∈ ℝ Definition 15.18. 35 Let 𝑆 be the Schwartz class of functions (Definition 15.17). 𝖽[⋅] is a tempered distribution if d 1. 𝖽[𝛼1 𝜙1 + 𝛼2 𝜙2 ] = 𝖽[𝛼1 𝜙1 ] + 𝖽[𝛼2 𝜙2 ] ∀𝜙1 ,𝜙2 ∈𝑆, 𝛼1 ,𝛼2 ∈ℝ (linear) e f (continuous) 2. lim 𝜙𝑛 = 𝜙 ⟹ lim 𝖽[𝜙𝑛 ] = 𝖽[𝜙] ∀𝜙1 ,𝜙2 ∈𝑆 𝑛→∞ 𝑛→∞

and

Definition 15.19. 36 Let 𝑆 be the Schwartz class of functions (Definition 15.17). d Two tempered distributions 𝖽1 and 𝖽2 are equal if e ∀𝜙1 , 𝜙2 ∈ 𝑆 𝖽[𝜙1 ] = 𝖽[𝜙2 ] f Theorem 15.16 (next) demonstrates that all continuous and what we might call “well behaved” functions generate a tempered distribution. Theorem 15.16. 𝖳𝖿 [𝜙] ≜ t h m

1. 2.

∫ ℝ

37

Let 𝖿 be a function in ℂℝ . Let 𝖳𝖿 be defined as

𝖿(𝑥)𝜙(𝑥) 𝖽𝑥.

𝖿 is continuous and ⟹ 𝖳𝖿 [𝜙] is a tempered distribution. 𝑛 ∃𝑛, 𝑀 such that |𝖿(𝑥)| ≤ 𝑀(1 + |𝑥|) ∀𝑥∈ℝ }

✎PROOF: 33

quote: 📘 Duistermaat and Kolk (2010) page ix image http://en.wikipedia.org/wiki/File:Giuseppe_Peano.jpg, public domain 34 📘 Vretblad (2003) page 200 35 📘 Vretblad (2003) pages 203–204 ⟨Definition 8.3⟩ 36 📘 Vretblad (2003) page 206 37 📘 Vretblad (2003) page 204 version 0.30


Daniel J. Greenhoe


page 263


1. Proof that 𝖳𝖿 is linear: 𝖳𝖿 [𝜙1 + 𝜙2 ] = =

∫ ℝ ∫ ℝ

𝖿 (𝑥)(𝜙1 (𝑥) + 𝜙2 (𝑥)) 𝖽𝑥

by definition of 𝖳𝖿

𝖿 (𝑥)𝜙1 (𝑥) 𝖽𝑥 +

by linearity of

∫ ℝ

𝖿 (𝑥)𝜙2 (𝑥) 𝖽𝑥

= 𝖳𝖿 [𝜙1 ] + 𝖳𝖿 [𝜙2 ]

∫


2. Proof that 𝖳𝖿 is cotinuous: 𝖿(𝑥)𝜙𝑛 (𝑥) 𝖽𝑥 − 𝖿(𝑥)𝜙(𝑥) 𝖽𝑥 lim |𝖳𝖿 [𝜙𝑛 ] − 𝖳𝖿 [𝜙]| = lim | 𝑛→∞ |∫ ∫

𝑛→∞

ℝ

= lim

𝑛→∞ |∫ ℝ

≤ lim

𝑛→∞ ∫ ℝ

=

∫ ℝ

𝖿(𝑥)(𝜙𝑛 (𝑥) − 𝜙(𝑥) 𝖽𝑥)

by linearity of

|

∫

𝑀(1 + |𝑥|)𝑚 |𝜙𝑛 (𝑥) − 𝜙(𝑥)| 𝖽𝑥

𝑀(1 + |𝑥|)𝑚+2 |𝜙𝑛 (𝑥) − 𝜙(𝑥)|

1 𝖽𝑥 (1 + |𝑥|)2

≤ lim max {𝑀(1 + |𝑥|)𝑚+2 |𝜙𝑛 (𝑥) − 𝜙(𝑥)|} 𝑛→∞


ℝ

𝑥

1 𝖽𝑥 2 ∫ ℝ (1 + |𝑥|)

=0

✏ Definition 15.20. 38 d The Dirac delta distribution 𝛿 ∈ ℂℝ is defined as e 𝛿[𝜙] ≜ 𝜙(0) f One could argue that a tempered distribution 𝖽 behaves as if it satisfies the following relation: 𝖽[𝜙] ≎

𝖽(𝑥)𝜙(𝑥) 𝖽𝑥. ∫ ℝ This is not technically correct because in general 𝖽 is not a function that can be evaluated at a given point 𝑥 (and hence the here undefined relation “≎”). But despite this failure, the notation is still very useful in that distributions do behave “as if” they are defined by the above integral relation. Using this notation, the Dirac delta distribution looks likes this: 𝛿[𝜙] ≜ 𝜙(0) ≎

∫ ℝ

𝛿(𝑥)𝜙(𝑥) 𝖽𝑥

We could also define another “scaled” and “translated” distribution 𝛿𝑎𝑏 such that 𝑥 𝛿𝑎𝑏 [𝜙] ≜ 𝑏𝜙(𝑎𝑏) ≎ 𝛿 ( − 𝑎)𝜙(𝑥) 𝖽𝑥 ∫ 𝑏 ℝ because 𝑥 𝛿 ( − 𝑎)𝜙(𝑥) 𝖽𝑥 = 𝛿(𝑢 − 𝑎)𝜙(𝑢𝑏)𝑏 𝖽𝑢 ∫ ∫ 𝑏 ℝ ℝ

where 𝑢 =

𝑥 𝑏

=𝑏

𝛿(𝑢 − 𝑎)𝜙(𝑢𝑏) 𝖽𝑢 ∫ ℝ = 𝑏𝜙(𝑎𝑏) 38

📘 Vretblad (2003) page 205 ⟨Example 8.13⟩, 📘 Friedlander and Joshi (1998) page 8


Daniel J. Greenhoe


version 0.30

page 264

15.4

15.4. LITERATURE

Literature

📚 Literature survey: 1. Reference books: 📘 Maddux (2006) 📘 Suppes (1972) ⟨0486616304⟩Chapter 3: Relations and Functions 📘 Kelley (1955) pages 6–13 2. Pioneering papers on relations: 📘 de Morgan (1864a) 📘 de Morgan (1864b) 📘 Peirce (1883a) 📘 Peirce (1883c) 📘 Peirce (1883b) 📘 Schröder (1895) 3. Axiomization of calculus of relations: 📘 Tarski (1941) 4. Historically oriented presentations: 📘 Maddux (1991) 📘 Pratt (1992) pages 248–254 5. Theory of Distributions 📘 Vretblad (2003) 📘 Hömander (2003) ⟨Referenced by Vretblad(2003) as a standard work.⟩ 📘 Knapp (2005) 6. Miscellaneous: 📘 Peirce (1870a) 📘 Peirce (1870b) 📘 Peirce (1870c)

📚

version 0.30


Daniel J. Greenhoe


APPENDIX A INTERVALS AND CONVEXITY

A.1 Intervals In the real number system, for 𝑎 ≤ 𝑏, the interval [𝑎 ∶ 𝑏] is the set 𝑎 and 𝑏 and all the numbers inbetween, as in [𝑎 ∶ 𝑏] ≜ {𝑥 ∈ ℝ|𝑎 ≤ 𝑥 ≤ 𝑏}. This concept can be easily generalized: In an ordered set (Definition 3.2 page 46), if two elements 𝑥 and 𝑦 are comparable and 𝑥 ≤ 𝑦, then we say that 𝑥 and 𝑦 and all the elements inbetween, as determined by the ordering relation ≤, are the interval [𝑎 ∶ 𝑏] (Definition A.1 page 265). In a lattice (Definition 4.3 page 61), the concept of the interval can be generalized even further. In an arbitrary ordered set, the interval [𝑥 ∶ 𝑦] of Definition A.1 is restricted to the case in which 𝑥 and 𝑦 are comparable (Definition 3.2 page 46). This restriction can be lifted (Definition A.2 page 265) with the additional structure of upper and lower bounds provided by lattices. A metric space in general has no order relation ≤ (Definition 3.2 page 46). But intervals can still be defined (Definition A.4 page 266) in a metric space in terms of the triangle inequality. A linear space over a real or complex field in general has no order relation that compares vectors in the space, but the standard order relation ≤ for real numbers ℝ can still be used (Definition A.5 page 266) to define an interval in a linear space. Definition A.1. 1 Let (𝑋, ≤) be a linearly ordered set. In an ordered set ( 𝑋, ≤) (Definition 3.2 page 46), the set [𝑥 ∶ 𝑦] ≜ {𝑧 ∈ 𝑋|𝑥 ≤ 𝑧 ≤ 𝑦} is called a closed interval d e the set (𝑥 ∶ 𝑦] ≜ {𝑧 ∈ 𝑋|𝑥 < 𝑧 ≤ 𝑦} is called a half-open interval f the set [𝑥 ∶ 𝑦) ≜ {𝑧 ∈ 𝑋|𝑥 ≤ 𝑧 < 𝑦} is called a half-open interval the set (𝑥 ∶ 𝑦) ≜ {𝑧 ∈ 𝑋|𝑥 < 𝑧 < 𝑦} is called an open interval.

and and and

Definition A.2 (intervals on lattices). 2 Let ( 𝑋, ∨, ∧ ; ≤) be a lattice. In a lattice ( 𝑋, ∨, ∧ ; ≤) (Definition 4.3 page 61), the set [𝑥 ∶ 𝑦] ≜ {𝑧 ∈ 𝑋|𝑥 ∧ 𝑦 ≤ 𝑧 ≤ 𝑥 ∨ 𝑦} is called a closed interval. d e the set (𝑥 ∶ 𝑦] ≜ {𝑧 ∈ 𝑋|𝑥 ∧ 𝑦 < 𝑧 ≤ 𝑥 ∨ 𝑦} is called a half-open interval. f the set [𝑥 ∶ 𝑦) ≜ {𝑧 ∈ 𝑋|𝑥 ∧ 𝑦 ≤ 𝑧 < 𝑥 ∨ 𝑦} is called a half-open interval. the set (𝑥 ∶ 𝑦) ≜ {𝑧 ∈ 𝑋|𝑥 ∧ 𝑦 < 𝑧 < 𝑥 ∨ 𝑦} is called an open interval. 1 2

📘 Apostol (1975) page 4, 📘 Ore (1935) page 409 📘 Duthie (1942) page 2, 📘 Ore (1935) page 425 ⟨quotient structures⟩

page 266

A.2. CONVEX SETS

When 𝑥 and 𝑦 are comparable and 𝑥 ≤ 𝑦, then Definition A.2 (previous) simplifies to Definition A.1 (page 265). Definition A.3. 3 Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice with dual 𝙇∗ . Let [𝑥 ∶ 𝑦] be a closed interval (Definition ∗ A.2 page 265) on set 𝑋. The sublattices 𝙇[𝑥 ∶ 𝑦] and 𝙇 [𝑥 ∶ 𝑦] are defined as follows: d 𝙇[𝑥 ∶ 𝑦] ≜ {𝑧 ∈ 𝙇|𝑧 ∈ [𝑥 ∶ 𝑦]} ∀𝑥,𝑦∈𝑋 e ∗ ∗ ∀𝑥,𝑦∈𝑋 f 𝙇 [𝑥 ∶ 𝑦] ≜ {𝑧 ∈ 𝙇 |𝑧 ∈ [𝑥 ∶ 𝑦]} Definition A.4. 4 In a metric space (𝑋, 𝖽), d the set [𝑎 ∶ 𝑏] is the closed interval from 𝑥 to 𝑦 and is defined as e [𝑥 ∶ 𝑦] ≜ {𝑧 ∈ 𝑋|𝖽(𝑥, 𝑧) + 𝖽(𝑧, 𝑦) = 𝖽(𝑥, 𝑦)}. f An element 𝑧 ∈ 𝑋 is geodesically between 𝑥 and 𝑦 if 𝑧 ∈ [𝑥 ∶ 𝑦]. Definition A.5. 5 In a linear space (𝑋, +, ⋅, (𝔽 , ∔, ⨰)), [𝒙 ∶ 𝒚] ≜ {𝜆𝒙 + (1 − 𝜆)𝒚 = 𝒛|0 ≤ 𝜆 ≤ 1} d e (𝒙 ∶ 𝒚] ≜ {𝜆𝒙 + (1 − 𝜆)𝒚 = 𝒛|0 < 𝜆 ≤ 1} f [𝒙 ∶ 𝒚) ≜ {𝜆𝒙 + (1 − 𝜆)𝒚 = 𝒛|0 ≤ 𝜆 < 1} (𝒙 ∶ 𝒚) ≜ {𝜆𝒙 + (1 − 𝜆)𝒚 = 𝒛|0 < 𝜆 < 1}

A.2

is called a closed interval is called a half-open interval is called a half-open interval is called an open interval.

and and and

Convex sets

Using the concept of the interval (previous section), we can define the convex set (next definition). Definition A.6. 6 Let 𝑋 be a set in an ordered set ( 𝑋, ≤), a lattice ( 𝑋, ∨, ∧ ; ≤), a metric space (𝑋, 𝖽), or a linear space (𝑋, +, ⋅, (𝔽 , ∔, ⨰)). d A subset 𝐷 ⊆ 𝑋 is a convex set in 𝑋 if e 𝑥, 𝑦 ∈ 𝐷 ⟹ [𝑥 ∶ 𝑦] ⊆ 𝐷. f A set that is not convex is concave. Example A.1. Consider the Euclidean space ℝ2 (a special case of a linear space). 𝜆𝒙 + (1 − 𝜆)𝒚 Q 𝒙 Q QQ s @ R @

𝒚

left is a ⇐ {The figure to the 2 convex set in ℝ . The figure to the right is a ⇒ } concave set in ℝ2 .

𝜆𝒙 + (1 − 𝜆)𝒚 Q Q 𝒙 QQ s 𝒚

Example A.2. In a metric space, examples of convex sets are convex balls. Examples include those balls generated by the following metrics: Taxi-cab metric Euclidean metric Sup metric Tangential metric 3

📘 Maeda and Maeda (1970), page 1 📘 van de Vel (1993) page 8 5 📘 Barvinok (2002) page 2 6 📘 Barvinok (2002) page 5 4

version 0.30


Daniel J. Greenhoe


page 267

APPENDIX A. INTERVALS AND CONVEXITY

Examples of metrics generating balls which are not convex include the following: Parabolic metric Exponential metric

A.3 Convex functions Definition A.7. 7 Let (𝑋, +, ⋅, (𝔽 , ∔, ⨰)) be a linear space and 𝐷 a convex set (Definition A.6 page 266) in 𝑋. A function 𝖿 ∈ 𝐹 𝐷 is convex if 𝖿 (𝜆𝒙 + [1 − 𝜆]𝒚) ≤ 𝜆𝖿(𝒙) + (1 − 𝜆) 𝖿(𝒚) ∀𝒙, 𝒚 ∈ 𝐷 and ∀𝜆 ∈ (0, 1) d A function 𝗀 ∈ 𝐹 𝐷 is strictly convex if e 𝗀(𝜆𝒙 + [1 − 𝜆]𝒚) = 𝜆𝗀(𝒙) + (1 − 𝜆) 𝗀(𝒚) ∀𝒙, 𝒚 ∈ 𝐷, 𝒙 ≠ 𝒚, and ∀𝜆 ∈ (0, 1) f 𝐷 A function 𝖿 ∈ 𝐹 is concave if −𝖿 is convex. A function 𝖿 ∈ 𝐹 𝐷 is affine if 𝖿 is convex and concave.

𝖿(𝑥) 𝜆𝖿 (𝑥) + (1 − 𝜆)𝖿 (𝑦) b 𝖿 (𝑦)

Example A.3. The function 𝖿(𝑥) = 2𝑥 is a convex function (Definition A.7 page 267), as illustrated to the right.

𝖿(𝑥) b

𝖿 (𝜆𝑥 + [1 − 𝜆]𝑦) 𝑦

𝑥

𝑥

Definition A.8. 8 Let (𝑋, +, ⋅, (ℝ, ∔, ⨰)) be a linear space. 𝙓 d The epigraph 𝖾𝗉𝗂(𝖿) and hypograph 𝗁𝗒𝗉(𝖿) of a functional 𝖿 ∈ ℝ are defined as e 𝖾𝗉𝗂(𝖿) ≜ {(𝑥, 𝑦) ∈ 𝑋 × ℝ|𝑦 ≥ 𝖿(𝑥)} f 𝗁𝗒𝗉(𝖿) ≜ {(𝑥, 𝑦) ∈ 𝑋 × ℝ|𝑦 ≤ 𝖿(𝑥)} Example A.4. 𝖾𝗉𝗂(𝑥2 )

2

e x

1 −1

1 1 − 𝑥2

𝑥2 0

1

𝖾𝗉𝗂(1 − 𝑥2 )

2

𝑥

−1

0

1

𝑥

Proposition A.1. 9 Let (𝑋, +, ⋅, (ℝ, ∔, ⨰)) be a linear space. Let 𝖿 be a functional in ℝ𝙓 . p 𝖿 is a 𝖾𝗉𝗂(𝖿) is a r ⟺ { convex set } p { convex function } Theorem A.1 (Jensen's Inequality).

10

Let (𝑋, +, ⋅, (𝔽 , ∔, ⨰)) be a linear space, 𝐷 a subset of 𝑋, and

7

📘 Simon (2011) page 2, 📘 Barvinok (2002) page 2, 📘 Bollobás (1999), page 3, 📘 Jensen (1906), page 176 📘 Beer (1993) page 13 ⟨§1.3⟩, 📘 Aubin and Frankowska (2009) page 222, 📘 Aubin (2011) page 223 9 📘 Udriste (1994) page 63, 📘 Kurdila and Zabarankin (2005) page 178 ⟨Proposition 6.1.1⟩, 📘 Rockafellar (1970) page 23 ⟨Section 4 Convex Functions⟩, 📘 Çinlar and Vanderbei (2013) page 86 ⟨5.4 Theorem⟩ 10 📘 Mitrinović et al. (2010) page 6, 📘 Bollobás (1999) page 3, 📘 Lay (1982) page 7, 📘 Jensen (1906) pages 179–180 8


Daniel J. Greenhoe


version 0.30

page 268

A.3. CONVEX FUNCTIONS

𝖿 a functional in 𝔽 𝐷 . Let ∑ be the summation operator. 1. 𝐷 is convex and ⎧ ⎫ ⎪ 2. 𝖿 is convex and ⎪ 𝘕 ⎛𝘕 ⎞ t ⎪ ⎪ ⎜ ⎟ h ⎨ ≤ 𝜆 𝖿 𝒙 ⟹ 𝖿 𝜆 𝒙 𝘕 ⎬ ∑ 𝑛 𝑛⎟ ∑ 𝑛 ( 𝑛) ⎜ m ⎪ 3. 𝜆 =1 (weights) ⎪ ⎝𝑛=1 ⎠ 𝑛=1 ∑ 𝑛 ⎪ ⎪ ⎩ 𝑛=1 ⎭ ✎PROOF:

∀𝒙𝑛 ∈ 𝐷, 𝘕 ∈ ℕ

Proof is by induction:

1. Proof that statement is true for 𝘕 = 1: ⎛𝘕 =1 ⎞ 𝖿⎜ 𝜆𝑛 𝒙𝑛 ⎟ = 𝖿 (𝜆1 𝒙1 ) ⎜∑ ⎟ ⎝ 𝑛=1 ⎠ ≤ 𝖿 (𝜆1 𝒙1 ) 𝘕 =1

=

𝜆𝑛 𝖿(𝒙𝑛 )

∑ 𝑛=1

2. Proof that statement is true for 𝘕 = 2: ⎛𝘕 =2 ⎞ 𝖿⎜ 𝜆𝑛 𝒙𝑛 ⎟ = 𝖿(𝜆1 𝒙1 + 𝜆2 𝒙2 ) ⎜∑ ⎟ ⎝ 𝑛=1 ⎠ ≤ 𝜆1 𝖿(𝒙1 ) + 𝜆2 𝖿 (𝒙2 )

by convexity hypothesis

𝘕 =2

=

𝜆𝑛 𝖿(𝒙𝑛 )

∑ 𝑛=1

3. Proof that if the statement is true for 𝘕 , then it is also true for 𝘕 + 1: 𝘕 ⎞ ⎛𝘕 +1 ⎟ ⎜ 𝜆𝑛 𝒙𝑛 + 𝜆𝘕 +1 𝒙𝘕 +1 𝜆𝑛 𝒙 𝑛 = 𝖿 𝖿 ⎟ ⎜∑ ) (∑ 𝑛=1 𝑛=1 ⎠ ⎝ 𝘕

=𝖿

(

[1 − 𝜆𝘕 +1 ]

𝜆𝑛 𝒙 + 𝜆𝘕 +1 𝒙𝘕 +1 ∑ 1 − 𝜆𝘕 +1 𝑛 ) 𝑛=1 𝘕

≤ [1 − 𝜆𝘕 +1 ]𝖿

𝜆𝑛 𝒙𝑛 + 𝜆𝘕 +1 𝖿(𝒙𝘕 +1 ) ∑ ( 𝑛=1 1 − 𝜆𝘕 +1 )

by convexity hypothesis

𝘕

≤ [1 − 𝜆𝘕 +1 ]

𝜆𝑛 𝖿 (𝒙𝑛 ) + 𝜆𝘕 +1 𝖿(𝒙𝘕 +1 ) ∑ 1 − 𝜆𝘕 +1

by “true for 𝘕 ” hypothesis

𝑛=1

𝘕

=

∑

𝜆𝑛 𝖿(𝒙𝑛 ) + 𝜆𝘕 +1 𝖿(𝒙𝘕 +1 )

𝑛=1

𝘕 +1

=

∑

𝜆𝑛 𝖿 (𝒙𝑛 )

𝑛=1

4. Since the statement is true for 𝘕 = 1, 𝘕 = 2, and true for 𝘕 ⟹ true for 𝘕 + 1, then it is true for 𝘕 = 1, 2, 3, 4, …

✏ version 0.30


Daniel J. Greenhoe


page 269

APPENDIX A. INTERVALS AND CONVEXITY

A.4 Literature 📚 Literature survey: 1. Abstract convexity: 📘 Edelman and Jamison (1985) 📘 van de Vel (1993) 📘 Hörmander (1994) 2. Order convexity (lattice theory): 📘 Edelman (1986) 3. Metric convexity: 📘 Menger (1928) 📘 Blumenthal (1970) page 41 ⟨?⟩ 📘 Khamsi and Kirk (2001) pages 35–38 4. Strictly convex: 📘 Clarkson (1936)

📚


Daniel J. Greenhoe


version 0.30

page 270

version 0.30

A.4. LITERATURE


Daniel J. Greenhoe


APPENDIX B NORMED ALGEBRAS

B.1 Algebras All linear spaces are equipped with an operation by which vectors in the spaces can be added together. Linear spaces also have an operation that allows a scalar and a vector to be “multiplied” together. But linear spaces in general have no operation that allows two vectors to be multiplied together. A linear space together with such an operator is an algebra.1 There are many many possible algebras—many more than one can shake a stick at, as indicated by Michiel Hazewinkel in his book, Handbook of Algebras: “Algebra, as we know it today (2005), consists of many different ideas, concepts and results. A reasonable estimate of the number of these different items would be somewhere between 50,000 and 200,000. Many of these have been named and many more could (and perhaps should) have a “name” or other convenient designation.”2

Definition B.1. 3 Let 𝑨 be an algebra. d e An algebra 𝑨 is unital if ∃𝑢 ∈ 𝑨 f

such that

𝑢𝑥 = 𝑥𝑢 = 𝑥

∀𝑥 ∈ 𝑨

Definition B.2. 4 Let 𝑨 be an unital algebra (Definition B.1 page 271) with unit 𝑒. 𝜎(𝑥) ≜ {𝜆 ∈ ℂ|𝜆𝑒 − 𝑥 is not invertible} . d The spectrum of 𝑥 ∈ 𝑨 is e The resolvent of 𝑥 ∈ 𝑨 is 𝜌 (𝜆) ≜ (𝜆𝑒 − 𝑥)−1 f The spectral radius of 𝑥 ∈ 𝑨 is 𝑥𝑟(𝑥) ≜ sup {|𝜆||𝜆 ∈ 𝜎(𝑥)} .

1

📘 Fuchs (1995), page 2 📘 Hazewinkel (2000), page v 3 📘 Folland (1995) page 1 4 📘 Folland (1995) pages 3–4 2

∀𝜆∉𝜎(𝑥).

page 272

B.2

B.2. STAR-ALGEBRAS

Star-Algebras

Definition B.3. 5 Let 𝑨 be an algebra. The pair (𝑨, ∗) is a ∗-algebra if ∗ 1. (𝑥 + 𝑦) = 𝑥∗ + 𝑦∗ ∀𝑥,𝑦∈𝑨 (distributive) ∗ d 2. (𝛼𝑥) = 𝛼𝑥 ̄ ∗ ∀𝑥∈𝑨, 𝛼∈ℂ (conjugate linear) e ∗ ∗ ∗ 3. (𝑥𝑦) = 𝑦 𝑥 ∀𝑥,𝑦∈𝑨 (antiautomorphic) f ∗∗ 4. 𝑥 = 𝑥 ∀𝑥∈𝑨 (involutory) The operator ∗ is called an involution on the algebra 𝑨. Proposition B.1. 6 Let (𝑨, ∗) be an unital ∗-algebra. ∗ p 1. 𝑥 is invertible r 𝑥 is invertible ⟹ { 2. (𝑥∗ )−1 = (𝑥−1 )∗ p ✎PROOF:

∀𝑥∈𝑨

and and and

and

∀𝑥∈𝑨

Let 𝑒 be the unit element of (𝑨, ∗).

1. Proof that 𝑒∗ = 𝑒: 𝑥 𝑒∗ = (𝑥 𝑒∗ )∗∗

by involutory property of Definition B.3 page 272

∗ ∗∗ ∗

= (𝑥 𝑒 ) = (𝑥∗ 𝑒)∗

by antiautomorphic property of Definition B.3 page 272 by involutory property of Definition B.3 page 272

∗ ∗

= (𝑥 ) =𝑥 ∗

∗

by definition of 𝑒 by involutory property of Definition B.3 page 272 ∗∗

𝑒 𝑥 = (𝑒 𝑥) = (𝑒∗∗ 𝑥∗ )∗

by involutory property of Definition B.3 page 272 by antiautomorphic property of Definition B.3 page 272

∗ ∗

= (𝑒 𝑥 ) = (𝑥∗ )∗


=𝑥


by definition of 𝑒

∗

2. Proof that (𝑥∗ )−1 = (𝑥−1 ) : ∗

−1 ∗ −1 (𝑥 ) (𝑥 ) = [𝑥 (𝑥 )] = 𝑒∗

∗

by item (1) page 272

=𝑒 ∗

−1 ∗

−1

(𝑥 ) (𝑥 ) = [𝑥 = 𝑒∗

∗

𝑥]

=𝑒

by item (1) page 272

✏ Definition B.4. 7 Let (𝑨, ‖⋅‖) be a ∗-algebra (Definition B.3 page 272). An element 𝑥 ∈ 𝑨 is hermitian or self-adjoint if 𝑥∗ = 𝑥. d e An element 𝑥 ∈ 𝑨 is normal if 𝑥𝑥∗ = 𝑥∗ 𝑥. f An element 𝑥 ∈ 𝑨 is a projection if 𝑥𝑥 = 𝑥 (involutory) and 𝑥∗ = 𝑥 (hermitian). 5

📘 Rickart (1960) page 178, 📓 Gelfand and Naimark (1964), page 241 📘 Folland (1995) page 5 7 📘 Rickart (1960) page 178, 📓 Gelfand and Naimark (1964), page 242 6

version 0.30


Daniel J. Greenhoe


page 273

APPENDIX B. NORMED ALGEBRAS 8

Let (𝑨, ‖⋅‖) be a ∗-algebra (Definition B.3 page 272). 𝑥 + 𝑦 = (𝑥 + 𝑦)∗ (𝑥 + 𝑦 is self adjoint) ⎧ ⎪ ∗ ∗ ∗ 𝑥 = (𝑥 ) (𝑥∗ is self adjoint) t ⎪ ∗ ∗ h ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑥 = 𝑥 and 𝑦 = 𝑦 ⟹ ⎨ ⏟⏟⏟⏟⏟⏟⏟ 𝑥𝑦 = (𝑥𝑦)∗ ⟺ 𝑥𝑦 = 𝑦𝑥 ⏟⏟⏟ m ⎪ 𝑥 and 𝑦 are hermitian ⎪ commutative ⎩ (𝑥𝑦) is hermitian

Theorem B.1.

✎PROOF: (𝑥 + 𝑦)∗ = 𝑥∗ + 𝑦∗

by Definition B.3 page 272

=𝑥+𝑦

by left hypothesis

∗ ∗ (𝑥 ) = 𝑥


Proof that 𝑥𝑦 = (𝑥𝑦)∗ ⟹ 𝑥𝑦 = 𝑦𝑥 𝑥𝑦 = (𝑥𝑦)∗

by left hypothesis

∗ ∗

=𝑦 𝑥


= 𝑦𝑥

by left hypothesis

Proof that 𝑥𝑦 = (𝑥𝑦)∗ ⟸ 𝑥𝑦 = 𝑦𝑥 (𝑥𝑦)∗ = (𝑦𝑥)∗

by left hypothesis

∗ ∗

=𝑥 𝑦


= 𝑥𝑦

by left hypothesis

✏ 9

Let (𝑨, ‖⋅‖) be a ∗-algebra (Definition B.3 page 272). 1 ℜ𝑥 ≜ 𝑥 + 𝑥∗ ) d The real part of 𝑥 is defined as 2( e f The imaginary part of 𝑥 is defined as ℑ𝑥 ≜ 1 𝑥 − 𝑥∗ ) 2𝑖 ( Theorem B.2. 10 Let (𝑨, ∗) be a ∗-algebra (Definition B.3 page 272). t ℜ𝑥 = (ℜ𝑥)∗ ∀𝑥∈𝑨 (ℜ𝑥 is hermitian) h ∗ ∀𝑥∈𝑨 (ℑ𝑥 is hermitian) m ℑ𝑥 = (ℑ𝑥) Definition B.5 (Hermitian components).

✎PROOF: ∗ 1 (ℜ𝑥)∗ = ( (𝑥 + 𝑥∗ )) 2 1 ∗ = (𝑥 + 𝑥∗∗ ) 2 1 = (𝑥∗ + 𝑥) 2 = ℜ𝑥 ∗ 1 (ℑ𝑥)∗ = ( (𝑥 − 𝑥∗ )) 2𝑖 1 ∗ = (𝑥 − 𝑥∗∗ ) 2𝑖 1 = (𝑥∗ − 𝑥) 2𝑖 = ℑ𝑥

by Definition B.5 page 273 by Definition B.3 page 272 by Definition B.3 page 272 by Definition B.5 page 273 by Definition B.5 page 273 by Definition B.3 page 272 by Definition B.3 page 272 by Definition B.5 page 273

8

📘 Michel and Herget (1993), page 429 📘 Michel and Herget (1993) page 430, 📘 Rickart (1960) page 179, 📓 Gelfand and Naimark (1964), page 242 10 📘 Michel and Herget (1993) page 430, 📘 Halmos (1998) page 42 9


Daniel J. Greenhoe


version 0.30

page 274

B.3. NORMED ALGEBRAS

✏ Theorem B.3 (Hermitian representation). 11 Let (𝑨, ∗) be a ∗-algebra (Definition B.3 page 272). t h 𝑎 = 𝑥 + 𝑖𝑦 ⟺ 𝑥 = ℜ𝑎 and 𝑦 = ℑ𝑎 m ✎PROOF: Proof that 𝑎 = 𝑥 + 𝑖𝑦 ⟹ 𝑥 = ℜ𝑎

and

𝑦 = ℑ𝑎:

𝑎 = 𝑥 + 𝑖𝑦

by left hypothesis

∗

𝑎 = (𝑥 + 𝑖𝑦)

⟹

∗

= 𝑥 − 𝑖𝑦

∗

by definition of adjoint

∗


= 𝑥 − 𝑖𝑦 ⟹

by Theorem B.2 page 273

𝑥 = 𝑎 − 𝑖𝑦

by solving for 𝑥 in 𝑎 = 𝑥 + 𝑖𝑦 equation

∗

by solving for 𝑥 in 𝑎∗ = 𝑥 − 𝑖𝑦 equation

𝑥 = 𝑎 + 𝑖𝑦 ⟹

𝑥 + 𝑥 = 𝑎 + 𝑎∗

⟹

∗

⟹

by adding previous 2 equations

2𝑥 = 𝑎 + 𝑎 1 𝑥 = (𝑎 + 𝑎∗ ) 2 = ℜ𝑎

by solving for 𝑥 in previous equation

by definition of ℜ (Definition B.5 page 273)

𝑖𝑦 = 𝑎 − 𝑥

by solving for 𝑖𝑦 in 𝑎 = 𝑥 + 𝑖𝑦 equation

∗

𝑖𝑦 = −𝑎 + 𝑥 ⟹ ⟹

by solving for 𝑖𝑦 in 𝑎 = 𝑥 + 𝑖𝑦 equation

∗

𝑖𝑦 + 𝑖𝑦 = 𝑎 − 𝑎 1 𝑦 = (𝑎 − 𝑎 ∗ ) 2𝑖 = ℑ𝑎

Proof that 𝑎 = 𝑥 + 𝑖𝑦 ⟸ 𝑥 = ℜ𝑎

and

by adding previous 2 equations by solving for 𝑖𝑦 in previous equations by definition of ℑ (Definition B.5 page 273) 𝑦 = ℑ𝑎:

𝑥 + 𝑖𝑦 = ℜ𝑎 + 𝑖 ℑ𝑎 1 1 = (𝑎 + 𝑎∗ ) + 𝑖 (𝑎 − 𝑎∗ ) ⏟⏟⏟⏟⏟⏟⏟ 2 2𝑖 ⏟⏟⏟⏟⏟⏟⏟ ℜ𝑎

(Definition B.4 page 272)

by right hypothesis by definition of ℜ and ℑ (Definition B.5 page 273)

ℑ𝑎

:0 1 1 1 ∗ 1 ∗ = ( 𝑎 + 𝑎) + 𝑎 − 𝑎 ( 2 2 2 2 ) =𝑎

✏

B.3

Normed Algebras

Definition B.6. 12 Let 𝑨 be an algebra. d The pair (𝑨, ‖⋅‖) is a normed algebra if e ∀𝑥, 𝑦 ∈ 𝑨 (multiplicative condition) ‖𝑥𝑦‖ ≤ ‖𝑥‖ ‖𝑦‖ f A normed algebra (𝑨, ‖⋅‖) is a Banach algebra if (𝑨, ‖⋅‖) is also a Banach space. 11 12

📘 Michel and Herget (1993) page 430, 📘 Rickart (1960) page 179, 📓 Gelfand and Neumark (1943b), page 7 📘 Rickart (1960) page 2, 📘 Berberian (1961) page 103 ⟨Theorem IV.9.2⟩

version 0.30


Daniel J. Greenhoe


page 275

APPENDIX B. NORMED ALGEBRAS

Proposition B.2. p r (𝑨, ‖⋅‖) is a normed algebra p

multiplication is continuous in (𝑨, ‖⋅‖)

⟹

✎PROOF: 1. Define 𝖿(𝑥) ≜ 𝑧𝑥. That is, the function 𝖿 represents multiplication of 𝑥 times some arbitrary value 𝑧. 2. Let 𝛿 ≜ ‖𝑥 − 𝑦‖ and 𝜖 ≜ ‖𝖿(𝑥) − 𝖿 (𝑦)‖. 3. To prove that multiplication (𝖿) is continuous with respect to the metric generated by ‖⋅‖, we have to show that we can always make 𝜖 arbitrarily small for some 𝛿 > 0. 4. And here is the proof that multiplication is indeed continuous in (𝑨, ‖⋅‖): ‖𝖿 (𝑥) − 𝖿(𝑦)‖ = ‖𝑧𝑥 − 𝑧𝑦‖

by definition of 𝖿

= ‖𝑧(𝑥 − 𝑦)‖ ≤ ‖𝑧‖ ‖𝑥 − 𝑦‖


= ‖𝑧‖ 𝛿

by definition of 𝛿

≤𝜖

for some value of 𝛿 > 0

✏ Theorem B.4 (Gelfand-Mazur Theorem). t (𝑨, ‖⋅‖) is a Banach algebra h m every nonzero 𝑥 ∈ 𝑨 is invertible }

13

Let ℂ be the field of complex numbers. ⟹

𝑨≡ℂ

(𝑨 is isomorphic to ℂ)

B.4 C* Algebras Definition B.7. 14 The triple (𝑨, ‖⋅‖ , ∗) is a 𝐶 ∗ algebra if d 1. (𝑨, ‖⋅‖) is a Banach algebra e 2. (𝑨, ∗) is a ∗-algebra f ∗ 2 3. ‖𝑥 𝑥‖ = ‖𝑥‖ ∀𝑥 ∈ 𝑨. Theorem B.5. 15 Let 𝑨 be an algebra. t h (𝑨, ‖⋅‖ , ∗) is a 𝐶 ∗ algebra ⟹ m

and and

∗ ‖𝑥 ‖ = ‖𝑥‖

13

📘 Folland (1995) page 4, 📃 Mazur (1938) ⟨(statement)⟩, 📃 Gelfand (1941) ⟨(proof)⟩ 📘 Folland (1995) page 1, 📓 Gelfand and Naimark (1964), page 241, 📃 Gelfand and Neumark (1943a), 📓 Gelfand and Neumark (1943b) 15 📘 Folland (1995) page 1, 📓 Gelfand and Neumark (1943b), page 4, 📃 Gelfand and Neumark (1943a) 14


Daniel J. Greenhoe


version 0.30

page 276

B.4. C* ALGEBRAS

✎PROOF: 1 ‖𝑥‖2 ‖𝑥‖ 1 = 𝑥∗ 𝑥‖ ‖𝑥‖ ‖ 1 ≤ 𝑥∗ ‖𝑥‖ ‖𝑥‖ ‖ ‖ = ‖𝑥∗ ‖

‖𝑥‖ =

∗ ∗∗ ‖𝑥 ‖ ≤ ‖𝑥 ‖ = ‖𝑥‖

by definition of 𝐶 ∗ -algebra Definition B.7 page 275 by definition of normed algebra Definition B.6 page 274

by previous result by involution property Definition B.3 page 272

✏

version 0.30


Daniel J. Greenhoe


APPENDIX C TRANSLATION SPACES

C.1 Translation C.1.1 Definitions Definition C.1. Let 𝑋 be a set and 𝐈 be the identity operator on 𝑋. 𝐓𝑥 is a translation operator on 𝑋 if 1. ∃0 ∈ 𝑋 such that 𝐓0 = 𝐈 ∀𝐴∈𝟚𝑋 (identity) 2. 𝐓𝑥 𝐓𝑦 = 𝐓𝑦 𝐓𝑥 ∀𝑥,𝑦∈𝑋 (commutative) 𝑋 3. 𝐓𝑥 𝐴 = 𝐓𝐴 ∀𝐴,𝑌 ∈𝟚 , 𝑥∈𝑋 (distributive over ∪) ⋃ 𝑖 ⋃ 𝑥 𝑖 d 𝑖∈𝐼 𝑖∈𝐼 e 4. 𝐓𝐴= 𝐓𝐵 ∀𝐴,𝐵∈𝟚𝑋 f ⋃ 𝑏 ⋃ 𝑎 𝑏∈𝐵

and and and and

𝑎∈𝐴

𝐓𝑥 (𝐴 ∩ 𝐵) = (𝐓𝑥 𝐴) ∩ (𝐓𝑥 𝐵 ) ∀𝐴,𝐵∈𝟚𝑋 , 𝑥∈𝑋 𝖼 6. 𝐓𝑥 (𝐴 ) = 𝖼(𝐓𝑥 𝐴) ∀𝐴,𝐵∈𝟚𝑋 , 𝑥∈𝑋. The pair (𝑋, 𝐓) is a translation space on 𝑋. 5.

and

Definition C.2. 1 Let 𝑋 be a set on which is defined the translation opertor 𝐓𝑥 . Minkowski addition ⊕ and Minkowski subtraction ⊖ is defined as follows: 𝐴⊕𝐵 = 𝐓𝐴 ∀ 𝐴,𝐵∈𝟚𝑋 (Minkowski addition) ⋃ 𝑏 d 𝑏∈𝐵 e 𝐓𝐴 ∀ 𝐴,𝐵∈𝟚𝑋 (Minkowski subtraction) f 𝐴⊖𝐵 = ⋂ 𝑏 𝑏∈𝐵

Theorem C.1 (next) shows a relationship between Minkowski addition and Minkowski subtraction. Theorem C.1 (de Morgan relations). 2 Let (𝑋, +) be a group with Minokowski addition operator ⊕ ∶ 𝑋 2 → 𝑋 and Minokowski subtraction operator ⊖ ∶ 𝑋 2 → 𝑋. t 𝖼(𝐴 ⊕ 𝐵) = 𝐴𝖼 ⊖ 𝐵 ∀ 𝐴,𝐵∈𝟚𝑋 h ∀ 𝐴,𝐵∈𝟚𝑋 m 𝖼(𝐴 ⊖ 𝐵) = 𝐴𝖼 ⊕ 𝐵 1 2

📘 Matheron (1975) page 17 📘 Lay (1982) page 7 📘 Pitas and Venetsanopoulos (1991), page 159

page 278

C.1. TRANSLATION

✎PROOF: 𝖼(𝐴 ⊕ 𝐵) = 𝖼 =

⋃ (𝑏∈𝐵

𝐓𝑏 𝐴

)

𝖼 𝐓𝐴 ⋂ ( 𝑏 )

by Definition C.2 page 277 by Demorgan relation page 277

𝑏∈𝐵

=

⋂

𝐓𝑏 (𝐴𝖼 )

𝑏∈𝐵 𝖼

=𝐴 ⊖𝐵

𝖼(𝐴 ⊖ 𝐵) = 𝖼 =

⋂ (𝑏∈𝐵

𝐓𝑏 𝐴

by Definition C.1 page 277 by Theorem C.2 page 280

)

𝖼 𝐓𝐴 ⋃ ( 𝑏 )

by Definition C.2 page 277 by Demorgan relation page 277

𝑏∈𝐵

=

⋃

𝐓𝑏 (𝐴𝖼 )

𝑏∈𝐵 𝖼

=𝐴 ⊕𝐵


✏

C.1.2

Examples

Example C.1 (Translation on groups). 3 Let ⊕ be the Minkowski addition operator defined in terms of the translation operator 𝐓. Let (𝑋, +) be a group. 𝑋 { 𝐓𝑥 𝐴 ≜ {𝑎 + 𝑥|𝑎 ∈ 𝐴} ∀𝐴∈𝟚 } ⟹ e 𝐓𝑥 is a translation operator and x { 𝐴 ⊕ 𝐵 = {𝑎 + 𝑏|𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵 } ∀ 𝐴, 𝐵 ∈ 𝟚𝑋 } ✎PROOF: 1. Proof that ∃0 ∈ 𝑋

such that

𝐓0 = 𝐈:

𝐓0 𝐴 = {𝑎 + 0|𝑎 ∈ 𝐴} = {𝑎|𝑎 ∈ 𝐴}

by definition of 𝐓𝑥 by additive identity property of groups

=𝐴 2. Proof that 𝐓𝑥 𝐓𝑦 = 𝐓𝑦 𝐓𝑥 : 𝐓𝑥 𝐓𝑦 𝐴 = 𝐓𝑥 {𝑎 + 𝑦|𝑎 ∈ 𝐴}

3

by definition of 𝐓𝑦

= {𝑎 + 𝑦 + 𝑥|𝑎 ∈ 𝐴}


= {𝑎 + 𝑥 + 𝑦|𝑎 ∈ 𝐴}

by commutative property of groups

= 𝐓𝑦 {𝑎 + 𝑥|𝑎 ∈ 𝐴}


= 𝐓𝑦 𝐓𝑥 {𝑎|𝑎 ∈ 𝐴}

by definition of 𝐓𝑥

📘 Matheron (1975) pages 16–17 📘 Pitas and Venetsanopoulos (1991) page 159 📘 Lay (1982) page 7

version 0.30


Daniel J. Greenhoe


page 279

APPENDIX C. TRANSLATION SPACES

3. Proof that 𝐓𝑥 ⋃𝑖∈𝐼 𝐴𝑖 = ⋃𝑖∈𝐼 𝐓𝑥 𝐴𝑖 : 𝐓𝑥

⋃

𝐴𝑖 =

𝑖

= =

{ {

𝑦 + 𝑥|𝑦 ∈

⋃

𝐴𝑖

𝑖

𝑦 + 𝑥|

⋁

}

𝑦 ∈ 𝐴𝑖

𝑖


}

𝑦 + 𝑥|𝑦 ∈ 𝐴𝑖 } ⋃{ 𝑖

=

⋃

𝐓𝑥 {𝑦|𝑦 ∈ 𝐴𝑖 }

⋃

𝐓𝑥 𝐴

𝑖

=

𝑖

4. Proof that ⋃𝑏∈𝐵 𝐓𝑏 𝐴 = ⋃𝑎∈𝐴 𝐓𝑎 𝐵: ⋃

𝐓𝑏 𝐴 =

𝑏∈𝐵

⋃

{𝑎 + 𝑏|𝑎 ∈ 𝐴}

by definition of 𝐓𝑥

𝑏∈𝐵

= {𝑎 + 𝑏|𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵 } = {𝑏 + 𝑎|𝑏 ∈ 𝐵 and 𝑎 ∈ 𝐴} =

⋃

{𝑏 + 𝑎|𝑏 ∈ 𝐵}

𝑎∈𝐴

=

⋃

𝐓𝑎 𝐵

𝑎∈𝐴

5. Proof that 𝐓𝑥 ⋂𝑖∈𝐼 𝐴𝑖 = ⋂𝑖∈𝐼 𝐓𝑥 𝐴𝑖 : 𝐓𝑥

⋂

𝐴𝑖 =

𝑖

{

𝑦 + 𝑥|𝑦 ∈

⋂

𝐴𝑖

𝑖

}

=

𝑦 + 𝑥| 𝑦 ∈ 𝐴𝑖 ⋀ { } 𝑖

=

⋂{


𝑦 + 𝑥|𝑦 ∈ 𝐴𝑖 }

𝑖

=

⋂

𝐓𝑥 {𝑦|𝑦 ∈ 𝐴𝑖 }

⋂

𝐓𝑥 𝐴

𝑖

=

𝑖

6. Proof that 𝐓𝑥 (𝐴𝖼 ) = 𝖼(𝐓𝑥 𝐴): 𝐓𝑥 𝖼𝐴 = 𝐓𝑥 {𝑎|𝑎 ∈ 𝐴𝖼 } = {𝑎 + 𝑥|𝑎 ∈ 𝐴𝖼 } = {𝑎 + 𝑥|𝑎 ∉ 𝐴} = {𝑎 + 𝑥|¬(𝑎 ∈ 𝐴)} = 𝖼 {𝑎 + 𝑥|𝑎 ∈ 𝐴} = 𝖼𝐓𝑥 𝐴

𝐴⊕𝐵 =

⋃

𝐓𝑏 𝐴

by Definition C.2 page 277

𝑏∈𝐵

= {𝑎 + 𝑏|𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵 }



Daniel J. Greenhoe


version 0.30

page 280

C.1. TRANSLATION 2 1s s s −2 −1 s 1 2 −1

⊕

−2 2 1s s s −2 −1 s 1 2 −1 −2

c

−2 −1 −1 −2 s

⊖

2 1

2 1

−2 −1 −1 −2

s

1 2 𝛥

1s 2

=

=

2 1 c c 𝛥c −2 −1 c 1 2 −1 𝛥 𝛥 −2 𝛥 2 1 −2 −1 −1 −2

s 1 2

Figure C.1: Illustration for Example C.2 (page 280) Example C.2. Let 𝐴 ≜ {(0, 0), (0, 1), (0, −1), (1, 1)} 𝐵 ≜ {(0, 0), (−2, 1), (1, −1)} Then 𝐴 ⊕ 𝐵 = {(0, 0) , (0, 1) , (0, −1) , (1, 1) , (−2, 1) , (−2, 2) , (−2, 0) , (−1, 2) , (1, −1) , (1, −2) , (2, 0)} 𝐴 ⊖ 𝐵 = {(1, 0)}

These relationships are illustrated in Figure C.1 (page 280).

3 2 1

𝐴⊕𝐵 𝐴 𝐵 1 2 3 4 5 6 7

6 m m m m m m m m m m m m 5 𝐴⊕𝐵 m m m m m m m m m m m m m m 4 } m m m m m m m m m m m m m 3

2 𝐴 1 𝐵 1 2 3 4 5 6 7

Figure C.2: Illustration for Example C.3 page 280 Example C.3.

C.1.3

4

Two more examples are illustrated in Figure C.2 (page 280).

Additive properties

Theorem C.2. 5 Let (𝑋, +) be a group with with Minokowski addition operator ⊕ ∶ 𝑋 2 → 𝑋. 𝐴 ⊕ {0} = 𝐴 ∀𝐴⊆𝑋 t 𝐴⊕𝐵 = 𝐵⊕𝐴 ∀𝐴,𝐵⊆𝑋 (commutative) h ∀𝐴,𝐵,𝐶⊆𝑋 (associative) m 𝐴 ⊕ (𝐵 ⊕ 𝐶) = (𝐴 ⊕ 𝐵) ⊕ 𝐶 𝐓𝑥 (𝐴 ⊕ 𝐵) = (𝐓𝑥 𝐴) ⊕ 𝐵 ∀𝐴,𝐵⊆𝑋, 𝑥∈𝑋 (translation invariant) 4 5

📘 Lay (1982) page 7 📘 Pitas and Venetsanopoulos (1991), pages 163–164

version 0.30


Daniel J. Greenhoe


page 281


✎PROOF: 𝐴 ⊕ {0} = 𝐴 ⊕ 𝐵|𝐵={0} =

⋃

𝐓𝑏 𝐴

𝑏∈𝐵

|

by Definition C.2 page 277 𝐵={0}

= 𝐓0 𝐴 =𝐴 𝐴⊕𝐵 =


⋃

𝐓𝑏 𝐴


⋃

𝐓𝑎 𝐵


𝑏∈𝐵

=

𝑎∈𝐴

=𝐵⊕𝐴 𝐴 ⊕ (𝐵 ⊕ 𝐶) =


𝐓𝑦 𝐴


⋃

𝐓𝑎 (𝐵 ⊕ 𝐶)


⋃

𝐓𝑎

⋃

𝑦∈𝐵⊕𝐶

=

𝑎∈𝐴

=

⋃ (𝑐∈𝐶

𝑎∈𝐴

= = =

⋃ (⋃

𝑎∈𝐴

𝑐∈𝐶

⋃

⋃

𝑎∈𝐴 (𝑐∈𝐶

⋃

𝐓𝑐

𝑐∈𝐶

= =

𝐓𝑐 𝐵

𝐓𝑎 𝐓𝑐 𝐵 𝐓𝑐 𝐓𝑎 𝐵

⋃ (𝑎∈𝐴

𝐓𝑎 𝐵

⋃

𝐓𝑐

𝑐∈𝐶

⋃ (𝑏∈𝐵

𝐓𝑏 𝐴

⋃

𝐓𝑐 (𝐴 ⊕ 𝐵)

) ) ) )

)

by Definition C.2 page 277 by Definition C.1 page 277 by Definition C.1 page 277 by Definition C.1 page 277 by Definition C.1 page 277 by Definition C.2 page 277

𝑐∈𝐶

= (𝐴 ⊕ 𝐵) ⊕ 𝐶 𝐓𝑥 (𝐴 ⊕ 𝐵) = 𝐓𝑥


𝐓𝑏 𝐴


⋃

𝐓𝑥 𝐓𝑏 𝐴


⋃

𝐓𝑏 𝐓𝑥 𝐴


⋃

𝑏∈𝐵

=

𝑏∈𝐵

=

𝑏∈𝐵

= (𝐓𝑥 𝐴) ⊕ 𝐵


✏ Theorem C.3. 6

6

Let (𝑋, +) be a group with with Minokowski addition operator ⊕ ∶ 𝑋 2 → 𝑋.

📘 Pitas and Venetsanopoulos (1991), page 163


Daniel J. Greenhoe


version 0.30

page 282

C.1. TRANSLATION

𝐴 ⊕ (𝐵 ∪ 𝐶) (𝐴 ∪ 𝐵) ⊕ 𝐶 𝐴 ⊕ (𝐵 ∩ 𝐶) (𝐴 ∩ 𝐵) ⊕ 𝐶

t h m

= = ⊆ ⊆

(𝐴 ⊕ 𝐵) ∪ (𝐴 ⊕ 𝐶) (𝐴 ⊕ 𝐶) ∪ (𝐵 ⊕ 𝐶) (𝐴 ⊕ 𝐵) ∩ (𝐴 ⊕ 𝐶) (𝐴 ⊕ 𝐶) ∩ (𝐵 ⊕ 𝐶)

∀𝐴,𝐵,𝐶⊆𝑋

(⊕ is left distributive over ∪)

∀𝐴,𝐵,𝐶⊆𝑋

(⊕ right distributive over ∪)

∀𝐴,𝐵,𝐶⊆𝑋 ∀𝐴,𝐵,𝐶⊆𝑋

✎PROOF: (𝐴 ∪ 𝐵) ⊕ 𝐶 =

⋃

𝐓𝑐 (𝐴 ∪ 𝐵)


𝑐∈𝐶

=

⋃ [(

𝐓𝑐 𝐴) ∪ (𝐓𝑐 𝐵 )]


𝑐∈𝐶

=

⋃ (𝑐∈𝐶

𝐓𝑐 𝐴

)

∪

⋃ (𝑐∈𝐶

𝐓𝑐 𝐵

)

= (𝐴 ⊕ 𝐶) ∪ (𝐵 ⊕ 𝐶)


𝐴 ⊕ (𝐵 ∪ 𝐶) = (𝐵 ∪ 𝐶) ⊕ 𝐴

by Theorem C.2 page 280

= (𝐵 ⊕ 𝐴) ∪ (𝐶 ⊕ 𝐴)

by previous result

= (𝐴 ⊕ 𝐵) ∪ (𝐴 ⊕ 𝐶)


𝐓𝑐 (𝐴 ∩ 𝐵)


(𝐴 ∩ 𝐵) ⊕ 𝐶 =

⋃

𝑐∈𝐶

=

⋃ [(

𝐓𝑐 𝐴) ∩ (𝐓𝑐 𝐵 )]


𝑐∈𝐶

⊆

⋃ (𝑐∈𝐶

𝐓𝑐 𝐴

)

∩

⋃ (𝑐∈𝐶

𝐓𝑐 𝐵

)

= (𝐴 ⊕ 𝐶) ∩ (𝐵 ⊕ 𝐶) 𝐴 ⊕ (𝐵 ∩ 𝐶) = (𝐵 ∩ 𝐶) ⊕ 𝐴

by minimax inequality page 64 by Theorem C.2 page 280 by Theorem C.2 page 280

⊆ (𝐵 ⊕ 𝐴) ∩ (𝐶 ⊕ 𝐴)

by previous result

= (𝐴 ⊕ 𝐵) ∩ (𝐴 ⊕ 𝐶)


✏

C.1.4

Subtractive properties

Let (𝑋, +) be a group with with Minokowski subtraction operator ⊖ ∶ 𝑋 2 → 𝑋. 𝐴 ⊖ {0} = 𝐴 ∀𝐴⊆𝑋 𝖼 𝖼 𝐴⊖𝐵 = 𝐵 ⊖𝐴 ∀𝐴,𝐵⊆𝑋 𝐓𝑥 (𝐴 ⊖ 𝐵) = (𝐓𝑥 𝐴) ⊖ 𝐵 ∀𝐴,𝐵⊆𝑋, 𝑥∈𝑋 (translation invariant) 𝐴⊆𝐵 ⟹ 𝐴⊖𝐶 ⊆ 𝐵⊖𝐶 ∀𝐴,𝐵,𝐶⊆𝑋 (increasing)

Theorem C.4. t h m

7

7

📘 Pitas and Venetsanopoulos (1991), pages 164–165

version 0.30


Daniel J. Greenhoe


page 283


✎PROOF: 𝐴 ⊖ {0} = 𝖼(𝐴𝖼 ⊕ {0}) 𝖼

= 𝖼(𝐴 )

by Theorem C.1 page 277 by Theorem C.2 page 280

=𝐴 𝐴 ⊖ 𝐵 = 𝖼𝖼(𝐴 ⊖ 𝐵) = 𝖼(𝐴𝖼 ⊕ 𝐵)


𝖼

= 𝖼(𝐵 ⊕ 𝐴 ) 𝖼

=𝐵 ⊖𝐴


𝖼


𝐓𝑥 (𝐴 ⊖ 𝐵) = 𝐓𝑥 𝖼(𝐴𝖼 ⊕ 𝐵 ) 𝖼


= 𝖼𝐓𝑥 (𝐴 ⊕ 𝐵 ) = 𝖼(𝐓𝑥 𝐴𝖼 ⊕ 𝐵 )


= 𝖼(𝖼𝐓𝑥 𝐴 ⊕ 𝐵 )


= 𝐓𝑥 𝐴 ⊖ 𝐵


𝐴⊖𝐶 =


⋂

𝐴𝑐


⋂

𝐵𝑐

by 𝐴 ⊆ 𝐵 hypothesis

𝑐∈𝐶

⊆

𝑐∈𝐶


=𝐵⊖𝐶

✏ Theorem C.5. 8 Let (𝑋, +) be a group with with Minokowski subtraction operator ⊖ ∶ 𝑋 2 → 𝑋. 𝐴 ⊖ (𝐵 ∪ 𝐶) = (𝐴 ⊖ 𝐵) ∩ (𝐴 ⊖ 𝐶) ∀𝐴,𝐵,𝐶⊆𝑋 (⊖ left distributive over ∪) t (𝐴 ∩ 𝐵) ⊖ 𝐶 = (𝐴 ⊖ 𝐶) ∩ (𝐵 ⊖ 𝐶) ∀𝐴,𝐵,𝐶⊆𝑋 (⊖ right distributive over ∩) h ∀𝐴,𝐵,𝐶⊆𝑋 m (𝐴 ∪ 𝐵) ⊖ 𝐶 ⊇ (𝐴 ⊖ 𝐶) ∪ (𝐵 ⊖ 𝐶) 𝐴 ⊖ (𝐵 ∩ 𝐶) ⊇ (𝐴 ⊖ 𝐵) ∪ (𝐴 ⊖ 𝐶) ∀𝐴,𝐵,𝐶⊆𝑋 ✎PROOF: 𝐴 ⊖ (𝐵 ∪ 𝐶) = 𝖼𝖼[ 𝐴 ⊖ (𝐵 ∪ 𝐶) ] = 𝖼[ 𝐴𝖼 ⊕ (𝐵 ∪ 𝐶) ]


= 𝖼[ (𝐴𝖼 ⊕ 𝐵) ∪ (𝐴𝖼 ⊕ 𝐶) ]


= [𝖼(𝐴𝖼 ⊕ 𝐵 )] ∩ [𝖼(𝐴𝖼 ⊕ 𝐶 )]

by Demorgan relation page 277

= (𝐴 ⊖ 𝐵) ∩ (𝐴 ⊖ 𝐶)


(𝐴 ∩ 𝐵) ⊖ 𝐶 = 𝖼[(𝐴 ∩ 𝐵) ⊖ 𝐶] = 𝖼[𝖼(𝐴 ∩ 𝐵) ⊕ 𝐶] 𝖼


𝖼

= 𝖼[(𝐴 ∪ 𝐵 ) ⊕ 𝐶 ] = 𝖼[(𝐴𝖼 ⊕ 𝐶) ∪ (𝐵 𝖼 ⊕ 𝐶)] 𝖼


𝖼

= 𝖼(𝐴 ⊕ 𝐶) ∩ 𝖼(𝐵 ⊕ 𝐶) = (𝐴 ⊖ 𝐶) ∩ (𝐵 ⊖ 𝐶) 8




Daniel J. Greenhoe


version 0.30

page 284

C.1. TRANSLATION

𝐴 ⊖ (𝐵 ∩ 𝐶) = 𝖼𝖼[ 𝐴 ⊖ (𝐵 ∩ 𝐶) ] = 𝖼[ 𝐴𝖼 ⊕ (𝐵 ∩ 𝐶) ]


⊇ 𝖼[(𝐴𝖼 ⊕ 𝐵) ∩ (𝐴𝖼 ⊕ 𝐶)] 𝖼

𝖼


= [𝖼(𝐴 ⊕ 𝐵)] ∪ [𝖼(𝐴 ⊕ 𝐶)]


= (𝐴 ⊖ 𝐵) ∪ (𝐴 ⊖ 𝐶)


(𝐴 ∪ 𝐵) ⊖ 𝐶 = 𝖼𝖼[(𝐴 ∪ 𝐵) ⊖ 𝐶] = 𝖼[𝖼(𝐴 ∪ 𝐵) ⊕ 𝐶] 𝖼


𝖼

= 𝖼[(𝐴 ∩ 𝐵 ) ⊕ 𝐶 ] ⊇ 𝖼[(𝐴𝖼 ⊕ 𝐶 ) ∩ (𝐵 𝖼 ⊕ 𝐶 )]


= 𝖼(𝐴𝖼 ⊕ 𝐶 ) ∪ 𝖼(𝐵 𝖼 ⊕ 𝐶 ) = (𝐴 ⊖ 𝐶) ∪ (𝐵 ⊖ 𝐶)


✏ Theorem C.6. 9 Let (𝑋, +) be a group with with Minokowski addition operator ⊕ ∶ 𝑋 2 → 𝑋 and Minokowski subtraction operator ⊖ ∶ 𝑋 2 → 𝑋. t 𝐴 ⊖ (𝐵 ⊕ 𝐶) = (𝐴 ⊖ 𝐵) ⊖ 𝐶 ∀𝐴,𝐵,𝐶⊆𝑋 h ∀𝐴,𝐵,𝐶⊆𝑋 m 𝐴 ⊕ (𝐵 ⊖ 𝐶) ⊆ (𝐴 ⊕ 𝐵) ⊖ 𝐶 ✎PROOF: 𝐴 ⊖ (𝐵 ⊕ 𝐶) = 𝖼𝖼[ 𝐴 ⊖ (𝐵 ⊕ 𝐶) ] = 𝖼[ 𝐴𝖼 ⊕ (𝐵 ⊕ 𝐶) ]


= 𝖼[ (𝐴𝖼 ⊕ 𝐵) ⊕ 𝐶 ]


= 𝖼(𝐴𝖼 ⊕ 𝐵) ⊖ 𝐶


= (𝐴 ⊖ 𝐵) ⊖ 𝐶


𝐴 ⊕ (𝐵 ⊖ 𝐶) = 𝐴 ⊕ = =

⋂ (𝑐∈𝐶 ⋃

𝑎∈𝐴

=

⋂ (𝑐∈𝐶 𝐓𝑐 𝐵

𝐓𝑎

𝐓𝑐 𝐵

)

⋂ (𝑐∈𝐶

)


⊕𝐴


𝐓𝑐 𝐵


)

⋃⋂

𝐓𝑎 𝐓𝑐 𝐵


⋂⋃

𝐓𝑎 𝐓𝑐 𝐵

by minimax inequality page 64

⋂⋃

𝐓𝑐 𝐓𝑎 𝐵


𝑎∈𝐴 𝑐∈𝐶

⊆

𝑐∈𝐶 𝑎∈𝐴

=

𝑐∈𝐶 𝑎∈𝐴 9


version 0.30


Daniel J. Greenhoe


page 285


= =

⋂

𝐓𝑐

𝑐∈𝐶

⋃ (𝑎∈𝐴

𝐓𝑎 𝐵

⋂

𝐓𝑐 (𝐵 ⊕ 𝐴)


)


𝑐∈𝐶

= (𝐵 ⊕ 𝐴) ⊖ 𝐶


= (𝐴 ⊕ 𝐵) ⊖ 𝐶


✏

C.2 Operations Definition C.3. 10 Let (𝑋, +) be a group. d e The symmetric set of 𝐴 is the set 𝐴̌ ≜ −𝐴 f

∀𝐴⊆𝑋

Definition C.4. 11 Let (𝑋, +) be a group with Minokowski addition operator ⊕ ∶ 𝑋 2 → 𝑋, Minokowski subtraction operator ⊖ ∶ 𝑋 2 → 𝑋, and 𝐷𝑠 be the symmetric set of set 𝐷. d The dilation of 𝐴 by 𝐷 is the operation 𝐴 ⊕ 𝐷̌ ∀𝐴,𝐷⊆𝑋. e ̌ ∀𝐴,𝐸⊆𝑋. f The erosion of 𝐴 by 𝐸 is the operation 𝐴 ⊖ 𝐸 Definition C.5. 12 Let (𝑋, +) be a group with Minokowski addition operator ⊕ ∶ 𝑋 2 → 𝑋, Minokowski subtraction operator ⊖ ∶ 𝑋 2 → 𝑋, and 𝐵 𝑠 be the symmetric set of a set 𝐵. The opening of 𝐴 with respect to 𝐵 is the set 𝐴𝐵 ≜ (𝐴 ⊖ 𝐵)̌ ⊕ 𝐵 ∀𝐴,𝐵⊆𝑋. ⏟⏟⏟ erosion ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

d e f

The closing of 𝐴 with respect to 𝐵 is the set

𝐴

𝐵

dilation

≜ (𝐴 ⊕ 𝐵)̌ ⊖ 𝐵 ⏟⏟⏟

∀𝐴,𝐵⊆𝑋.

dilation ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ erosion

Theorem C.7. 13 Let (𝑋, +) be a group with 𝐴𝐵 representing the opening of a set 𝐴 with respect to a set 𝐵 and 𝐴𝐵 representing the closing of a set 𝐴 with respect to a set 𝐵. t (complement of the opening)→ 𝖼(𝐴𝐵 ) = (𝐴𝖼 )𝐵 ←(closing of the complement) ∀𝐴,𝐵⊆𝑋 h 𝐵 𝖼 (complement of the closing)→ 𝖼(𝐴 ) = (𝐴 )𝐵 ←(opening of the complement) ∀𝐴,𝐵⊆𝑋 m ✎PROOF: 𝖼(𝐴𝐵 ) = 𝖼[ (𝐴 ⊖ 𝐵)̌ ⊕ 𝐵 ] = 𝖼(𝐴 ⊖ 𝐵)̌ ⊖ 𝐵

10 11 12 13


= 𝖼(𝐴 ⊖ 𝐵)̌ ⊖ 𝐵 = (𝐴𝖼 ⊕ 𝐵)̌ ⊖ 𝐵


= (𝐴𝖼 )𝐵



📘 Matheron (1975), page 17 📘 Pitas and Venetsanopoulos (1991), page 161 📘 Serra (1982), page 50 📘 Serra (1982), page 51


Daniel J. Greenhoe


version 0.30

page 286

C.2. OPERATIONS

𝖼(𝐴𝐵 ) = 𝖼[ (𝐴 ⊕ 𝐵)̌ ⊖ 𝐵 ] = 𝖼(𝐴 ⊕ 𝐵)̌ ⊕ 𝐵


= 𝖼(𝐴 ⊕ 𝐵)̌ ⊕ 𝐵 = (𝐴𝖼 ⊖ 𝐵)̌ ⊕ 𝐵

by Theorem C.1 page 277 by Theorem C.1 page 277

𝖼

= (𝐴 )𝐵


✏ Example C.4. 𝐴 ≜ {(0, 0) , (1, 0) , (2, 0) , (0, 1)} ⟹ 𝐴(−2,1) = {(−2, 1) , (−1, 1) , (0, 1) , (−2, 2)} { 𝐴̌ = {(0, 0) , (−1, 0) , (−2, 0) , (0, −1)} e x

𝐴 2 1c s 𝛥 −2 −1 1 2 −1 −2

𝐓 𝐴 c (−2,1) 2 s 1𝛥 −2 −1 −1 −2

1 2

and

}

−𝐴 2 1 𝛥 s −2 −1 c 1 2 −1 −2

Example C.5. An example similar to Example C.4 (page 286) but using solid shapes is illustrated next: e x

𝐴 2 1 −3 −2 −1 −1 −2

version 0.30

𝐴(−3,−2) 2 1 1 2 3

−3 −2 −1 −1 −2

1 2 3


2 1 −3 −2 −1 −1 −2

𝐴̌

1 2 3

Daniel J. Greenhoe


page 287


Back Matter ❝It

appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils.❞ Niels Henrik Abel (1802–1829), Norwegian mathematician

14

❝When

evening comes, I return home and go to my study. On the threshold I strip naked, taking off my muddy, sweaty workaday clothes, and put on the robes of court and palace, and in this graver dress I enter the courts of the ancients and am welcomed by them, and there I taste the food that alone is mine, and for which I was born. And there I make bold to speak to them and ask the motives of their actions, and they, in their humanity reply to me. And for the space of four hours I forget the world, remember no vexation, fear poverty no more, tremble no more at death; I pass indeed into their world.❞ 15

Niccolò Machiavelli (1469–1527), Italian political philosopher, in a 1513 letter to friend Francesco Vettori.

16

ancient library of Alexandria 14

The Book Worm by Carl Spitzweg, circa 1850

15

quote: 📘 Simmons (2007), page 187. image: http://en.wikipedia.org/wiki/Image:Niels_Henrik_Abel.jpg, public domain quote: 📘 Machiavelli (1961), page 139?. image: http://commons.wikimedia.org/wiki/File:Santi_di_Tito_-_Niccolo_Machiavelli%27s_portrait_headcrop.jpg, public

16

domain http://en.wikipedia.org/wiki/File:Ancientlibraryalex.jpg, public domain http://en.wikipedia.org/wiki/File:Carl_Spitzweg_021.jpg,


Daniel J. Greenhoe


version 0.30

page 288

C.2. OPERATIONS

❝To sit alone in the lamplight with a book spread out before you, and hold intimate converse with men of unseen generations— such is a pleasure beyond compare.❞ Yoshida Kenko (Urabe Kaneyoshi) (1283? – 1350?), Japanese author and Buddhist monk

17

quote: image:

version 0.30

17

📘 Kenko (circa 1330) http://en.wikipedia.org/wiki/Yoshida_Kenko Sets, Relations, and Order Structures

Daniel J. Greenhoe


BIBLIOGRAPHY

Jan Łukasiewicz. On three-valued logic. In Storrs McCall, editor, Polish Logic, 1920–1939, pages 15–18. Oxford University Press, 1920. ISBN 9780198243045. URL http://books.google.com/ books?vid=ISBN0198243049&pg=PA15. collection published in 1967. M. E. Adams. Uniquely complemented lattices. In Kenneth P. Bogart, Ralph S. Freese, and Joseph P.S. Kung, editors, The Dilworth theorems: selected papers of Robert P. Dilworth, pages 79– 84. Birkhäuser, Boston, 1990. ISBN 0817634347. URL http://books.google.com/books?vid= ISBN0817634347. Donald J. Albers and Gerald L. Alexanderson. Mathematical People: Profiles and Interviews. Birkhäuser, Boston, 1985. ISBN 0817631917. URL http://books.google.com/books?vid= ISBN0817631917. Charalambos D. Aliprantis and Owen Burkinshaw. Principles of Real Analysis. Acedemic Press, London, 3 edition, 1998. ISBN 9780120502578. URL http://www.amazon.com/dp/0120502577. Charalambos D. Aliprantis and Owen Burkinshaw. Positive Operators. Springer, Dordrecht, 2006. ISBN 9781402050077. URL http://books.google.com/books?vid=ISBN1402050070. reprint of Academic Press 1985 edition. Herbert Amann and Joachim Escher. Analysis I. Birkhäuser, Basel, 2005. ISBN 3764371536. URL http://books.google.com/books?vid=ISBN3764371536. translated from German. Tom M. Apostol. Mathematical Analysis. Addison-Wesley series in mathematics. Addison-Wesley, Reading, 2 edition, 1975. ISBN 986-154-103-9. URL http://books.google.com/books?vid= ISBN0201002884. Aristotle. Metaphysics book iv. In Aristotle: Metaphysics, Books I–IX, number 271 in Loeb Classical Library, pages 146–207. Harvard University Press (1933), Cambridge MA. ISBN 0674992997. URL http://www.perseus.tufts.edu/cgi-bin/ptext?lookup=Aristot.+Met.+4.1003a. V.A. Artamonov. Varieties of algebras. In Michiel Hazewinkel, editor, Handbook of Algebras, volume 2, pages 545–576. North-Holland, Amsterdam, 1 edition, 2000. ISBN 044450396X. URL http: //books.google.com/books?vid=ISBN044450396X&pg=PA545. Jean-Pierre Aubin. Applied Functional Analysis, volume 47 of Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts. John Wiley & Sons, 2 edition, September 30 2011. ISBN 9781118030974. URL http://books.google.com/books?vid=ISBN1118030974.

page 290

BIBLIOGRAPHY

Jean-Pierre Aubin and Hélène Frankowska. Set-Valued Analysis. Modern Birkhäuser Classics. Springer, March 2 2009. ISBN 9780817648480. URL http://books.google.com/books?vid= ISBN0817648488. Arnon Avron. Natural 3-valued logics–characterization and proof theory. The Journal of Symbolic Logic, 56(1):276–294, March 1991. URL http://www.jstor.org/stable/2274919. R. W. Bagley. On the characterization of the lattice of topologies. Journal of the London Math Society, 30:247–249, 1955. URL http://jlms.oxfordjournals.org/cgi/reprint/s1-30/2/247. MR 16,788. Kirby A. Baker. Equational classes of modular lattices. Pacific Journal of Mathematics, 28(1):9–15, 1969. URL http://projecteuclid.org/euclid.pjm/1102983605. Raymond Balbes. Projective and injective distributive lattices. Pacific Journal of Mathematics, 21 (3):405–420, 1967. URL http://projecteuclid.org/euclid.pjm/1102992388. Raymond Balbes and Philip Dwinger. Distributive Lattices. University of Missouri Press, Columbia, February 1975. ISBN 0826201636. URL http://books.google.com/books?vid=ISBN098380110X. 2011 reprint edition available (ISBN 9780983801108). Raymond Balbes and Alfred Horn. Projective distributive lattices. Pacific Journal of Mathematics, 33(2):273–279, 1970. URL http://projecteuclid.org/euclid.pjm/1102976963. Stefan Banach. Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales (on abstract operations and their applications to the integral equations). Fundamenta Mathematicae, 3:133–181, 1922. URL http://matwbn.icm.edu.pl/ksiazki/fm/fm3/ fm3120.pdf. Hans-J. Bandelt and Jarmila Hedlíková. Median algebras. Discrete Mathematics, 45(1):1–30, 1983. URL http://www.sciencedirect.com/science/journal/0012365X. Robert G. Bartle. A Modern Theory of Integration, volume 32 of Graduate studies in mathematics. American Mathematical Society, Providence, R.I., 2001. ISBN 0821808451. URL http://books. google.com/books?vid=ISBN0821808451. Alexander Barvinok. A Course in Convexity, volume 54 of Graduate studies in mathematics. American Mathematical Society, 2002. ISBN 9780821872314. URL http://books.google.com/books? vid=ISBN0821872311. Gerald Beer. Topologies on Closed and Closed Convex Sets, volume 268 of Mathematics and Its Applications. October 31 1993. ISBN 9780792325314. URL http://books.google.com/books?vid= ISBN0792325311. Eric Temple Bell. Exponential numbers. The American Mathematical Monthly, 41(7):411–419, August–September 1934. URL http://www.jstor.org/stable/2300300. Rirchard Bellman and Magnus Giertz. On the analytic formalism of the theory of fuzzy sets. Information Sciences, 5:149–156, 1973. doi: 10.1016/0020-0255(73)90009-1. URL http://www. sciencedirect.com/science/article/pii/0020025573900091. Nuel D. Belnap, Jr. A useful four-valued logic. In John Michael Dunn and George Epstein, editors, Modern Uses of Multiple-valued Logic: Invited Papers from the 5. International Symposium on Multiple-Valued Logic, Held at Indiana University, Bloomington, Indiana, May 13 - 16, 1975 ; with a Bibliography of Many-valued Logic by Robert G. Wolf, volume 2 of Episteme, pages 8–37. D. Reidel, 1977. ISBN 9789401011617. URL http://www.amazon.com/dp/9401011613. version 0.30


Daniel J. Greenhoe


page 291

BIBLIOGRAPHY

Ladislav Beran. Three identities for ortholattices. Notre Dame Journal of Formal Logic, 17(2):251– 252, 1976. doi: 10.1305/ndjfl/1093887530. URL http://projecteuclid.org/euclid.ndjfl/ 1093887530. Ladislav Beran. Boolean and orthomodular lattices — a short characterization via commutativity, volume 23. Czech Republic, 1982. URL http://journalseek.net/cgi-bin/journalseek/ journalsearch.cgi?field=issn&query=0001-7140. Ladislav Beran. Orthomodular Lattices: Algebraic Approach. Mathematics and Its Applications (East European Series). D. Reidel Publishing Company, Dordrecht, 1985. ISBN 90-277-1715-X. URL http://books.google.com/books?vid=ISBN902771715X. Sterling Khazag Berberian. Introduction to Hilbert Space. Oxford University Press, New York, 1961. URL http://books.google.com/books?vid=ISBN0821819127. Yurij M. Berezansky, Zinovij G. Sheftel, and Georgij F. Us. Functional Analysis: Volume I (Operator Theory, Advances and Applications, Volume 85), volume 85 of Operator Theory Advances and Applications. Birkhäuser, Basel, 1996. ISBN 3764353449. URL http://books.google.com/books? vid=ISBN3764353449. translated into English from Russian. Gustav Bergman. Zur axiomatik der elementargeometrie. Monatshefte für Mathematik, 36(1): 269–284, December 1929. ISSN 0026-9255. URL http://www.springerlink.com/content/ n30211355u2k/. Benjamin Abram Bernstein. A complete set of postulates for the logic of classes expressed in terms of the operation “exception,” and a proof of the independence of a set of postulates due to del ré. University of California Publications on Mathematics, 1(4):87–96, May 15 1914. URL http: //www.archive.org/details/113597_001_004. Benjamin Abram Bernstein. A simplification of the whitehead-huntington set of postulates for boolean algebras. Bulletin of the American Mathematical Society, 22:458–459, 1916. ISSN 00029904. doi: 10.1090/S0002-9904-1916-02831-X. URL http://www.ams.org/bull/1916-22-09/ S0002-9904-1916-02831-X/. Benjamin Abram Bernstein. Simplification of the set of four postulates for boolean algebras in terms of rejection. Bulletin of the American Mathematical Society, 39:783–787, October 1933. ISSN 0002-9904. doi: 10.1090/S0002-9904-1933-05738-5. URL http://www.ams.org/bull/ 1933-39-10/S0002-9904-1933-05738-5/. Benjamin Abram Bernstein. A set of four postulates for boolean algebra in terms of the “implicative” operation. Transactions of the American Mathematical Society, 36(4):876–884, October 1934. URL http://www.jstor.org/stable/1989830. Benjamin Abram Bernstein. Postulates for boolean algebra involving the operation of complete disjunction. The Annals of Mathematics, 37(2):317–325, April 1936. URL http://www.jstor. org/stable/1968444. Garrett Birkhoff. On the combination of subalgebras. Mathematical Proceedings of the Cambridge Philosophical Society, 29:441–464, October 1933a. doi: 10.1017/S0305004100011464. URL http: //adsabs.harvard.edu/abs/1933MPCPS..29..441B. Garrett Birkhoff. On the combination of subalgebras by garrett birkhoff. In Garrett Birkhoff, GianCarlo Rota, and Joseph S. Oliveira, editors, Selected Papers on Algebra and Topology, Contemporary mathematicians, pages 9–32. Birkhäuser, Boston, 1933b. ISBN 0817631143. URL http: //books.google.com/books?vid=ISBN0817631143. This book published in 1987 by Birkhäuser. 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 292

BIBLIOGRAPHY

Garrett Birkhoff. On the combination of topologies. Fundamenta Mathematicae, 26:156–166, 1936a. ISSN 0016-2736. URL http://matwbn.icm.edu.pl/ksiazki/fm/fm26/fm26116.pdf. Garrett Birkhoff. The logic of quantum mechanics. Annals of Mathematics, 37(4):823–843, October 1936b. URL http://www.jstor.org/stable/1968621. Garrett Birkhoff. Rings of sets. Duke Math. J., 3(3):443–454, 1937. doi: 10.1215/ S0012-7094-37-00334-X. URL http://projecteuclid.org/euclid.dmj/1077490201. Garrett Birkhoff. Lattices and their applications. Bulletin of the American Mathematical Society, 44:1:793–800, 1938. doi: 10.1090/S0002-9904-1938-06866-8. URL http://www.ams.org/bull/ 1938-44-12/S0002-9904-1938-06866-8/. Garrett Birkhoff. Lattice Theory. American Mathematical Society, New York, 1 edition, 1940. URL http://www.worldcat.org/oclc/1241388. Garrett Birkhoff. Lattice Theory. American Mathematical Society, New York, 2 edition, 1948. URL http://books.google.com/books?vid=ISBN3540120440. Garrett Birkhoff. Lattice Theory, volume 25 of Colloquium Publications. American Mathematical Society, Providence, 3 edition, 1967. ISBN 0-8218-1025-1. URL http://books.google.com/ books?vid=ISBN0821810251. Garrett Birkhoff and P. Hall. Applications of lattice algebra. Mathematical Proceedings of the Cambridge Philosophical Society, 30(2):115–122, 1934. doi: 10.1017/S0305004100016522. URL http://adsabs.harvard.edu/abs/1934MPCPS..30..115B. Garrett Birkhoff and S.A. Kiss. A ternary operation in distributive lattices. Bulletin of the American Mathematical Society, 53:749–752, 1947a. ISSN 1936-881X. doi: 10.1090/S0002-9904-1947-08864-9. URL http://www.ams.org/bull/1947-53-08/ S0002-9904-1947-08864-9. Garrett Birkhoff and S.A. Kiss. A ternary operation in distributive lattices. In J.S. Oliveira and G.C. Rota, editors, Selected Papers on Algebra and Topology by Garrett Birkhoff, Contemporary mathematicians, pages 107–110. Birkhäuser (1987), Boston, 1947b. ISBN 0817631143. URL http://books.google.com/books?vid=ISBN0817631143. Garrett Birkhoff and Saunders MacLane. A Survey of Modern Algebra. A K Peters, 1996. ISBN 1568810687. URL http://books.google.com/books?vid=ISBN1568810687. Garrett Birkhoff and John Von Neumann. The logic of quantum mechanics. The Annals of Mathematics, 37(4):823–843, October 1936. URL http://www.jstor.org/stable/1968621. Garrett Birkhoff and Morgan Ward. A characterization of boolean algebras. The Annals of Mathematics, 40(3):609–610, July 1939a. URL http://www.jstor.org/stable/1968945. Garrett Birkhoff and Morgan Ward. A characterization of boolean algebras. In J.S. Oliveira and G.C. Rota, editors, Selected Papers on Algebra and Topology by Garrett Birkhoff, Contemporary mathematicians, pages 89–90. Birkhäuser (1987), Boston, 1939b. ISBN 0817631143. URL http: //books.google.com/books?vid=ISBN0817631143. Garrett Birkhoff and Morgan Ward. Selected Papers on Algebra and Topology by Garrett Birkhoff. Contemporary mathematicians. Birkhäuser (1987), Boston, 1987. ISBN 0817631143. URL http: //books.google.com/books?vid=ISBN0817631143. version 0.30


Daniel J. Greenhoe


page 293

BIBLIOGRAPHY

George D. Birkhoff and Garrett Birkhoff. Distributive postulates for systems like boolean algebras. Transactions of the American Mathematical Society, 60(1):3–11, July 1946. URL http: //www.jstor.org/stable/1990239. Leonard Mascot Blumenthal. Theory and Applications of Distance Geometry. Chelsea Publishing Company, Bronx, New York, USA, 2 edition, 1970. ISBN 0-8284-0242-6. URL http://books. google.com/books?vid=ISBN0828402426. Thomas Scott Blyth. Lattices and ordered algebraic structures. Springer, London, 2005. ISBN 1852339055. URL http://books.google.com/books?vid=ISBN1852339055. Béla Bollobás. Linear Analysis; an introductory course. Cambridge mathematical textbooks. Cambridge University Press, Cambridge, 2 edition, March 1 1999. ISBN 978-0521655774. URL http://books.google.com/books?vid=ISBN0521655773. George Boole. The Mathematical Analysis of Logic. Macmillan, Barclay, & Macmillan, Cambridge, 1847. URL http://www.archive.org/details/mathematicalanal00booluoft. George Boole. An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities. Walton and Maberly, London, 1854. URL http://www.archive. org/details/investigationofl00boolrich. Umberto Bottazzini. The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass. Springer-Verlag, New York, 1986. ISBN 0-387-96302-2. URL http://books.google. com/books?vid=ISBN0387963022. Bourbaki. Éléments de mathématique. Prewmieère partie: Les structures fondamentales de l'analyse. Livre I: Théorie des ensembles (Fascicule des résultats). Hermann & Cie, Paris, 1939. Bourbaki. Theory of Sets. Elements of Mathematics. Hermann, Paris, 1968. URL http://www. worldcat.org/oclc/281383. R.B. Braithwaite. Characterisations of finite boolean lattices and related algebras. Journal of the London Mathematical Society, 17:180–192, 1942. URL http://jlms.oxfordjournals.org/cgi/ reprint/s1-17/3/180. Gunnar Brinkmann and Brendan D. McKay. Posets on up to 16 points. Order, 19(2):147–179, June 2002. ISSN 0167-8094 (print) 1572-9273 (online). doi: 10.1023/A:1016543307592. URL http: //www.springerlink.com/content/d4dbce7pmctuenmg/. Jason I. Brown and Stephen Watson. Self complementary topologies and preorders. Order, 7(4): 317–328, 1991. ISSN 0167-8094 (print) 1572-9273 (online). doi: 10.1007/BF00383196. URL http: //www.springerlink.com/content/t164x9114754w4lq/. Jason I. Brown and Stephen Watson. The number of complements of a topology on n points is at least 2𝑛 (except for some special cases). Discrete Mathematics, 154(1–3):27–39, 15 June 1996. doi: 10.1016/0012-365X(95)00004-G. URL http://dx.doi.org/10.1016/0012-365X(95)00004-G. Lucas N. H. Bunt, Phillip S. Jones, and Jack D. Bedient. The Historical Roots of Elementary Mathematics. Dover Publications, Mineola, 1988. ISBN 0-486-25563-8. URL http://books.google. com/books?vid=ISBN0486255638. Stanley Burris. The laws of boole's thought. Apil 4 2000. URL www.math.uwaterloo.ca/~snburris/ htdocs/MYWORKS/TALKS/ams-boole.ps. 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 294

BIBLIOGRAPHY

Stanley Burris and Hanamantagida Pandappa Sankappanavar. A Course in Universal Algebra. Number 78 in Graduate texts in mathematics. Springer-Verlag, New York, 1 edition, 1981. ISBN 0387-90578-2. URL http://books.google.com/books?vid=ISBN0387905782. 2000 edition available for free online. Stanley Burris and Hanamantagida Pandappa Sankappanavar. A course in universal algebra. Retypeset and corrected version of the 1981 edition, 2000. URL http://www.math.uwaterloo.ca/ ~snburris/htdocs/ualg.html. Lee Byrne. Two brief formulations of boolean algebra. Bulletin of the American Mathematical Society, 52:269–272, 1946. ISSN 1936-881X. doi: 10.1090/S0002-9904-1946-08556-0. URL http: //www.ams.org/bull/1946-52-04/S0002-9904-1946-08556-0/. Lee Byrne. Boolean algebra in terms of inclusion. American Journal of Mathematics, 70(1):139–143, January 1948. URL http://www.jstor.org/stable/2371939. Lee Byrne. Short formulations of boolean algebra. Canadian Journal of Mathematics, 3(1):31–33, 1951. Florian Cajori. A history of mathematical notations; notations mainly in higher mathematics. In A History of Mathematical Notations; Two Volumes Bound as One, volume 2. Dover, Mineola, New York, USA, 1993. ISBN 0-486-67766-4. URL http://books.google.com/books?vid= ISBN0486677664. reprint of 1929 edition by The Open Court Publishing Company. Georg Cantor. Ueber unendliche, lineare punktmannichfaltigkeiten 3 (on infinite linear manifold of points 3). Mathematische Annalen, 20:113–121, 1882. URL http://dz-srv1.sub. uni-goettingen.de/cache/toc/D35867.html. Gerolamo Cardano. Artis magnae, sive de regulis algebraicis. Nürnberg, 1545a. URL http: //www.filosofia.unimi.it/cardano/testi/operaomnia/vol_4_s_4.pdf. Also known as Ars Magna (The Great Art). Gerolamo Cardano. Ars Magna or the Rules of Algebra. Dover Publications, Mineola, New York, 1545b. ISBN 0486458733. URL http://www.amazon.com/dp/0486458733. English translation of the Latin Ars Magna edition, published in 2007. N.L. Carothers. Real Analysis. Cambridge University Press, Cambridge, 2000. 0521497565. URL http://books.google.com/books?vid=ISBN0521497566.

ISBN 978-

J.C. Carrega. Exclusion d'algèbres. Comptes Rendus des Seances de l'Academie des Sciences, 295: 43–46, 1982. Sirie I: Mathematique. Gianpiero Cattaneo and Davide Ciucci. Lattices with interior and closure operators and abstract approximation spaces. In James F. Peters and Andrzej Skowron, editors, Transactions on Rough Sets X, volume 5656 of Lecture notes in computer science, pages 67–116. Springer, 2009. ISBN 9783642032813. Arthur Cayley. A memoir on the theory of matrices. Philosophical Transactions of the Royal Society of London, 148:17–37, 1858. ISSN 1364-503X. URL http://www.jstor.org/view/02610523/ ap000059/00a00020/0. S. D. Chatterji. The number of topologies on 𝑛 points. Technical Report N67-31144, National Aeronautics and Space Administration, July 1967. URL http://ntrs.nasa.gov/archive/nasa/casi. ntrs.nasa.gov/19670021815_1967021815.pdf. techreport. version 0.30


Daniel J. Greenhoe


page 295

BIBLIOGRAPHY

Gustave Choquet. Theory of capacities. Annales de l'institut Fourier, 5:131–295, 1954. doi: 10.5802/ aif.53. URL http://aif.cedram.org/item?id=AIF_1954__5__131_0. Roberto Cignoli. Injective de morgan and kleene algebras. Proceedings of the American Mathematical Society, 47(2):269–278, February 1975. URL http://www.ams.org/journals/proc/ 1975-047-02/S0002-9939-1975-0357259-4/S0002-9939-1975-0357259-4.pdf. Erhan Çinlar and Robert J Vanderbei. Real and Convex Analysis. Undergraduate Texts in Mathematics. Springer, January 4 2013. ISBN 1461452570. URL http://books.google.com/books? vid=ISBN1461452570. James A. Clarkson. Uniformly convex spaces. Transactions of the American Mathematical Society, 40(3):396–414, December 1936. URL http://www.jstor.org/stable/1989630. David W. Cohen. An Introduction to Hilbert Space and Quantum Logic. Problem Books in Mathematics. Springer-Verlag, New York, 1989. ISBN 0-387-96870-9. URL http://books.google.com/ books?vid=ISBN1461388430. Louis Comtet. Recouvrements, bases de filtre et topologies d'un ensemble fini. Comptes rendus de l'Acade'mie des sciences, 262(20):A1091–A1094, 1966. Recoveries, bases and filter topologies of a finite set. Louis Comtet. Advanced combinatorics: the art of finite and infinite. D. Reidel Publishing Company, Dordrecht, 1974. ISBN 978-9027704412. URL http://books.google.com/books?vid= ISBN9027704414. translated and corrected version of the 1970 French edition. Corneliu Constantinescu. Spaces of measures. Walter De Gruyter, Berlin, 1984. ISBN 3110087847. URL http://books.google.com/books?vid=ISBN3110087847. Edward Thomas Copson. Metric Spaces. Number 57 in Cambridge tracts in mathematics and mathematical physics. Cambridge University Press, London, 1968. ISBN 978-0521047227. URL http://books.google.com/books?vid=ISBN0521047226. René Cori and Daniel Lascar. Mathematical Logic: A Course with Exercises. Part II Recursion theory, Gödel's Theorems, Set Theory, Model Theory. Oxford University Press, 2001. ISBN 0198500513. URL http://books.google.com/books?vid=ISBN0198500513. Peter Crawley and Robert Palmer Dilworth. Algebraic Theory of Lattices. Prentice-Hall, January 1973. ISBN 0130222690. URL http://books.google.com/books?vid=ISBN0130222690. Brian A. Davey and Hilary A. Priestley. Introduction to Lattices and Order. Cambridge mathematical text books. Cambridge University Press, Cambridge, 2 edition, May 6 2002. ISBN 978-0521784511. URL http://books.google.com/books?vid=ISBN0521784514. Anne C. Davis. A characterization of complete lattices. Pacific Journal of Mathematics, 5(2):311– 319, 1955. URL http://projecteuclid.org/euclid.pjm/1103044539. Sheldon W. Davis. Topology. McGraw Hill, Boston, 2005. ISBN 007-124339-9. URL http://www. worldcat.org/isbn/0071243399. William H. E. Day. The complexity of computing metric distances between partitions. Mathematical Social Sciences, 1:269–287, May 1981. ISSN 0165-4896. doi: 10.1016/0165-4896(81)90042-1. URL http://www.sciencedirect.com/science/article/ B6V88-4582D9S-27/2/4e955bb32fd3bfeb49850b2014b4ca2d. 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 296

BIBLIOGRAPHY

Charles Jean de la Vallée-Poussin. Sur l'intégrale de lebesgue. Transactions of the American Mathematical Society, 16(4):435–501, October 1915. URL http://www.jstor.org/stable/1988879. Augustus de Morgan. On the syllogism, no. iv. and on the logic of relations. Transactions of the Cambridge Philosophical Society, 10:331–358, 1864a. read 1860 April 23, reprinted by Heath. Augustus de Morgan. On the syllogism, no. iv. and on the logic of relations. In Peter Lauchlan Heath, editor, On the Syllogism: And Other Logical Writings, Rare masterpieces of philosophy and science, pages 208–246. Routledge & Kegan Paul (1966), London, 1864b. URL http://books. google.com/books?ei=__WRSbTLG5WYkwSe0_3OCg&id=YNENAAAAIAAJ. Andreas de Vries. Algebraic hierarchy of logics unifying fuzzy logic and quantum logic. The registered submission date for this paper is 2007 July 14, but the date appearing on paper proper is 2009 December 6. The latest year in the references is 2006, July 14 2007. URL http://arxiv.org/ abs/0707.2161. Richard Dedekind. Stetigkeit und irrationale zahlen. In Robert Fricke, Emmy Noether, and /:Oystein Ore, editors, Gesammelte mathematische Werke, pages 315–334. Druck and Verlag von Friedr. Vieweg and Sohn Akt.-Ges., Braunschweig, 1872a. URL http://resolver.sub. uni-goettingen.de/purl?PPN23569441X. Continuity and irrational numbers. Richard Dedekind. Continuity and irrational numbers. In Essays on the Theory of Numbers, pages 1–13. The Open Court Publishing Company, Chicago, 1872b. URL http://www.gutenberg.org/ etext/21016. 1901 English translation of Stetigkeit und irrationale Zahlen. Richard Dedekind. Continuity and irrational numbers. In Essays on the Theory of Numbers. Dover, 1872c. ISBN 0486210103. URL http://books.google.com/books?vid=ISBN0486210103. 1901 English translation and 1963 Dover republication of Stetigkeit und irrationale Zahlen. Richard Dedekind. Continuity and irrational numbers. In William Ewald, editor, From Kant to Hilbert: A Source Book in the Foundations of Mathematics, volume 2, pages 765–778? Oxford University Press, Oxford, 1872d. ISBN 019850537X. URL http://books.google.com/books?vid= ISBN019850537X. Published in 1996, English translation of Stetigkeit und irrationale Zahlen. Richard Dedekind. Was sind und was sollen die zahlen? In Robert Fricke, Emmy Noether, and /:Oystein Ore, editors, Gesammelte mathematische Werke, pages 335–391. Druck and Verlag von Friedr. Vieweg and Sohn Akt.-Ges., Braunschweig, 1888a. URL http://resolver.sub. uni-goettingen.de/purl?PPN23569441X. What are and what should be numbers? Richard Dedekind. The nature and meaning of numbers. In Essays on the Theory of Numbers, pages 14–58. The Open Court Publishing Company, Chicago, 1888b. URL http://www.gutenberg.org/ etext/21016. 1901 English translation of Was sind und was sollen die Zahlen? Richard Dedekind. The nature and meaning of numbers. In Essays on the Theory of Numbers. Dover, 1888c. ISBN 0486210103. URL http://books.google.com/books?vid=ISBN0486210103. 1901 English translation and 1963 Dover republication of Was sind und was sollen die Zahlen? Richard Dedekind. Ueber die von drei moduln erzeugte dualgruppe. Mathematische Annalen, 53:371–403, January 8 1900. URL http://resolver.sub.uni-goettingen.de/purl/ ?GDZPPN002257947. Regarding the Dual Group Generated by Three Modules. Augustus DeMorgan. A Budget of Paradoxes. Ayer Publishing, Freeport, 2 edition, 1872. ISBN 0836951190. URL http://www.archive.org/details/budgetofparadoxe00demouoft. René Descartes. Regulae ad directionem ingenii. 1684a. URL http://www.fh-augsburg.de/ ~harsch/Chronologia/Lspost17/Descartes/des_re00.html. version 0.30


Daniel J. Greenhoe


page 297

BIBLIOGRAPHY

René Descartes. Rules for Direction of the Mind. 1684b. URL http://en.wikisource.org/wiki/ Rules_for_the_Direction_of_the_Mind. D. Devidi. Negation: Philosophical aspects. In Keith Brown, editor, Encyclopedia of Language & Linguistics, pages 567–570. Elsevier, 2 edition, April 6 2006. ISBN 9780080442990. URL http: //www.sciencedirect.com/science/article/pii/B0080448542012025. D. Devidi. Negation: Philosophical aspects. In Alex Barber and Robert J Stainton, editors, Concise Encyclopedia of Philosophy of Language and Linguistics, pages 510–513. Elsevier, April 6 2010. ISBN 9780080965017. URL http://books.google.com/books?vid=ISBN0080965016&pg=PA510. Elena Deza and Michel-Marie Deza. Dictionary of Distances. Elsevier Science, Amsterdam, 2006. ISBN 0444520872. URL http://books.google.com/books?vid=ISBN0444520872. Michel-Marie Deza and Elena Deza. Encyclopedia of Distances. Springer, 2009. ISBN 3642002331. URL http://www.uco.es/users/ma1fegan/Comunes/asignaturas/vision/ Encyclopedia-of-distances-2009.pdf. Michel Marie Deza and Monique Laurent. Geometry of Cuts and Metrics, volume 15 of Algorithms and Combinatorics. Springer, Berlin/Heidelberg/New York, May 20 1997. ISBN 354061611X. URL http://books.google.com/books?vid=ISBN354061611X. A. H. Diamond. The complete existential theory of the whitehead-huntington set of postulates for the algebra of logic. Transactions of the American Mathematical Society, 35(4):940–948, October 1933. URL http://www.jstor.org/stable/1989601. A. H. Diamond. Simplification of the whitehead-huntington set of postulates for the algebra of logic. Bulletin of the American Mathematical Society, 40:599–601, 1934. ISSN 00029904. doi: 10.1090/S0002-9904-1934-05925-1. URL http://www.ams.org/bull/1934-40-08/ S0002-9904-1934-05925-1/. A. H. Diamond and J.C.C. McKinsey. Algebras and their subalgebras. Bulletin of the American Mathematical Society, 53:959–962, 1947. ISSN 0002-9904. doi: 10.1090/S0002-9904-1947-08916-3. URL http://www.ams.org/bull/1947-53-10/S0002-9904-1947-08916-3/. Emmanuele DiBenedetto. Real Analysis. Birkhäuser Advanced Texts. Birkhäuser, Boston, 2002. ISBN 0817642315. URL http://books.google.com/books?vid=ISBN0817642315. Jean Alexandre Dieudonné. Foundations of Modern Analysis. Academic Press, New York, 1969. ISBN 1406727911. URL http://books.google.com/books?vid=ISBN1406727911. R.P. Dilworth. On complemented lattices. Tôhoku Mathematical Journal, 47:18–23, 1940. ISSN 0040-8735. URL http://projecteuclid.org/tmj. R.P. Dilworth. Lattices with unique complements. Transactions of the American Mathematical Society, 57(1):123–154, January 1945. URL http://www.jstor.org/stable/1990171. R.P. Dilworth. A decomposition theorem for partially ordered sets. Annals of Mathematics, 51(1): 161–166, January 1950a. doi: 10.2307/1969503. URL http://www.jstor.org/stable/1969503. R.P. Dilworth. A decomposition theorem for partially ordered sets. In Kenneth P. Bogart, Ralph S. Freese, and Joseph P.S. Kung, editors, The Dilworth theorems: selected papers of Robert P. Dilworth, page ? Birkhäuser (1990), Boston, 1950b. ISBN 0817634347. URL http://books.google. com/books?vid=ISBN0817634347. 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 298

BIBLIOGRAPHY

R.P. Dilworth. The role of order in lattice theory. In Ivan Rival, editor, Ordered sets: proceedings of the NATO Advanced Study Institute held at Banff, Canada, August 28 to September 12, 1981, volume 83 of NATO advanced study institutes series, Series C, Mathematical and physical sciences, pages 333–353. D. Reidel Pub. Co., 1982. ISBN 9027713960. URL http://books.google.com/books? vid=ISBN9027713960. R.P. Dilworth. Aspects of distributivity. Algebra Universalis, 18(1):4–17, February 1984. ISSN 0002-5240. doi: 10.1007/BF01182245. URL http://www.springerlink.com/content/ l4480658xw08pp71/. R.P. Dilworth. On complemented lattices. In Kenneth P. Bogart, Ralph S. Freese, and Joseph P.S. Kung, editors, The Dilworth theorems: selected papers of Robert P. Dilworth, pages 73– 78? Birkhäuser, Boston, 1990. ISBN 0817634347. URL http://books.google.com/books?vid= ISBN0817634347. Maurice d'Ocagne. Sur une classe de nombres remarquables. American Journal of Mathematics, 9 (4):353–380, June 1887. URL http://www.jstor.org/stable/2369478. John Doner and Alfred Tarski. An extended arithmetic of ordinal numbers. Fundamenta Mathematicae, 65:95–127, 1969. URL http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=65. J. J. Duistermaat and J. A. C. Kolk. Distributions: Theory and Applications. Cornerstones. Birkhäuser, Basel, 2010. ISBN 0817646728. URL http://books.google.com/books?vid= ISBN0817646728. Nelson Dunford and Jacob T. Schwartz. Linear operators. Part 1, General Theory, volume 7 of Pure and applied mathematics. Interscience Publishers, New York, 1957. ISBN 0471226394. URL http://www.amazon.com/dp/0471608483. with the assistance of William G. Bade and Robert G. Bartle. J. Michael Dunn. Intuitive semantics for first-degree entailments and `coupled trees'. Philosophical Studies, 29(3):149–168, 1976. URL http://link.springer.com/article/10.1007/BF00373152. J. Michael Dunn. Generalized ortho negation. In Heinrich Wansing, editor, Negation: A Notion in Focus, volume 7 of Perspektiven der Analytischen Philosophie / Perspectives in Analytical Philosophy, pages 3–26. De Gruyter, January 1 1996. ISBN 9783110876802. URL http: //books.google.com/books?vid=ISBN3110876809. J. Michael Dunn. A comparative study of various model-theoretic treatments of negation: A history of formal negation. In Dov M. Gabbay and Heinrich Wansing, editors, What is Negation?, volume 13 of Applied Logic Series, pages 23–52. De Gruyter, 1999. ISBN 9780792355694. URL http:// books.google.com/books?vid=ISBN0792355695. John R. Durbin. Modern Algebra; An Introduction. John Wiley & Sons, Inc., 4 edition, 2000. ISBN 0-471-32147-8. URL http://www.worldcat.org/isbn/0471321478. W.D. Duthie. Segments of ordered sets. Transactions of the American Mathematical Society, 51(1): 1–14, January 1942. doi: 10.2307/1989978. URL http://www.jstor.org/stable/1989978. Philip Dwinger. Introduction to Boolean algebras, volume 40 of Hamburger mathematische Einzelschriften. Physica-Verlag, Würzburg, 1 edition, 1961. URL http://books.google.com/ books?id=en6WOgAACAAJ. Philip Dwinger. Introduction to Boolean algebras, volume 40 of Hamburger mathematische Einzelschriften. Physica-Verlag, Würzburg, 2 edition, 1971. URL http://www.amazon.com/dp/ 3790800864. version 0.30


Daniel J. Greenhoe


page 299

BIBLIOGRAPHY

Paul H. Edelman. Abstract convexity and meet-distributive lattices. In Ivan Rival, editor, Combinatorics and ordered sets: Proceedings of the AMS-IMS-SIAM joint summer research conference, held August 11–17, 1985, volume 57 of Contemporary Mathematics, pages 127–150, Providence RI, 1986. American Mathematical Society. ISBN 0821850512. URL http://books.google.com/ books?vid=ISBN0821850512. conference held in Arcata California. Paul H. Edelman and Robert E. Jamison. The theory of convex geometries. Geometriae Dedicata, 19(3):247–270, December 1985. ISSN 0046-5755. doi: 10.1007/BF00149365. URL http://www. springerlink.com/content/n4344856887387gw/. Harold M. Edwards. Essays in Constructive Mathematics. Springer, 2005. ISBN 9780387219783. URL http://books.google.com/books?vid=ISBN0387219781. Charles Elkan, H.R. Berenji, B. Chandrasekaran, C.J.S. de Silva, Y. Attikiouzel, D. Dubois, H. Prade, P. Smets, C. Freksa, O.N. Garcia, G.J. Klir, Bo Yuan, E.H. Mamdani, F.J. Pelletier, E.H. Ruspini, B. Turksen, N. Vadiee, M. Jamshidi, Pei-Zhuang Wang, Sie-Keng Tan, Shaohua Tan, R.R. Yager, and L.A. Zadeh. The paradoxical success of fuzzy logic. IEEE Expert, 9(4):3–49, August 1994. URL http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=336150. “see also IEEE Intelligent Systems and Their Applications”. Paul Erdös and A. Tarski. On families of mutually exclusive sets. Annals of Mathematics, pages 315–329, 1943. URL http://www.renyi.hu/~p_erdos/1943-04.pdf. Marcel Erné, Jobst Heitzig, and Jürgen Reinhold. On the number of distributive lattices. The Electronic Journal of Combinatorics, 9(1), April 2002. URL http://www.emis.de/journals/EJC/ Volume_9/Abstracts/v9i1r24.html. Euclid. Elements. circa 300BC. URL http://farside.ph.utexas.edu/euclid.html. Leonhard Euler. Vollstandige Anleitung zur Algebra. 1770a. URL http://math.dartmouth.edu/ ~euler/pages/E387.html. (Complete instruction in algebra, book 1). Leonhard Euler. Complete instruction in algebra, book 1. 3 edition, 1770b. URL http://math. dartmouth.edu/~euler/pages/E387.html. English translation of Vollstandige Anleitung zur Algebra. Leonhard Euler. De formulis differentialibus angularibus maxime irrationalibus, quas tamen per logarithmos et arcus circulares integrare licet. 1777. URL http://math.dartmouth.edu/~euler/ pages/E671.html. Elliot Evans. Median lattices and convex subalgebras. In Eligius Tamás Schmidt B. Csákány, Ervin Fried, editor, Universal Algebra, volume 29 of Colloquia mathematica Societatis János Bolyai, Amsterdam, June 27 – July 1 1977. Proceedings of the Colloquium on Universal Algebra, Esztergom 1977, North-Holland (1982). ISBN 0444854053. URL http://books.google.com/books? vid=ISBN0444854053. J.W. Evans, Frank Harary, and M.S. Lynn. On the computer enumeration of finite topologies. Communications of the ACM — Association for Computing Machinery, 10:295–297, 1967. ISSN 00010782. URL http://portal.acm.org/citation.cfm?id=363282.363311. David Ewen. The Book of Modern Composers. Alfred A. Knopf, New York, 1950. URL http://books. google.com/books?id=yHw4AAAAIAAJ. David Ewen. The New Book of Modern Composers. Alfred A. Knopf, New York, 3 edition, 1961. URL http://books.google.com/books?id=bZIaAAAAMAAJ. 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 300

BIBLIOGRAPHY

Jonathan David Farley. Chain decomposition theorems for ordered sets and other musings. African Americans in Mathematics DIMACS Workshop, 34:3–14, June 26–28 1996. URL http://books. google.com/books?vid=ISBN0821806785. Jonathan David Farley. Chain decomposition theorems for ordered sets and other musings. arXiv.org preprint, pages 1–12, July 16 1997. URL http://arxiv.org/abs/math/9707220. Gy. Fáy. Transitivity of implication in orthomodular lattices. Acta Scientiarum Mathematicarum, 28(3–4):267–270, 1967. ISSN 0001-6969. URL http://www.acta.hu/acta/. William Feller. An Introduction to Probability Theory and its Applications Volume II. John Wiley & Sons/ Mei Ya Publications, New York/Taipei, 2 edition, 1971. P. D. Finch. Quantum logic as an implication algebra. Bulletin of the Australian Mathematical Soceity, 2:101–106, 1970. URL http://dx.doi.org/10.1017/S0004972700041642. János Fodor and Ronald R. Yager. Fuzzy set-theoretic operators and quantifiers. In Didier Dubois and Henri Padre, editors, Fundamentals of Fuzzy Sets, volume 7 of The Handbooks of Fuzzy Sets, pages 125–195. Springer Science & Business Media, 2000. ISBN 9780792377320. URL http:// books.google.com/books?vid=ISBN079237732X. Gerald B. Folland. A Course in Abstract Harmonic Analysis. Studies in Advanced Mathematics. CRC Press, Boca Raton, 1995. ISBN 0-8493-8490-7. URL http://books.google.com/books?vid= ISBN0849384907. David J. Foulis. A note on orthomodular lattices. Portugaliae Mathematica, 21(1):65–72, 1962. ISSN 0032-5155. URL http://purl.pt/2387. Abraham Fraenkel. Zu den grundlagen der cantor-zermeloschen mengenlehre. Mathematische Annalen, 86:230–237, 1922. URL http://dz-srv1.sub.uni-goettingen.de/cache/toc/D36920. html. Abraham Adolf Fraenkel. Abstract Set Theory. Studies in logic and the foundations of mathematics. North-Holland Publishing Company, Amsterdam, 1953. URL http://books.google.com/books? ei=Z_irR5OrOI7AiQHCqfmmBg&id=E_NLAAAAMAAJ. Marice René Fréchet. Abstract sets, abstract spaces, and general analysis. Mathematics Magazine, 24:147–155, 1950. URL http://www.jstor.org/stable/3029090. Maurice René Fréchet. Sur quelques points du calcul fonctionnel (on some points of functional calculation). Rendiconti del Circolo Matematico di Palermo, 22:1–74, 1906. Rendiconti del Circolo Matematico di Palermo (Statements of the Mathematical Circle of Palermo). Maurice René Fréchet. Les Espaces abstraits et leur théorie considérée comme introduction a l'analyse générale. Borel series. Gauthier-Villars, Paris, 1928. URL http://books.google.com/ books?id=9czoHQAACAAJ. Abstract spaces and their theory regarded as an introduction to general analysis. Gottlob Frege. Grundgesetze der Arithmetik (Basic Laws of Arithmetic), volume 2. Verlag von Hermann Pohle, Jena, 1903. Gottlob Frege. Translations from the Philosophical Writings of Gottlob Frege. Blackwell, Oxford, 1952. URL http://books.google.com/books?id=ox44AAAAMAAJ. Gottlob Frege. The Frege Reader. Blackwell Publishing, Oxford, July 1997. ISBN 0631194452. URL http://books.google.com/books?vid=ISBN0631194452. version 0.30


Daniel J. Greenhoe


page 301

BIBLIOGRAPHY

Friedrich Gerard Friedlander and Mark Suresh Joshi. Introduction to the Theory of Distributions. Cambridge University Press, Cambridge, 2 edition, 1998. ISBN 9780521649711. URL http:// books.google.com/books?vid=ISBN0521649714. Orrin Frink, Jr. Representations of boolean algebras. Bulletin of the American Mathematical Society, 47:755–756, 1941. ISSN 0002-9904. doi: 10.1090/S0002-9904-1941-07554-3. URL http://www. ams.org/bull/1941-47-10/S0002-9904-1941-07554-3/. Otto Frölich. Das halbordnungssystem der topologischen räume auf einer menge. Mathematische Annalen, 156:79–95, 1964. URL http://resolver.sub.uni-goettingen.de/purl? GDZPPN002293501. Jürgen Fuchs. Affine Lie Algebras and Quantum Groups: An Introduction, With Applications in Conformal Field Theory. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 1995. ISBN 052148412X. URL http://books.google.com/books?vid=ISBN052148412X. Paul Abraham Fuhrmann. A Polynomial Approach to Linear Algebra. Springer Science+Business Media, LLC, 2 edition, 2012. ISBN 978-1461403371. URL http://books.google.com/books?vid= ISBN1461403375. Haim Gaifman. The lattice of all topologies on a denumerable set. Notices of the American Mathematical Society, 8(356), 1961. ISSN 0002-9920 (print) 1088-9477 (electronic). Haim Gaifman. Remarks on complementation in the lattice of all topologies. Canadian Journal of Mathematics, 18(1):83–88, 1966. URL http://books.google.com/books?id=eLgzWbwnW2QC. Israel M. Gelfand. Normierte ringe. Mat. Sbornik, 9(51):3–24, 1941. Israel M. Gelfand and Mark A. Naimark. Normed Rings with an Involution and their Representations, pages 240–274. Chelsea Publishing Company, Bronx, 1964. ISBN 0821820222. URL http:// books.google.com/books?vid=ISBN0821820222. Israel M. Gelfand and Mark A. Neumark. On the imbedding of normed rings into the ring of operators in hilbert space. Mat. Sbornik, 12(54:2):197–217, 1943a. Israel M. Gelfand and Mark A. Neumark. On the imbedding of normed rings into the ring of operators in Hilbert Space, pages 3–19. 1943b. ISBN 0821851756. URL http://books.google.com/books? vid=ISBN0821851756. F. Gerrish. The independence of ”huntington's axioms” for boolean algebra. The Mathematical Gazette, 62(419):35–40, March 1978. URL http://www.jstor.org/stable/3617622. John Robilliard Giles. Introduction to the Analysis of Metric Spaces. Number 3 in Australian Mathematical Society lecture series. Cambridge University Press, Cambridge, 1987. ISBN 0521359287. URL http://books.google.com/books?vid=ISBN0521359287. John Robilliard Giles. Introduction to the Analysis of Normed Linear Spaces. Number 13 in Australian Mathematical Society lecture series. Cambridge University Press, Cambridge, 2000. ISBN 0-52165375-4. URL http://books.google.com/books?vid=ISBN0521653754. Steven Givant and Paul Halmos. Introduction to Boolean Algebras. Undergraduate Texts in Mathematics. Springer, 2009. ISBN 0387402934. URL http://books.google.com/books?vid= ISBN0387402934. 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 302

BIBLIOGRAPHY

Kurt Gödel. Die vollständigkeit der axiome des logischen funktionenkalküls. Monatshefte für Mathematik und Physik, 37(1):349–360, December 1 1930a. doi: 10.1007/BF01696781. URL http://link.springer.com/article/10.1007/BF01696781. Kurt Gödel. The completeness of the axioms of the functional calculus of logic. In Jean Van Heijenoort, editor, From Frege to G odel : A Source Book in Mathematical Logic, 1879-1931, pages 582–591. Harvard University Press (1967), Cambridge, Massachusetts, 1930b. ISBN 0674324498. URL http://www.amazon.com/dp/0674324498. Siegfried Gottwald. Many-valued logic and fuzzy set theory. In Ulrich Höhle and S.E. Rodabaugh, editors, Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, volume 3 of The Handbooks of Fuzzy Sets, pages 5–90. Kluwer Academic Publishers, 1999. ISBN 9780792383888. URL http://books.google.com/books?vid=ISBN0792383885. George A. Grätzer. Lattice Theory; first concepts and distributive lattices. A Series of books in mathematics. W. H. Freeman & Company, San Francisco, June 1971. ISBN 0716704420. URL http://books.google.com/books?vid=ISBN0716704420. George A. Grätzer. General Lattice Theory. Birkhäuser Verlag, Basel, 2 edition, 1998. ISBN 0-81765239-6. URL http://books.google.com/books?vid=ISBN0817652396. George A. Grätzer. General Lattice Theory. Birkhäuser Verlag, Basel, 2 edition, January 17 2003. ISBN 3-7643-6996-5. URL http://books.google.com/books?vid=ISBN3764369965. George A. Grätzer. Two problems that shaped a century of lattice theory. Notices of the American Mathematical Society, 54(6):696–707, June/July 2007. URL http://www.ams.org/notices/ 200706/. George A. Grätzer. Universal Algebra. Springer, 2 edition, July 2008. ISBN 0387774866. URL http: //books.google.com/books?vid=ISBN0387774866. A.A. Grau. Ternary boolean algebra. Bulletin of the American Mathematical Society, 53:567–572, 1947. ISSN 1936-881X. doi: 10.1090/S0002-9904-1947-08834-0. URL http://www.ams.org/ bull/1947-53-06/S0002-9904-1947-08834-0. Stanley Gudder. Quantum Probability. Probability and Mathematical Statistics. Academic Press, August 28 1988. ISBN 0123053404. URL http://books.google.com/books?vid= ISBN0123053404. Norman B. Haaser and Joseph A. Sullivan. Real Analysis. Dover Publications, New York, 1991. ISBN 0-486-66509-7. URL http://books.google.com/books?vid=ISBN0486665097. Hans Hahn and Arthur Rosenthal. Set Functions. University of New Mexico Press, 1948. ISBN 111422295X. URL http://books.google.com/books?vid=ISBN111422295X. Theodore Hailperin. Boole's algebra isn't boolean algebra. Mathematics Magazine, 54(4):173–184, September 1981. URL http://www.jstor.org/stable/2689628. Paul R. Halmos. Measure Theory. The University series in higher mathematics. D. Van Nostrand Company, New York, 1950. URL http://www.amazon.com/dp/0387900888. 1976 reprint edition available from Springer with ISBN 9780387900889. Paul R. Halmos. Lectures in Boolean Algebras. Van Nostrand Reinhold, London, New York, 1972. URL http://books.google.com/books?id=1s99IAAACAAJ. version 0.30


Daniel J. Greenhoe


page 303

BIBLIOGRAPHY

Paul R. Halmos. I Want to Be a Mathematician: An Automathography. Springer-Verlag, New York, 1 edition, December 31 1985. ISBN 3540960783. URL http://books.google.com/books?vid= ISBN3540960783. Paul R. Halmos. Intoduction to Hilbert Space and the Theory of Spectral Multiplicity. Chelsea Publishing Company, New York, 2 edition, 1998. ISBN 0821813781. URL http://books.google.com/ books?vid=ISBN0821813781. Paul Richard Halmos. Naive Set Theory. The University Series in Undergraduate Mathematics. D. Van Nostrand Company, Inc., Princeton, New Jersey, 1960. ISBN 0387900926. URL http: //books.google.com/books?vid=isbn0387900926. William Rowan Hamilton. Theory of conjugate functions, or algebraic couples; with a preliminary and elementary essay on algebra as the science of pure time. Transactions of the Royal Irish Academy, 17:293–422, 1837. URL http://www.emis.de/classics/Hamilton/index.html. Gary M. Hardegree. The conditional in abstract and concrete quantum logic. In Cliff A. Hooker, editor, The Logico-Algebraic Approach to Quantum Mechanics: Volume II: Contemporary Consolidation, The Western Ontario Series in Philosophy of Science, Ontario University of Western Ontario, pages 49–108. Kluwer, May 31 1979. ISBN 9789027707079. URL http://www.amazon. com/dp/9027707073. Godfrey H. Hardy. A Mathematician's Apology. Cambridge University Press, Cambridge, 1940. URL http://www.math.ualberta.ca/~mss/misc/A%20Mathematician's%20Apology.pdf. Juris Hartmanis. On the lattice of topologies. Canadian Journal of Mathematics, 10(4):547–553, 1958. URL http://books.google.com/books?&id=OPDcFxeiBesC. Felix Hausdorff. Grundzüge der Mengenlehre. Von Veit, Leipzig, 1914. URL http://books.google. com/books?id=KTs4AAAAMAAJ. Properties of Set Theory. Felix Hausdorff. Grundzüge der Mengenlehre. Gruyter, Berlin, 2 edition, 1927. ??? Felix Hausdorff. Set Theory. Chelsea Publishing Company, New York, 3 edition, 1937. ISBN 0828401195. URL http://books.google.com/books?vid=ISBN0828401195. 1957 translation of the 1937 German Grundzüge der Mengenlehre. Michiel Hazewinkel, editor. Handbook of Algebras, volume 2. North-Holland, Amsterdam, 1 edition, 2000. ISBN 044450396X. URL http://books.google.com/books?vid=ISBN044450396X. Jean Van Heijenoort. From Frege to Gödel : A Source Book in Mathematical Logic, 1879-1931. Harvard University Press, Cambridge, Massachusetts, 1967. URL http://www.hup.harvard.edu/ catalog/VANFGX.html. Jobst Heitzig and Jürgen Reinhold. Counting finite lattices. Journal Algebra Universalis, 48(1):43– 53, August 2002. ISSN 0002-5240 (print) 1420-8911 (online). doi: 10.1007/PL00013837. URL http://citeseer.ist.psu.edu/486156.html. Leon Albert Henkin. The completeness of the first-order functional calculus. Journal of Symbolic Logic, 14(3):159–166, September 1949. URL http://www.jstor.org/pss/2267044. Edwin Hewitt and Kenneth A. Ross. Abstract Harmonic Analysis. Springer, New York, 2 edition, 1994. ISBN 0387941908. URL http://books.google.com/books?vid=ISBN0387941908. 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 304

BIBLIOGRAPHY

Arend Heyting. Die formalen regeln der intuitionistischen logik i. In Sitzungsberichte der Preussischen Akademie der Wissenschaften, pages 42–56, 1930a. English translation of title: “The formal rules of intuitionistic logic I”. English translation of text in Mancosu 1998 pages 311–327. Arend Heyting. Die formalen regeln der intuitionistischen logik ii. In Sitzungsberichte der Preussischen Akademie der Wissenschaften, pages 57–71, 1930b. English translation of title: “The formal rules of intuitionistic logic II”. Arend Heyting. Die formalen regeln der intuitionistischen logik iii. In Sitzungsberichte der Preussischen Akademie der Wissenschaften, pages 158–169, 1930c. English translation of title: “The formal rules of intuitionistic logic III”. Arend Heyting. Sur la logique intuitionniste. Bulletin de la Classe des Sciences, 16:957–963, 1930d. English translation of title: “On intuitionistic logic”. English translation of text in Mancosu 1998 pages 306–310. David Hilbert, Lothar Nordheim, and John von Neumann. über die grundlagen der quantenmechanik (on the bases of quantum mechanics). Mathematische Annalen, 98:1–30, 1927. ISSN 0025-5831 (print) 1432-1807 (online). URL http://dz-srv1.sub.uni-goettingen.de/cache/ toc/D27776.html. Solomon Hoberman and J. C. C. McKinsey. A set of postulates for boolean algebra. Bulletin of the American Mathematical Society, 43:588–592, 1937. ISSN 0002-9904. doi: 10.1090/S0002-9904-1937-06611-0. URL http://www.ams.org/bull/1937-43-08/ S0002-9904-1937-06611-0/. Ernest William Hobson. The Theory of Functions of a Real Variable and the Theory of Fourier's Series, Volume 1, volume 1. The University Press, 2 edition, 1926. URL http://www.archive. org/details/theoryfunctreal01hobsrich. Ulrich Höhle. Probabilistic uniformization of fuzzy topologies. Fuzzy Sets and Systems, 1(4):311– 332, October 1978. URL http://dx.doi.org/10.1016/0165-0114(78)90021-0. Samuel S. Holland, Jr. A radon-nikodym theorem in dimension lattices. Transactions of the American Mathematical Society, 108(1):66–87, July 1963. URL http://www.jstor.org/stable/ 1993826. Samuel S. Holland, Jr. The current interest in orthomodular lattices. In James C. Abbott, editor, Trends in Lattice Theory, pages 41–126. Van Nostrand-Reinhold, New York, 1970. URL http://books.google.com/books?id=ZfA-AAAAIAAJ. from Preface: “The present volume contains written versions of four talks on lattice theory delivered to a symposium on Trends in Lattice Theory held at the United States Naval Academy in May of 1966.”. Lars Hömander. The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis. Classics in Mathematics. Springer, Berlin, 2003. ISBN 3540006621. URL http: //books.google.com/books?vid=ISBN3540006621. Lars Hörmander. Notions of Convexity. Birkhäuser, Boston, 1994. ISBN 0817637990. URL http: //books.google.com/books?vid=ISBN0817637990. Laurence R. Horn. A Natural History of Negation. The David Hume Series: Philosophy and Cognitive Science Reissues. CSLI Publications, reissue edition, 2001. URL http://emilkirkegaard. dk/en/wp-content/uploads/A-natural-history-of-negation-Laurence-R.-Horn.pdf. Alfred Edward Housman. More Poems. Alfred A. Knopf, 1936. URL http://books.google.com/ books?id=rTMiAAAAMAAJ. version 0.30


Daniel J. Greenhoe


page 305

BIBLIOGRAPHY

Edward Vermilye Huntington. Sets of independent postulates for the algebra of logic. Transactions of the American Mathematical Society, 5(3):288–309, July 1904. ISSN 00029947. URL http://www. jstor.org/stable/1986459. Edward Vermilye Huntington. New sets of independent postulates for the algebra of logic, with special reference to whitehead and russell's principia mathematica. Transactions of the American Mathematical Society, 35(1):274–304, January 1933. doi: 10.2307/1989325. URL http://www. jstor.org/stable/1989325. K Husimi. Studies on the foundations of quantum mechanics i. Proceedings of the PhysicoMathematical Society of Japan, 19:766–789, 1937. John R. Isbell. Median algebra. Transactions of the American Mathematical Society, 260(2):319–362, August 1980. URL http://www.jstor.org/stable/1998007. Chris J. Isham. Modern Differential Geometry for Physicists. World Scientific Publishing, New Jersey, 2 edition, 1999. ISBN 9810235623. URL http://books.google.com/books?vid=ISBN9810235623. C.J. Isham. Quantum topology and quantisation on the lattice of topologies. Classical and Quantum Gravity, 6:1509–1534, November 1989. doi: 10.1088/0264-9381/6/11/007. URL http: //www.iop.org/EJ/abstract/0264-9381/6/11/007. Vasile I. Istrǎţescu. Inner Product Structures: Theory and Applications. Mathematics and Its Applications. D. Reidel Publishing Company, 1987. ISBN 9789027721822. URL http://books.google. com/books?vid=ISBN9027721823. Luisa Iturrioz. Ordered structures in the description of quantum systems: mathematical progress. In Methods and applications of mathematical logic: proceedings of the VII Latin American Symposium on Mathematical Logic held July 29-August 2, 1985, volume 69, pages 55–75, Providence Rhode Island, July 29–August 2 1985. Sociedade Brasileira de Lógica, Sociedade Brasileira de Matemática, and the Association for Symbolic Logic, AMS Bookstore (1988). ISBN 0821850768. S. Jaskowski. Investigations into the system of intuitionistic logic. In Storrs McCall, editor, Polish Logic, 1920–1939, pages 259–263. Oxford University Press, 1936. ISBN 9780198243045. URL http: //books.google.com/books?vid=ISBN0198243049&pg=PA259. collection published in 1967. Barbara Jeffcott. The center of an orthologic. The Journal of Symbolic Logic, 37(4):641–645, December 1972. doi: 10.2307/2272407. URL http://www.jstor.org/stable/2272407. S. Jenei. Structure of girard monoids on [0,1]. In Stephen Ernest Rodabaugh and Erich Peter Klement, editors, Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, volume 20 of Trends in Logic, pages 277–308. Springer, 2003. ISBN 9781402015151. URL http://books.google.com/books?vid=ISBN1402015151. J. L. W. V. Jensen. Sur les fonctions convexes et les ine'galite's entre les valeurs moyennes (on the convex functions and the inequalities between the average values). Acta Mathematica, 30 (1):175–193, December 1906. ISSN 0001-5962. doi: 10.1007/BF02418571. URL http://www. springerlink.com/content/r55q1411g840j446/. William Stanley Jevons. Pure Logic or the Logic of Quality Apart from Quantity; with Remarks on Boole's System and the Relation of Logic and Mathematics. Edward Stanford, London, 1864. URL http://books.google.com/books?id=WVMOAAAAYAAJ. William Stanley Jevons. Letters & Journal of W. Stanley Jevons. Macmillan and Co., London, 1886. URL http://oll.libertyfund.org/index.php?option=com_staticxt&staticfile=show.php% 3Ftitle=2079. 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 306

BIBLIOGRAPHY

Peter Jipsen and Henry Rose. Varieties of Lattices. Number 1533 in Lecture notes in mathematics. Springer Verlag, New York, 1992. ISBN 3540563148. URL http://www1.chapman.edu/~jipsen/ JipsenRoseVoL.html. available for free online. Peter Johnstone. Stone Spaces. Cambridge University Press, London, 1982. ISBN 0-521-23893-5. URL http://books.google.com/books?vid=ISBN0521337798. Library QA611. K. D. Joshi. Foundations of Discrete Mathematics. New Age International, New Delhi, 1989. ISBN 8122401201. URL http://books.google.com/books?vid=ISBN8122401201. Young Bae Jun, Yang Xu, and Keyun Qin. Positive implicative and associative filters of lattice implication algebras. Bulletin of the Korean Mathematical Soceity, pages 53–61, 1998. ISSN 1015-8634 (print), 2234-3016 (online). URL http://www.mathnet.or.kr/mathnet/kms_tex/31983.pdf. JA Kalman. Two axiom definition for lattices. Revue Roumaine de Mathematiques Pures et Appliquees, 13:669–670, 1968. ISSN 0035-3965. Gudrun Kalmbach. Orthomodular logic. In Proceedings of the University of Houston, pages 498–503, Houston, Texas, USA, 1973. Lattice Theory Conference. URL http://www.math.uh.edu/~hjm/ 1973_Lattice/p00498-p00503.pdf. Gudrun Kalmbach. Orthomodular logic. Mathematical Logic Quarterly, 20(25–27):395–406, 1974. doi: 10.1002/malq.19740202504. URL http://onlinelibrary.wiley.com/doi/10.1002/malq. 19740202504/abstract. Gudrun Kalmbach. Orthomodular Lattices. Academic Press, London, New York, 1983. ISBN 0123945801. URL http://books.google.com/books?vid=ISBN0123945801. Norihiro Kamide. On natural eight-valued reasoning. Multiple-Valued Logic (ISMVL), 2013 IEEE 43rd International Symposium on, pages 231–236, May 22–24 2013. ISSN 0195-623X. doi: 10.1109/ISMVL.2013.43. URL http://ieeexplore.ieee.org/xpl/articleDetails.jsp? arnumber=6524669. Alexander Karpenko. Łukasiewicz's Logics and Prime Numbers. Luniver Press, Beckington, Frome BA11 6TT UK, January 1 2006. ISBN 9780955117039. URL http://books.google.com/books? vid=ISBN0955117038. John L. Kelley and T. P. Srinivasan. Measure and Integral, volume 116 of Graduate texts in mathematics. Springer, New York, 1988. ISBN 0387966331. URL http://books.google.com/books? vid=ISBN0387966331. John Leroy Kelley. General Topology. University Series in Higher Mathematics. Van Nostrand, New York, 1955. ISBN 0387901256. URL http://books.google.com/books?vid=ISBN0387901256. Republished by Springer-Verlag, New York, 1975. Yoshida Kenko. The Tsuredzure Gusa of Yoshida No Kaneyoshi. Being the meditations of a recluse in the 14th Century (Essays in Idleness). circa 1330. URL http://www.humanistictexts.org/kenko. htm. 1911 translation of circa 1330 text. Mohamed A. Khamsi and W.A. Kirk. An Introduction to Metric Spaces and Fixed Point Theory. John Wiley, New York, 2001. ISBN 978-0471418252. URL http://books.google.com/books?vid= isbn0471418250. Stephen Cole Kleene. On notation for ordinal numbers. The Journal of Symbolic Logic, 3(4), December 1938. URL http://www.jstor.org/stable/2267778. version 0.30


Daniel J. Greenhoe


page 307

BIBLIOGRAPHY

Stephen Cole Kleene. Introduction to Metamathematics. North-Holland publishing C∘ , 1952. D. Kleitman and B. Rothschild. The number of finite topologies. Proceedings of the American Mathematical Society, 25(2):276–282, June 1970. URL http://www.jstor.org/stable/2037205. Morris Kline. Mathematical Thought From Ancient To Modern Times, volume 1. Oxford University Press, New York, 1990. ISBN 0195061357. URL http://books.google.com/books?vid= ISBN0195061357. Anthony W Knapp. Advanced Real Analysis. Cornerstones. Birkhäuser, Boston, Massachusetts, USA, 1 edition, July 29 2005. ISBN 0817643826. URL http://books.google.com/books?vid= ISBN0817643826. Andrei Nikolaevich Kolmogorov and Sergei Vasil'evich Fomin. Introductory Real Analysis. Dover Publications, New York, 1975. ISBN 0486612260. URL http://books.google.com/books?vid= ISBN0486612260. “unabridged, slightly corrected republication of the work originally published by Prentice-Hall, Inc., Englewood, N.J., in 1970”. Andrei Nikolaevich Kolmogorov and Sergei Vasil'evich Fomin. Elements of the Theory of Functions and Functional Analysis: Volumes 1 and 2, Two Volumes Bound as One. Dover Publications, New York, 1999. ISBN 0486406830. URL http://books.google.com/books?vid=ISBN0486406830. Michiro Kondo and Wieslaw A. Dudek. On bounded lattices satisfying elkan's law. Soft Computing, 12(11):1035–1037, September 2008. ISSN 1432-7643. doi: 10.1007/s00500-007-0270-z. URL http://www.springerlink.com/content/e36576406345uw1t/. A. Korselt. Bemerkung zur algebra der logik. Mathematische Annalen, 44(1):156–157, March 1894. ISSN 0025-5831. doi: 10.1007/BF01446978. URL http://www.springerlink.com/content/ v681m56871273j73/. referenced by Birkhoff(1948)p.133. V. Krishnamurthy. On the number of topologies on a finite set. The American Mathematical Monthly, 73(2):154–157, February 1966. URL http://www.jstor.org/stable/2313548. Carlos S. Kubrusly. The Elements of Operator Theory. Springer, 2 edition, 2011. 9780817649975. URL http://books.google.com/books?vid=ISBN0817649972.

ISBN

Kazimierz Kuratowski. Sur la notion d'ordre dans la theorie des ensembles (on the concept of order in set theory). Fundamenta Mathematicae, 2:161–171, 1921. URL http://matwbn.icm.edu.pl/ ksiazki/fm/fm2/fm2122.pdf. Kazimierz Kuratowski. Introduction to Set Theory and Topology, volume 13 of International Series on Monographs on Pure and Applied Mathematics. Pergamon Press, New York, 1961. URL http: //www.worldcat.org/oclc/1023273. Andrew Kurdila and Michael Zabarankin. Convex Functional Analysis. Systems & Control: Foundations & Applications. Birkhäuser, Boston, 2005. ISBN 9783764321987. URL http://books. google.com/books?vid=ISBN3764321989. Shoji Kyuno. An inductive algorithm to construct finite lattices. Mathematics of Computation, 33 (145):409–421, January 1979. URL http://www.jstor.org/stable/2006054. Edmund Landau. Foundations of analysis: the arithmetic of whole, rational, irrational, and complex numbers. A supplement to textbooks on the differential and integral calculus. AMS Chelsea Publishing, Providence, Rhode Island, USA, 3 edition, 1966. URL http://books.google.com/ books?vid=ISBN082182693X. 1966 English translation of the German text Grundlagen der Analysis. 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 308

BIBLIOGRAPHY

R.E. Larson and S.J. Andima. The lattice of topologies: a survey. Rocky Mountain Journal of Mathematics, 5:177–198, 1975. URL http://rmmc.asu.edu/rmj/rmj.html. Steven R. Lay. Convex Sets and their Applications. John Wiley & Sons, New York, 1982. ISBN 0-47109584-2. URL http://books.google.com/books?vid=isbn0471095842. Azriel Levy. Basic Set Theory. Dover, New York, 2002. ISBN 0486420795. URL http://books. google.com/books?vid=ISBN0486420795. Rudolf Lidl and Günter Pilz. Applied Abstract Algebra. Undergraduate texts in mathematics. Springer, New York, 1998. ISBN 0387982906. URL http://books.google.com/books?vid= ISBN0387982906. Lynn H. Loomis. The Lattice Theoretic Background of the Dimension Theory of Operator Algebras, volume 18 of Memoirs of the American Mathematical Society. American Mathematical Society, Providence RI, 1955. ISBN 0821812181. URL http://books.google.com/books?id=P3Vl_ lXCFRkC. Niccolò Machiavelli. The Literary Works of Machiavelli: Mandragola, Clizia, A Dialogue on Language, and Belfagor, with Selections from the Private Correspondence. Oxford University Press, 1961. ISBN 0313212481. URL http://www.worldcat.org/isbn/0313212481. Saunders MacLane and Garrett Birkhoff. Algebra. Macmillan, New York, 1 edition, 1967. URL http://www.worldcat.org/oclc/350724. Saunders MacLane and Garrett Birkhoff. Algebra. AMS Chelsea Publishing, Providence, 3 edition, 1999. ISBN 0821816462. URL http://books.google.com/books?vid=isbn0821816462. M. Donald MacLaren. Atomic orthocomplemented lattices. Pacific Journal of Mathematics, 14(2): 597–612, 1964. URL http://projecteuclid.org/euclid.pjm/1103034188. Roger Duncan Maddux. The origin of relation algebras in the development and axiomatization of the calculus of relations. Studia Logica, 50(3–4):421–455, September 1991. ISSN 0039-3215. doi: 10.1007/BF00370681. URL http://eprints.kfupm.edu.sa/70735/1/70735.pdf. Roger Duncan Maddux. Relation Algebras. Elsevier Science, 1 edition, July 27 2006. 0444520139. URL http://books.google.com/books?vid=ISBN0444520139.

ISBN

Fumitomo Maeda. Kontinuierliche Geometrien, volume 95 of Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen. Springer-Verlag, Berlin, 1958. Fumitomo Maeda and Shûichirô Maeda. Theory of Symmetric lattices, volume 173 of Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen. Springer-Verlag, Berlin/New York, 1970. URL http://books.google.com/books?id=4oeBAAAAIAAJ. Shûichirô Maeda. On conditions for the orthomodularity. Proceedings of the Japan Academy, 42(3):247–251, 1966. ISSN 0021-4280. URL http://joi.jlc.jst.go.jp/JST.Journalarchive/ pjab1945/42.247. Paolo Mancosu, editor. From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s. Oxford University Press, 1998. ISBN 9780195096323. URL http://www.amazon.com/ dp/0195096320. Georges Matheron. Random Sets and Integral Geometry. John Wiley & Sons, New York, 1975. ISBN 0-471-57621-2. URL http://www.worldcat.org/isbn/0471576212. Library QA273.5.M37. version 0.30


Daniel J. Greenhoe


page 309

BIBLIOGRAPHY

Stefan Mazur. Sur les anneaux linéaires. Comptes rendus de l'Académie des sciences, 207:1025–1027, 1938. W. McCune and R. Padmanabhan. Automated deduction in equational logic and cubic curves. Number 1095 in Leture Notes in Artificial Intelligence. Springer, Berlin, 1996. ISBN 3540613986. URL http://books.google.com/books?vid=ISBN3540613986. William McCune, Ranganathan Padmanabhan, and Robert Veroff. Yet another single law for lattices. Algebra Universalis, 50(2):165–169, December 2003a. ISSN 0002-5240 (print) 1420-8911 (online). doi: 10.1007/s00012-003-1832-2. William McCune, Ranganathan Padmanabhan, and Robert Veroff. Yet another single law for lattices. pages 1–5, July 21 2003b. URL http://arxiv.org/abs/math/0307284. Ralph N. McKenzie. Equational bases for lattice theories. Mathematica Scandinavica, 27:24–38, December 1970. ISSN 0025-5521. URL http://www.mscand.dk/article.php?id=1973. Ralph N. McKenzie. Equational bases and nonmodular lattice varieties. Transactions of the American Mathematical Society, 174:1–43, December 1972. URL http://www.jstor.org/stable/ 1996095. J. E. McLaughlin. Atomic lattices with unique comparable complements. Proceedings of the American Mathematical Society, 7(5):864–866, October 1956. URL http://www.jstor.org/stable/ 2033551. Karl Menger. Untersuchungen über allgemeine metrik. Mathematische Annalen, 100:75–163, 1928. ISSN 0025-5831. URL http://link.springer.com/article/10.1007/BF01455705. (Investigations on general metric). Claudia Menini and Freddy Van Oystaeyen. Abstract Algebra; A Comprehensive Treatment. Marcel Dekker Inc, New York, April 2004. ISBN 0-8247-0985-3. URL http://books.google.com/books? vid=isbn0824709853. Anthony N. Michel and Charles J. Herget. Applied Algebra and Functional Analysis. Dover Publications, Inc., 1993. ISBN 0-486-67598-X. URL http://books.google.com/books?vid= ISBN048667598X. original version published by Prentice-Hall in 1981. D. G. Miller. Postulates for boolean algebra. The American Mathematical Monthly, 59(2):93–96, February 1952. URL http://www.jstor.org/stable/2307107. Dragoslav S. Mitrinović, J. E. Pečarić, and Arlington M. Fink. Classical and New Inequalities in Analysis, volume 61 of Mathematics and its Applications (East European Series). Kluwer Academic Publishers, Dordrecht, Boston, London, 2010. ISBN 978-90-481-4225-5. URL http: //www.amazon.com/dp/0792320646. P. Mittelstaedt. Quantenlogische interpretation orthokomplementärer quasimodularer verbände. Zeitschrift für Naturforschung A, 25:1773–1778, 1970. URL http://www.znaturforsch.com/. English translation of title: “Quantum Logical interpretation ortho complementary quasi modular organizations”. Ilya S. Molchanov. Theory of Random Sets. Probability and Its Applications. Springer, 2005. ISBN 185233892X. URL http://books.google.com/books?vid=ISBN185233892X. James Donald Monk. Handbook of Boolean Algebras. North-Holland, Amsterdam, 1989. ISBN 0444872914. URL http://books.google.com/books?vid=ISBN0444872914. 3 volumes. 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 310

BIBLIOGRAPHY

Richard Montague and Jan Tarski. On bernstein's self-dual set of postulates for boolean algebras. Proceedings of the American Mathematical Society, 5(2):310–311, April 1954. URL http://www. jstor.org/stable/2032243. Karl Eugen Müller. Abriss der Algebra der Logik (Summary of the Algebra of Logic). B. G. Teubner, 1909. URL http://projecteuclid.org/euclid.bams/1183421830. “bearbeitet im auftrag der Deutschen Mathematiker-Vereinigung” (produced on behalf of the German Mathematical Society). “In drei Teilen” (In three parts). Markus Müller-Olm. 2. complete boolean lattices. In Modular Compiler Verification: A RefinementAlgebraic Approach Advocating Stepwise Abstraction, volume 1283 of Lecture Notes in Computer Science, chapter 2, pages 9–14. Springer, September 12 1997. ISBN 978-3-540-69539-4. URL http://link.springer.com/chapter/10.1007/BFb0027455. Chapter 2. James R. Munkres. Topology. Prentice Hall, Upper Saddle River, NJ, 2 edition, 2000. 0131816292. URL http://www.amazon.com/dp/0131816292.

ISBN

Masahiro Nakamura. The permutability in a certain orthocomplemented lattice. Kodai Math. Sem. Rep., 9(4):158–160, 1957. doi: 10.2996/kmj/1138843933. URL http://projecteuclid.org/ euclid.kmj/1138843933. H. Nakano and S. Romberger. Cluster lattices. Bulletin De l'Académie Polonaise Des Sciences, 19: 5–7, 1971. URL books.google.com/books?id=gkUSAQAAMAAJ. M.H.A. Newman. A characterisation of boolean lattices and rings. Journal of the London Mathematical Society, 16(4):256–272, 1941. URL http://jlms.oxfordjournals.org/cgi/reprint/s1-16/ 4/256. Hung T. Nguyen and Elbert A. Walker. A First Course in Fuzzy Logic. Chapman & Hall/CRC, 3 edition, 2006. ISBN 1584885262. URL http://books.google.com/books?vid=ISBN1584885262. Yves Nievergelt. Foundations of logic and mathematics: applications to computer science and cryptography. Birkhäuser, Boston, 2002. URL http://books.google.com/books?vid= ISBN0817642498. Vilém Novák, Irina Perfilieva, and Jiří Močkoř. Mathematical Principles of Fuzzy Logic. The Springer International Series in Engineering and Computer Science. Kluwer Academic Publishers, Boston, 1999. ISBN 9780792385950. URL http://books.google.com/books?vid=ISBN0792385950. Oystein Ore. On the foundation of abstract algebra. i. The Annals of Mathematics, 36(2):406–437, April 1935. URL http://www.jstor.org/stable/1968580. Oystein Ore. Remarks on structures and group relations. Vierteljschr. Naturforsch. Ges. Zürich, 85: 1–4, 1940. S. V. Ovchinnikov. General negations in fuzzy set theory. Journal of Mathematical Analysis and Applications, 92(1):234–239, March 1983. doi: 10.1016/0022-247X(83)90282-2. URL http:// www.sciencedirect.com/science/article/pii/0022247X83902822. James G. Oxley. Matroid Theory, volume 3 of Oxford graduate texts in mathematics. Oxford University Press, Oxford, 2006. ISBN 0199202508. URL http://books.google.com/books?vid= ISBN0199202508. R. Padmanabhan. Two identities for lattices. Proceedings of the American Mathematical Society, 20(2):409–412, February 1969. doi: 10.2307/2035665. URL http://www.jstor.org/stable/ 2035665. version 0.30


Daniel J. Greenhoe


page 311

BIBLIOGRAPHY

R. Padmanabhan and S. Rudeanu. Axioms for Lattices and Boolean Algebras. World Scientific, Hackensack, NJ, 2008. ISBN 9812834540. URL http://www.worldscibooks.com/mathematics/ 7007.html. Alessandro Padoa. La Logique Déductive dans sa Dernière Phase de Développment. Gauthier-Villars, Paris, 1912. URL http://books.google.com/books?&id=Z-0MJw8K8CgC. Endre Pap. Null-Additive Set Functions, volume 337 of Mathematics and Its Applications. Kluwer Academic Publishers, 1995. ISBN 0792336585. URL http://www.amazon.com/dp/0792336585. Mladen Pavičić and Norman D. Megill. Is quantum logic a logic? pages 1–24, December 15 2008. URL https://arxiv.org/abs/0812.2698v1. Note: this paper also appears in the collection “Handbook of Quantum Logic and Quantum Structures: Quantum Logic” (2009). Giuseppe Peano. Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle operazioni della logica deduttiva. Fratelli Bocca Editori, Torino, 1888a. Geometric Calculus: According to the Ausdehnungslehre of H. Grassmann. Giuseppe Peano. Geometric Calculus: According to the Ausdehnungslehre of H. Grassmann. Springer (2000), 1888b. ISBN 0817641262. URL http://books.google.com/books?vid= isbn0817641262. originally published in 1888 in Italian. Giuseppe Peano. Árithmetices principa, nova methodo exposita. Fratres Bocca, 1889a. URL https: //archive.org/details/arithmeticespri00peangoog. The principles of arithmetic presented by a new method. Giuseppe Peano. The principles of arithmetic, presented by a new method. In Jean Van Heijenoort, editor, From Frege to G odel : A Source Book in Mathematical Logic, 1879-1931, pages 85–97. Harvard University Press (1967), Cambridge, Massachusetts, 1889b. ISBN 0674324498. URL http://www.amazon.com/dp/0674324498. translation of Árithmetices principa, nova methodo exposita. Michael Pedersen. Functional Analysis in Applied Mathematics and Engineering. Chapman & Hall/CRC, New York, 2000. ISBN 9780849371691. URL http://books.google.com/books?vid= ISBN0849371694. Library QA320.P394 1999. Charles S. Peirce. Logic notebook. In Writitings of Charles S. Peirce, pages 337–350. 0253372011. URL http://books.google.com/books?vid=ISBN0253372011.

ISBN

Charles S. Peirce, December 24 1903. 1903 December 24 letter to Huntington, not known to still be in existence. Charles S. Peirce, 1904. 1904 February 14 letter to Huntington. Charles Sanders Peirce. Notation for the Logic of Relatives resulting from an amplification of the conceptions of Boole's Calculus of Logic. Welch, Bigelow, and Company, Cambridge, 1870a. URL http://www.archive.org/details/descriptionanot00peirgoog. Charles Sanders Peirce. Description of a notation for the logic of relatives, resulting from an amplification of the conceptions of boole's calculus of logic. In C. Hartshorne and P. Weiss, editors, Collected Papers of Charles Sanders Peirce. Harvard University Press (1958), 1870b. ISBN 0674138007. URL http://books.google.com/books?vid=ISBN0674138007. 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 312

BIBLIOGRAPHY

Charles Sanders Peirce. Description of a notation for the logic of relatives, resulting from an amplification of the conceptions of boole's calculus of logic. In Edward C. Moore, editor, Writings of Charles S. Peirce: A Chronological Edition, 1867–1871, pages 359–429. Indiana University Press (1984 January), 1870c. ISBN 025337202X. URL http://books.google.com/books?vid= ISBN025337202X. Charles Sanders Peirce. A boolean algebra with one constant. In Christian J.W. Kloesel, editor, Writings of Charles S. Peirce, A Chronological Edition, 1879–1884, volume 4, pages 218– 221. Indiana University Press, Bloomington, 2 edition, 1880a. ISBN 0253372046. URL http: //books.google.com/books?vid=ISBN0253372046. collection published in October 1989. Charles Sanders Peirce. Note b: the logic of relatives. In Studies in Logic by Members of the Johns Hopkins University, pages 187–203. Little, Brown, and Co., Boston, 1883a. URL http: //www.archive.org/details/studiesinlogic00peiruoft. Charles Sanders Peirce. Note b: the logic of relatives. In Studies in Logic by Members of the Johns Hopkins University, pages 187–. Adamant Media Corporation (2005 November 30), 1883b. ISBN 140219966X. URL http://books.google.com/books?vid=ISBN140219966X. reprint of the Little Brown and Co. edition. Charles Sanders Peirce. Note b: the logic of relatives. In Studies in Logic by Members of the Johns Hopkins University. J. Benjamins (1983), Boston, 1883c. URL http://books.google.com/books? id=YES2HAAACAAJ. reprint of the Little Brown and Co. edition. Charles Sanders Peirce. The simplest mathematics. In C. Hartshorne and P. Weiss, editors, Collected Papers of Charles Sanders Peirce, volume 4, pages 189–262. Harvard University Press, 1902. URL http://books.google.com/books?id=ZhcPOwAACAAJ. C.S. Peirce. On the algebra of logic. American Journal of Mathematics, 3(1):15–57, March 1880b. URL http://www.jstor.org/stable/2369442. Don Pigozzi. Equational logic and equational theories of algebras. Technical Report 135, Purdue University, Indiana, March 1975. URL http://www.cs.purdue.edu/research/technical_ reports/#1975. 187 pages. Ioannis Pitas and Anastasios N. Venetsanopoulos. Nonlinear Digital Filters: Principles and Applications. The Kluwer international series in engineering and computer science. Kluwer Academic Publishers, Boston, 1991. ISBN 0792390490. URL http://books.google.com/books?vid= ISBN0792390490. Plato. Sophist. In Plato in Twelve Volumes, volume 12. Harvard University Press, Cambridge, MA, USA, circa 360 B.C. URL http://data.perseus.org/texts/urn:cts:greekLit:tlg0059. tlg007.perseus-eng1. Henri Poincaré. Science et méthode. E. Flammarion, 1908a. URL http://www.ac-nancy-metz. fr/enseign/philo/textesph/Scienceetmethode.pdf. (Science and Method) http://fr. wikisource.org/wiki/Science_et_m%C3%A9thode. Henri Poincaré. Science and Method. Courier Dover Publications (2003), Mineola, New York, 1908b. ISBN 0486432696. URL http://books.google.com/books?vid=ISBN0486432696. translation of Science et méthode. Henri Poincaré, Ferdinand Lindemann, and Frau Lisbeth Küssner Lindemann. Wissenschaft und Hypothese. Autorisierte deutsche Ausgabe mit erläuternden Anmerkungen. B.G. Teubner, 1904. URL http://www.archive.org/details/wissenschaftund00lindgoog. translation of La Science et l'hypothèse (1902). version 0.30


Daniel J. Greenhoe


page 313

BIBLIOGRAPHY

Robert Potts. Euclid's Elements of Geometry, the first six books, chiefly from the text of Dr. Simson, with explanatory notes; a series of questions on each book; and a selection of geometrical exercises from the senate-house and college examination papers; with hints and &c. John W. Parker and Son, London, 5 edition, 1860. URL http://books.google.com/books?id=eck2AAAAMAAJ. Vaughan Pratt. Origins of the calculus of binary relations. In Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science, number 22–25 in LICS '92., pages 248–254, Santa Cruz, California, June 22–25 1992. IEEE Computer Society Technical Committee on Mathematical Foundations of Computing, Symposium on Logic in Computer Science, IEEE computer society Technical committee on mathematical foundations of computing, IEEE Computer Society Press (Los Alamitos, California). ISBN 0-8186-2735-2. doi: 10.1109/LICS.1992.185537. URL http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=185537. free downloadable version available at http://boole.stanford.edu/pub/ocbr.pdf. Pavel Pudlák and Jiří Tůma. Every finite lattice can be embedded in a finite partition lattice (preliminary communication). Commentationes Mathematicae Universitatis Carolinae, 18(2):409–414, 1977. URL http://www.dml.cz/dmlcz/105785. Pavel Pudlák and Jiří Tůma. Every finite lattice can be embedded in a finite partition lattice. Algebra Universalis, 10(1):74–95, December 1980. ISSN 0002-5240 (print) 1420-8911 (online). doi: 10. 1007/BF02482893. URL http://www.springerlink.com/content/4r820875g8314806/. Charles Chapman Pugh. Real Mathematical Analysis. Undergraduate texts in mathematics. Springer, New York, 2002. ISBN 0-387-95297-7. URL http://books.google.com/books?vid= ISBN0387952977. Willard V. Quine. Mathematical Logic. Harvard University Press, Cambridge, Mass., 10 edition, 1979. ISBN 0674554515. URL http://books.google.com/books?vid=ISBN0674554515. Malempati Madhusudana Rao. Measure Theory and Integration. Number 265 in Monographs and textbooks in pure and applied mathematics. Marcel Dekker, New York, 2 edition, January 2004. ISBN 0-8247-5401-8. URL http://books.google.com/books?vid=ISBN0824754018. Marlon C. Rayburn. On the lattice of 𝜎-algebras. Canadian Journal of Mathematics, 21(3):755–761, 1969. URL http://books.google.com/books?id=wjBXeqo3az0C. Robert Recorde. The Whetstone of Witte: Whiche is the Seconde Parte of Arithmetike, Containyng Thextraction of Rootes, the Cossike Practise, with the Rule of Equation, and the Woorkes of Surde Nombers. By Ihon Kyngstone, 1557. URL http://books.google.com/books?id=Y5WEAAAAIAAJ. E. Renedo, E. Trillas, and C. Alsina. On the law (𝑎 ⋅ 𝑏′ )′ = 𝑏 + 𝑎′ ⋅ 𝑏′ in de morgan algebras and orthomodular lattices. Soft Computing, 8(1):71–73, October 2003. ISSN 1432-7643. doi: 10. 1007/s00500-003-0264-4. URL http://www.springerlink.com/content/7gdjaawe55l11260/. Greg Restall. An Introduction to Substructural Logics. Routledge, 2000. ISBN 9780415215343. URL http://books.google.com/books?vid=ISBN041521534X. Greg Restall. Laws of non-contradiction, laws of the excluded middle, and logics. July 20 2001. URL http://consequently.org/papers/lnclem.pdf. Greg Restall. Laws of non-contradiction, laws of the excluded middle, and logics. In Graham Priest, J. C. Beall, and Bradley Armour-Garb, editors, The Law of Non-contradiction, chapter 4, pages 73–84. Oxford University Press, 2004. ISBN 978-0199265176. URL http://books.google.com/ books?vid=ISBN0199265178&pg=PA73. 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 314

BIBLIOGRAPHY

Charles Earl Rickart. General Theory of Banach Algebras. University series in higher mathematics. D. Van Nostrand Company, Yale University, 1960. URL http://books.google.com/books?id= PVrvAAAAMAAJ. Frigyes Riesz. Stetigkeitsbegriff und abstrakte mengenlehre. In Guido Castelnuovo, editor, Atti del IV Congresso Internazionale dei Matematici, volume II, pages 18–24, Rome, 1909. Tipografia della R. Accademia dei Lincei. URL http://www.mathunion.org/ICM/ICM1908.2/Main/icm1908. 2.0018.0024.ocr.pdf. 1908 April 6–11. Frigyes Riesz. Les systèmes d'équations linéaires à une infinité d'inconnues (The linear systems of equations containing an infinite number of unknowns). Collection de monographies sur la théorie des fonctions. Gauthier-Villars, Paris, 1913. URL http://www.worldcat.org/oclc/ 1374913. J. Riečan. K axiomatike modulárnych sväzov. Acta Fac. Rer. Nat. Univ. Comenian, 2:257–262, 1957. URL http://www.mat.savba.sk/KTO_SME/riecan/riecan_publikacie.html. R. Tyrrell Rockafellar. Convex Analysis. Princeton landmarks in mathematics. Princeton University Press, 1970. ISBN 9780691015866. URL http://books.google.com/books?vid=ISBN0691015864. Steven Roman. Lattices and Ordered Sets. Springer, New York, 1 edition, 2008. ISBN 0387789006. URL http://books.google.com/books?vid=ISBN0387789006. Gian-Carlo Rota. The number of partitions of a set. The American Mathematical Monthly, 71(5): 498–504, May 1964. URL http://www.jstor.org/stable/2312585. Gian-Carlo Rota. The many lives of lattice theory. Notices of the American Mathematical Society, 44 (11):1440–1445, December 1997. URL http://www.ams.org/notices/199711/comm-rota.pdf. Ron M. Roth. Introduction to Coding Theory. Cambridge University Press, 2006. ISBN 0521845041. URL http://books.google.com/books?vid=ISBN0521845041. S. Rudeanu. On definition of boolean algebras by means of binary operations. Revue Roumaine de Mathématiques Pures et Appliquées, 6:171–183, 1961. referenced in Sikorski 1969. Walter Rudin. Principles of Mathematical Analysis. McGraw-Hill, New York, 3 edition, 1976. ISBN 007054235X. URL http://books.google.com/books?vid=ISBN007054235X. Library QA300.R8 1976. Walter Rudin. Functional Analysis. McGraw-Hill, New York, 2 edition, 1991. ISBN 0-07-118845-2. URL http://www.worldcat.org/isbn/0070542252. Library QA320.R83 1991. Bertrand Russell. The Autobiography Of Bertrand Russell. Little, Brown and Company, 1951. URL http://www.archive.org/details/autobiographyofb017701mbp. Víac̀heslav Nikolaevich Saliǐ. Lattices with Unique Complements, volume 69 of Translations of mathematical monographs. American Mathematical Society, Providence, 1988. ISBN 0821845225. URL http://books.google.com/books?vid=ISBN0821845225. translation of Reshetki s edinstvennymi dopolneniíam̀i. Usa Sasaki. Orthocomplemented lattices satisfying the exchange axiom. Journal of Science of the Hiroshima University, 17:293–302, 1954. ISSN 0386-3034. URL http://journalseek.net/ cgi-bin/journalseek/journalsearch.cgi?field=issn&query=0386-3034. Paul S. Schnare. Multiple complementation in the lattice of topologies. Fundamenta Mathematicae, 62, 1968. URL http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=62. version 0.30


Daniel J. Greenhoe


page 315

BIBLIOGRAPHY

Bernd Siegfried Walter Schröder. Ordered Sets: An Introduction. Birkhäuser, Boston, 2003. ISBN 0817641289. URL http://books.google.com/books?vid=ISBN0817641289. Ernst Schröder. Vorlesungen über die Algebra der Logik: Exakte Logik, volume 1. B. G. Teubner, Leipzig, 1890. URL http://www.archive.org/details/vorlesungenberd02mlgoog. Ernst Schröder. Vorlesungen über die Algebra der Logik: Exakte Logik, volume 3. B. G. Teubner, Leipzig, 1895. URL http://www.archive.org/details/vorlesungenberd03mlgoog. Jean Serra. Image Analysis and Mathematical Processing, volume 1. Academic Press Limited, San Diego, 1982. ISBN 0-12-637240-3. Henry Maurice Sheffer. A set of five independent postulates for boolean algebra, with application to logical constants. Transactions of the American Mathematical Society, 14(4):481–488, October 1913. URL http://www.jstor.org/stable/1988701. Henry Maurice Sheffer. Review of “a survey of symbolic logic” by c. i. lewis. The American Mathematical Monthly, 27(7/9):309–311, July–September 1920. URL http://www.jstor.org/stable/ 2972257. Alexander Shen and Nikolai Konstantinovich Vereshchagin. Basic Set Theory, volume 17 of Student mathematical library. American Mathematical Society, Providence, July 9 2002. ISBN 0821827316. URL http://books.google.com/books?vid=ISBN0821827316. translated from Russian. Sajjan G. Shiva. Introduction to Logic Design. CRC Press, 2 edition, 1998. ISBN 0824700821. URL http://books.google.com/books?vid=ISBN0824700821. Marlow Sholander. Postulates for distributive lattices. Canadian Journal of Mathematics, pages 28–30, 1951. URL http://books.google.com/books?hl=en&lr=&id=dKDdYkMCfAIC&pg=PA28. Yaroslav Shramko and Heinrich Wansing. Some useful 16-valued logics: How a computer network should think. Journal of Philosophical Logic, 34(2):121–153, April 2005. ISSN 15730433. doi: 10.1007/s10992-005-0556-5. URL http://link.springer.com/article/10.1007/ s10992-005-0556-5. Roman Sikorski. Boolean Algebras, volume 25 of Ergebnisse der Mathematik und ihrer Grenzbiete. Springer-Verlag, New York, 3 edition, 1969. URL http://www.worldcat.org/oclc/11243. George Finlay Simmons. Calculus Gems: Brief Lives and Memorable Mathematicians. Mathematical Association of America, Washington DC, 2007. ISBN 0883855615. URL http://books. google.com/books?vid=ISBN0883855615. Barry Simon. Convexity: An Analytic Viewpoint, volume 187 of Cambridge Tracts in Mathematics. Cambridge University Press, May 19 2011. ISBN 9781107007314. URL http://books.google. com/books?vid=ISBN1107007313. Neil J. A. Sloane. On-line encyclopedia of integer sequences. World Wide Web, 2014. URL http: //oeis.org/. Sonja Smets. From intuitionistic logic to dynamic operational quantum logic. In Jacek Malinowski and Andrzej Pietruszczak, editors, Essays in Logic and Ontology, volume 91 of Poznań studies in the philosophy of the sciences and the humanities, pages 257–276. Rodopi, 2006. ISBN 9789042021303. URL http://books.google.com/books?vid=ISBN9042021306&pg=PA257. 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 316

BIBLIOGRAPHY

Boleslaw Sobociński. Axiomatization of a partial system of three-value calculus of propositions. Journal of Computing Systems, 1:23–55, 1952. Boleslaw Sobociński. Equational two axiom bases for boolean algebras and some other lattice theories. Notre Dame Journal of Formal Logic, 20(4):865–875, October 1979. URL http: //projecteuclid.org/euclid.ndjfl/1093882808. Richard P. Stanley. Enumerative Combinatorics, volume 49 of Cambridge studies in advanced mathematics. Cambridge University Press, 1 edition, 1997. ISBN 0-521-55309-1. URL http: //books.google.com/books?vid=ISBN0521663512. Lynn Arthur Steen and J. Arthur Seebach. Counterexamples in Topology. Springer-Verlag, 2, revised edition, 1978. URL http://books.google.com/books?vid=ISBN0486319296. A 1995 “unabridged and unaltered republication” Dover edition is available. Anne K. Steiner. The lattice of topologies: Structure and complementation. Transactions of the American Mathematical Society, 122(2):379–398, April 1966. URL http://www.jstor.org/ stable/1994555. Manfred Stern. Semimodular Lattices: Theory and Applications, volume 73 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, May 13 1999. ISBN 0521461057. URL http://books.google.com/books?vid=ISBN0521461057. John Stillwell. Mathematics and Its History. Undergraduate texts in mathematics. Springer, New York, 2 edition, 2002. ISBN 0387953361. URL http://books.google.com/books?vid= ISBN0387953361. M. H. Stone. Postulates for boolean algebras and generalized boolean algebras. American Journal of Mathematics, 57(4):703–732, October 1935. URL http://www.jstor.org/stable/2371008. M. H. Stone. The theory of representation for boolean algebras. Transactions of the American Mathematical Society, 40(1):37–111, July 1936. URL http://www.jstor.org/stable/1989664. Lutz Straßburger. What is logic, and what is a proof? In Jean-Yves Beziau, editor, Logica Universalis: Towards a General Theory of Logic, Mathematics and Statistics, pages 135–145. Birkhäuser, 2005. ISBN 9783764373047. URL http://books.google.com/books?vid=ISBN3764373040. Robert S. Strichartz. The Way of Analysis. Jones and Bartlett Publishers, Boston-London, 1995. ISBN 978-0867204711. URL http://books.google.com/books?vid=ISBN0867204710. Daniel W. Stroock. A Concise Introduction to the Theory of Integration. Birkhäuser, Boston, 3 edition, 1999. ISBN 0817640738. URL http://books.google.com/books?vid=ISBN0817640738. Patrick Suppes. Axiomatic Set Theory. Dover Publications, New York, 1972. ISBN 0486616304. URL http://books.google.com/books?vid=ISBN0486616304. Karl Svozil. Randomness & Undecidability in Physics. World Scientific, 1994. ISBN 981020809X. URL http://books.google.com/books?vid=ISBN981020809X. William W. Tait. Cantor's grundlagen and the paradoxes of set theory. In Richard L. Tieszen Gila Sher, editor, Between Logic and Intuition: Essays in Honor of Charles Parsons, pages 269–290. Cambridge University Press, May 1 2000. ISBN 0521650763. URL http://home.uchicago.edu/ ~wwtx/cantor.pdf. version 0.30


Daniel J. Greenhoe


page 317

BIBLIOGRAPHY

Gaisi Takeuti. Proof Theory, volume 81 of Studies in logic and the foundations of mathematics. North-Holland Publishing Company/American Elsevier Publishing Company, Amsterdam-New York, 1975. ISBN 978-0444104922. URL http://books.google.com/books?vid=ISBN0444104925. Saburo Tamura. Two identities for lattices, distributive lattices and modular lattices with a constant. Notre Dame Journal of Formal Logic, 16(1):137–140, 1975. URL http://projecteuclid. org/euclid.ndjfl/1093891622. Terence Tao. Epsilon of Room, I: Real Analysis: pages from year three of a mathematical blog, volume 117 of Graduate Studies in Mathematics. American Mathematical Society, 2010. ISBN 9780821852781. URL http://books.google.com/books?vid=ISBN0821852787. Terence Tao. An Introduction to Measure Theory, volume 126 of Graduate Studies in Mathematics. American Mathematical Society, 2011. ISBN 9780821869192. URL http://books.google.com/ books?vid=ISBN0821869191. Alfred Tarski. On the calculus of relations. The Journal of Symbolic Logic, 6(3):73–89, September 1941. URL http://www.jstor.org/stable/2268577. Alfred Tarski. Equational logic and equational theories of algebras. In Helmut J. Thiele H. Arnold Schmidt, Kurt Schütte, editor, Contributions to Mathematical Logic: Proceedings of the Logic Colloquium, Studies in logic and the foundations of mathematics, pages 275–288, Hannover, August 1966. International Union of the History and Philosophy of Science, Division of Logic, Methodology and Philosophy of Science, North-Holland Publishing Company (1968). URL http://books.google.com/books?id=W7tLAAAAMAAJ. James Sturdevant Taylor. A set of five postulates for boolean algebras in terms of the operation “exception”. 1(12):241–248, April 12 1920. URL http://www.archive.org/details/113597_001_ 012. Walter Taylor. Equational logic. Houston Journal of Mathematics, pages i–iii,1–83, 1979. Walter Taylor. Equational Logic, chapter Appendix 4, pages 378–400. Springer, New York, 2008. ISBN 978-0-387-77486-2. doi: 10.1007/978-0-387-77487-9. URL http://www.springerlink. com/content/rp1374214u122546/. an “abridged” version of Taylor 1979. N. K. Thakare, M. M. Pawar, and B. N. Waphare. A structure theorem for dismantlable lattices and enumeration. Journal Periodica Mathematica Hungarica, 45(1–2):147–160, September 2002. ISSN 0031-5303 (print) 1588-2829 (online). doi: 10.1023/A:1022314517291. URL http://www.springerlink.com/content/p6r26p872j603285/. Hugh Ansfrid Thurston. The Number-System. Blackie, London, 1956. URL http://books.google. com/books?vid=ISBN0486458067. 2007 Dover republication also available. Hugh Ansfrid Thurston. The Number-System. Dover, Mineola, 2007. ISBN 0486458067. URL http: //books.google.com/books?vid=ISBN0486458067. republication of 1956 edition. Heinrich Franz Friedrich Tietze. Beiträge zur allgemeinen topologie i. Mathematische Annalen, 88 (3–4):290–312, 1923. URL http://link.springer.com/article/10.1007/BF01579182. E. Trillas, E. Renedo, and C. Alsina. On three laws typical of booleanity. Fuzzy Information, 2004. Processing NAFIPS '04. IEEE Annual Meeting of the, 2:520–523, 27–30 June 2004. doi: 10.1109/ NAFIPS.2004.1337354. URL http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber= 1337354. 2016 December 27 Tuesday UTC

Daniel J. Greenhoe


version 0.30

page 318

BIBLIOGRAPHY

A.S. Troelstra and D. van Dalen. Constructivism in Mathematics: An Introduction, volume 121 of Studies in Logic and the Foundations of Mathematics. North Holland/Elsevier, Amsterdam/New York/Oxford/Tokyo, 1988. ISBN 0080570887. URL http://books.google.com/books? vid=ISBN0080570887. Constantin Udriste. Convex Functions and Optimization Methods on Riemannian Manifolds, volume 297 of Mathematics and Its Applications. Springer, July 31 1994. ISBN 0792330021. URL http://books.google.com/books?vid=ISBN0792330021. R. Vaidyanathaswamy. Treatise on set topology, Part I. Indian Mathematical Society, Madras, 1947. MR 9, 367. R. Vaidyanathaswamy. Set Topology. Chelsea Publishing, 2 edition, 1960. ISBN 0486404560. URL http://www.amazon.com/dp/0486404560. note: 978-0486404561 is a Dover edition: “This Dover edition, first published in 1999, is an unabridged republication of the work originally published in 1960 by Chelsea Publishing Company.”. M.L.J. van de Vel. Theory of Convex Structures, volume 50 of North-Holland Mathematical Library. North-Holland, Amsterdam, 1993. ISBN 978-0444815057. URL http://books.google. com/books?vid=ISBN0444815058. A.C.M. van Rooij. The lattice of all topologies is complemented. Canadian Journal of Mathematics, 20(805–807), 1968. URL http://books.google.com/books?id=24hsmjEDbNUC. V. S. Varadarajan. Geometry of Quantum Theory. Springer, 2 edition, 1985. ISBN 9780387493862. URL http://books.google.com/books?vid=ISBN0387493867. Nikolaj K. Vereščagin and A. Shen. Basic Set Theory, volume 17 of Student Mathematical Library. American Mathematical Society, 2002. ISBN 9780821827314. URL http://books.google.com/ books?vid=ISBN0821827316. Denis Artemevich Vladimirov. Boolean Algebras in Analysis. Mathematics and Its Applications. Kluwer Academic, Dordrecht, March 31 2002. URL http://books.google.com/books?vid= ISBN140200480X. John von Neumann. Allgemeine eigenwerttheorie hermitescher funktionaloperatoren. Mathematische Annalen, 102(1):49–131, 1929. ISSN 0025-5831 (print) 1432-1807 (online). URL http://resolver.sub.uni-goettingen.de/purl?GDZPPN002273535. General eigenvalue theory of Hermitian functional operators. John von Neumann. Continuous Geometry. Princeton mathematical series. Princeton University Press, Princeton, 1960. URL http://books.google.com/books?id=3bjqOgAACAAJ. Anders Vretblad. Fourier Analysis and Its Applications, volume 223 of Graduate texts in mathematics. Springer, 2003. ISBN 0387008365. URL http://books.google.com/books?vid= ISBN0387008365. Stephen Watson. The number of complements in the lattice of topologies on a fixed set. Topology and its Applications, 55(2):101–125, 26 January 1994. URL http://www.sciencedirect.com/ science/journal/01668641. Alfred North Whitehead. A Treatise on Universal Algebra with Applications, volume 1. University Press, Cambridge, 1898. URL http://resolver.library.cornell.edu/math/1927624. version 0.30


Daniel J. Greenhoe


page 319

BIBLIOGRAPHY

Albert Whiteman. Postulates for boolean algebra in terms of ternary rejection. Bulletin of the American Mathematical Society, 43:293–298, 1937. ISSN 0002-9904. doi: 10.1090/S0002-9904-1937-06538-4. URL http://www.ams.org/bull/1937-43-04/ S0002-9904-1937-06538-4/. Eldon Whitesitt. Boolean Algebra and Its Applications. Dover, New York, 1995. ISBN 0486684830. URL http://books.google.com/books?vid=ISBN0486684830. Philip M. Whitman. Lattices, equivalence relations, and subgroups. Bulletin of the American Mathematical Society, 52:507–522, 1946. ISSN 0002-9904. doi: 10.1090/S0002-9904-1946-08602-4. URL http://www.ams.org/bull/1946-52-06/S0002-9904-1946-08602-4/. Norbert Wiener. A simplification of the logic of relations. Proceedings of the Cambridge Philosophical Society, 17:387–390, 1914. URL http://books.google.com/books?vid=ISBN0674324498&pg= PA224. reprinted in Jean Van Heijenoort's From Frege to G odel : A Source Book in Mathematical Logic, 1879-1931 page 224–227. J. B. Wilker. Rings of sets are really rings. The American Mathematical Monthly, 89(3):211–211, March 1982. URL http://www.jstor.org/stable/2320207. Robert S. Wolf. Proof, Logic, and Conjecture; The Mathematician's Toolbox. W.H. Freeman and Company, 1998. ISBN 0716730502. URL http://www.worldcat.org/isbn/0716730502. Yang Xu. Lattice implication algebras and mv-algebras. Chinese Quarterly Journal of Mathematics, 3:24–32, 1999. URL http://www.polytech.univ-savoie.fr/fileadmin/polytech_autres_ sites/sites/listic/busefal/Papers/77.zip/77_05.pdf. Yang Xu, Da Ruan, Keyun Qin, and Jun Liu. Lattice-Valued Logic: An Alternative Approach to Treat Fuzziness and Incomparability, volume 132 of Studies in Fuzziness and Soft Computing. Springer, July 15 2003. ISBN 9783540401759. URL http://www.amazon.com/dp/354040175X/. Ronald R. Yager. On the measure of fuzziness and negation part i: Membership in the unit interval. International Journal of General Systems, 5(4):221–229, 1979. doi: 10.1080/03081077908547452. URL http://www.tandfonline.com/doi/abs/10.1080/03081077908547452. Ronald R. Yager. On the measure of fuzziness and negation ii: Lattices. Information and Control, 44(3):236–260, March 1980. doi: 10.1016/S0019-9958(80)90156-4. URL http://www. sciencedirect.com/science/article/pii/S0019995880901564. Ernst Zermelo. Untersuchungen über die grundlagen der mengenlehre i. Mathematische Annalen, 65:261–281, 1908a. URL http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader? ht=VIEW&did=D38200. Ernst Zermelo. Investigations in the foundations of set theory. In Jean Van Heijenoort, editor, From Frege to G odel : A Source Book in Mathematical Logic, 1879-1931, pages 199–215. Harvard University Press, 1967, Cambridge, Massechusettes, 1908b. URL http://www.amazon.com/dp/ 0674324498. Eustachy Żyliński. Some remarks concerning the theory of deduction. Fundamenta Mathematicae, 7:203–209, 1925. URL http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=7.


Daniel J. Greenhoe


version 0.30

page 320

version 0.30

BIBLIOGRAPHY


Daniel J. Greenhoe


REFERENCE INDEX

Aliprantis and Burkinshaw (2006), 53 Aliprantis and Burkinshaw (1998), 7, 10, 27, 28, 154, 202, 204, 214, 221, 232, 234, 258 Adams (1990), 111, 112 Albers and Alexanderson (1985), 76 Amann and Escher (2005), 22 Apostol (1975), 10, 25, 26, 29, 30, 35, 37, 250, 265 Aristotle, 157 Artamonov (2000), 112 Aubin and Frankowska (2009), 267 Aubin (2011), 267 Avron (1991), 166, 180, 181 Bagley (1955), 234 Baker (1969), 76 Balbes (1967), 112 Balbes and Horn (1970), 112 Balbes and Dwinger (1975), 90, 112 Banach (1922), 12 Bandelt and Hedlíková (1983), 112 Bartle (2001), 216 Barvinok (2002), 266, 267 Beer (1993), 267 Bell (1934), 218 Bellman and Giertz (1973), 158, 159 Belnap (1977), 168, 169, 182 Beran (1976), 143 Beran (1982), 132 Beran (1985), 46, 63, 68, 82, 93, 132, 140, 142, 143, 159, 160, 162 Berberian (1961), 37, 251, 260, 274 Bergman (1929), 112 Bernstein (1914), 124, 129,

137 Bernstein (1916), 137 Bernstein (1933), 137 Bernstein (1934), 124, 137, 185 Bernstein (1936), 137 Birkhoff (1933b), 76 Birkhoff (1933a), 47, 61, 62, 65, 76, 90 Birkhoff and Hall (1934), 94, 97 Birkhoff (1936b), 193 Birkhoff (1936a), 66 Birkhoff (1937), 234 Birkhoff (1938), 62, 113 Birkhoff and Ward (1939b), 112 Birkhoff and Ward (1939a), 112 Birkhoff (1940), 76, 112 Birkhoff and Birkhoff (1946), 100 Birkhoff and Kiss (1947b), 112 Birkhoff and Kiss (1947a), 112 Birkhoff (1948), 47, 61, 64, 65, 75, 76, 90, 93, 94, 100, 105, 109 Birkhoff (1967), 54, 59, 66, 72, 76, 80, 153, 154 Birkhoff and Ward (1987), 112 Birkhoff and MacLane (1996), 42 Birkhoff and Neumann (1936), 140, 142, 162 Bunt et al. (1988), 13 Blumenthal (1970), 269 Blyth (2005), 97, 153, 154 Bollobás (1999), 267 Boole (1847), 138

Boole (1854), 138 Bottazzini (1986), 31, 249 Bourbaki (1939), 8, 249 Bourbaki (1968), 6 Braithwaite (1942), 137 Brinkmann and McKay (2002), 51 Brown and Watson (1991), 45, 234 Brown and Watson (1996), 218, 227, 234, 235 Berezansky et al. (1996), 215, 222, 231 Burris and Sankappanavar (1981), 62, 65, 66, 68, 82, 90, 92–94, 105 Burris (2000), 138 Burris and Sankappanavar (2000), 51, 53, 68, 72, 250 Byrne (1946), 134, 136, 137 Byrne (1948), 137 Byrne (1951), 138 Cajori (1993), ix, 173 Cantor (1882), 4 Cardano (1545b), 31 Cardano (1545a), 31 Carothers (2000), 27–29 Carrega (1982), 149 Cattaneo and Ciucci (2009), 159 Cayley (1858), 259 Chatterji (1967), 218 Choquet (1954), 201 Cignoli (1975), 169 Çinlar and Vanderbei (2013), 267 Clarkson (1936), 269 Cohen (1989), 142, 162 Comtet (1966), 218 Comtet (1974), 51, 217, 218, 250, 252 Constantinescu (1984), 234

page 322

REFERENCE INDEX

Copson (1968), 153, 221 Cori and Lascar (2001), 14 Crawley and Dilworth (1973), 76 Davey and Priestley (2002), 65, 149 Davis (1955), 89 Davis (2005), 251, 255, 256 Day (1981), 234 Dedekind (1872c), 14, 42 Dedekind (1872d), 14, 42 Dedekind (1872b), 14, 24, 42 Dedekind (1872a), 14, 42 Dedekind (1888c), 42 Dedekind (1888b), 14–20, 42 Dedekind (1888a), 14, 15, 42 Dedekind (1900), 46, 62, 65, 76, 82 de Morgan (1864b), 264 de Morgan (1864a), 264 DeMorgan (1872), 115 Descartes (1684b), ix Descartes (1684a), ix Devidi (2006), 158 Devidi (2010), 158 de Vries (2007), 158, 159, 169, 179 Deza and Laurent (1997), 153, 154 Deza and Deza (2006), 75, 153, 234 Deza and Deza (2009), 153 Diamond (1933), 137 Diamond (1934), 137 Diamond and McKinsey (1947), 131 DiBenedetto (2002), 211 Dieudonné (1969), 153, 221 Dilworth (1990), 150 Dilworth (1940), 150 Dilworth (1945), 110–112 Dilworth (1950b), 54 Dilworth (1950a), 54 Dilworth (1982), 84 Dilworth (1984), 90, 112 Doner and Tarski (1969), 63 Duistermaat and Kolk (2010), 262 Dunford and Schwartz (1957), 49, 66 Dunn (1976), 169 Dunn (1996), 157, 158 Dunn (1999), 157, 158 Durbin (2000), 110, 254 Duthie (1942), 265 Dwinger (1961), 137 Dwinger (1971), 137 Edelman and Jamison (1985), 269 Edelman (1986), 269 Edwards (2005), 23 version 0.30

Elkan et al. (1994), 131 Erdös and Tarski (1943), 234 Erné et al. (2002), 107, 108 Euclid (circa 300BC), 153 Euler (1770b), 31 Euler (1770a), 31 Euler (1777), 31 Evans et al. (1967), 218 Evans (1977), 112 Ewen (1950), viii Ewen (1961), viii Farley (1996), 66 Farley (1997), 54, 57, 66 Fáy (1967), 159, 160 Feller (1971), 258 Finch (1970), 175, 182, 183 Fodor and Yager (2000), 159, 163, 164 Folland (1995), 271, 272, 275 Foulis (1962), 89, 148, 149, 196 Fraenkel (1922), 4 Fraenkel (1953), 254 Fréchet (1906), 11, 12, 153 Fréchet (1928), 11, 153 Fréchet (1950), 12 Frege (1903), 15 Frege (1997), 15 Frege (1952), 15 Friedlander and Joshi (1998), 263 Frink (1941), 137 Frölich (1964), 227 Fuchs (1995), 271 Fuhrmann (2012), 242, 252 Gaifman (1961), 226, 234, 235 Gaifman (1966), 234 Gelfand (1941), 275 Gelfand and Neumark (1943b), 274, 275 Gelfand and Neumark (1943a), 275 Gelfand and Naimark (1964), 272, 273, 275 Gerrish (1978), 131, 137 Giles (1987), 50, 154 Giles (2000), 53, 261 Givant and Halmos (2009), 63, 116, 120, 124, 125, 127, 131–134, 137, 185, 186 Gödel (1930b), 10 Gödel (1930a), 10 Gottwald (1999), 158 Grätzer (1971), 82, 112 Grätzer (1998), 46, 76 Grätzer (2003), 54, 65, 110, 113, 214 Grätzer (2007), 111–113, 217 Grätzer (2008), 112 Grau (1947), 112 Gudder (1988), 140


Daniel J. Greenhoe

Haaser and Sullivan (1991), 234 Hahn and Rosenthal (1948), 201 Hailperin (1981), 138 Halmos (1950), 201, 214–216, 222, 231 Halmos (1960), 4–7, 14, 50, 55, 210, 237 Halmos (1972), 137 Halmos (1985), 8 Halmos (1998), 273 Hamilton (1837), 31 Hardegree (1979), 174 Hardy (1940), viii Hartmanis (1958), 226, 227, 234, 235 Hausdorff (1914), 211 Hausdorff (1927), 234 Hausdorff (1937), 4, 5, 10, 50, 55, 153, 211, 214, 215, 234, 258 Hazewinkel (2000), 271 Heijenoort (1967), viii Heitzig and Reinhold (2002), 67, 86, 107 Henkin (1949), 10 Hewitt and Ross (1994), 234 Heyting (1930a), 167, 182 Heyting (1930b), 167, 182 Heyting (1930c), 167, 182 Heyting (1930d), 167, 182 Hilbert et al. (1927), 259 Hoberman and McKinsey (1937), 137 Hobson (1926), 23 Höhle (1978), 158 Holland (1963), 140, 148, 149, 192–194 Holland (1970), 62, 140, 193, 196 Hörmander (1994), 269 Hömander (2003), 264 Horn (2001), 157 Housman (1936), viii Huntington (1904), 112, 113, 116, 120, 131, 137, 138 Huntington (1933), 120, 124, 131, 133, 137 Husimi (1937), 148, 158 Isbell (1980), 112 Isham (1989), 67, 211, 225, 226 Isham (1999), 49, 66, 67, 211, 225, 226 Istrǎţescu (1987), 75, 153 Iturrioz (1985), 148–150 Jaskowski (1936), 167, 182 Jeffcott (1972), 197 Jenei (2003), 158 Jensen (1906), 267 2016 December 27 Tuesday UTC

page 323

REFERENCE INDEX Jevons (1864), 77, 115, 138 Jevons (1886), 173 Jipsen and Rose (1992), 76, 112, 113 Johnstone (1982), 167, 182 Joshi (1989), 116, 117, 121, 131, 214 Jun et al. (1998), 174, 182 Kalman (1968), 76 Kalmbach (1973), 174, 175 Kalmbach (1974), 175 Kalmbach (1983), 140, 146, 148–150, 158, 171, 174, 175, 193, 196 Kamide (2013), 184 Karpenko (2006), 167, 182 Kelley (1955), 5, 8, 10, 241– 243, 247–249, 260, 264 Kelley and Srinivasan (1988), 223, 234 Kenko (circa 1330), 288 Khamsi and Kirk (2001), 153, 269 Kleene (1938), 166, 180 Kleene (1952), 166, 180 Kleitman and Rothschild (1970), 218 Kline (1990), 13 Knapp (2005), 264 Kolmogorov and Fomin (1975), 216, 234 Kolmogorov and Fomin (1999), 234 Kondo and Dudek (2008), 131, 138 Korselt (1894), 46, 65, 90, 93 Krishnamurthy (1966), 218 Kubrusly (2011), 211 Kuratowski (1921), 4, 6 Kuratowski (1961), 5, 6 Kurdila and Zabarankin (2005), 267 Kyuno (1979), 67, 68 Landau (1966), 14–19, 21, 24, 31–39, 42 Larson and Andima (1975), 227, 234 Lay (1982), 267, 277, 278, 280 Levy (2002), 214 Lidl and Pilz (1998), 148, 158 Loomis (1955), 140, 191 Łukasiewicz (1920), 166, 181 Machiavelli (1961), 287 MacLane and Birkhoff (1967), 47, 76 MacLane and Birkhoff (1999), 46, 47, 49, 59–62, 66, 106, 107, 111, 115, 120, 185, 234 MacLaren (1964), 196 Maddux (1991), 264 2016 December 27 Tuesday UTC

Maddux (2006), 237, 264 Maeda (1958), 193 Maeda (1966), 79, 140 Maeda and Maeda (1970), 61, 79, 80, 87, 89, 149, 266 Mancosu (1998), 167, 182 Matheron (1975), 277, 278, 285 Mazur (1938), 275 McCune and Padmanabhan (1996), 70, 77 McCune et al. (2003b), 70, 77 McCune et al. (2003a), 70, 77 McKenzie (1970), 68, 76 McKenzie (1972), 76 McLaughlin (1956), 84 Menger (1928), 269 Menini and Oystaeyen (2004), 49, 210 Michel and Herget (1993), 49, 50, 154, 221, 252, 259, 261, 273, 274 Miller (1952), 138 Mitrinović et al. (2010), 267 Mittelstaedt (1970), 175, 182, 183 Molchanov (2005), 201 Monk (1989), 137 Montague and Tarski (1954), 138 Müller (1909), 120, 185 Müller-Olm (1997), 65 Munkres (2000), 6, 40, 53, 211, 216, 243 Nakamura (1957), 149, 171, 194 Nakano and Romberger (1971), 160 Newman (1941), 137 Nguyen and Walker (2006), 158 Nievergelt (2002), 254 Novák et al. (1999), 158, 179, 184 d'Ocagne (1887), 218 Ore (1935), 47, 61, 62, 76, 81, 90, 265 Ore (1940), 90 Ovchinnikov (1983), 164 Oxley (2006), 66 Padmanabhan (1969), 77 Padmanabhan and Rudeanu (2008), 63, 68, 76, 81, 84, 100, 137 Padoa (1912), 173 Pap (1995), 201 Pavičić and Megill (2008), 168, 174, 182, 183 Peano (1888b), 10, 233 Peano (1888a), 10, 12, 233 Peano (1889b), 7, 14, 185 Daniel J. Greenhoe

Peano (1889a), 14 Pedersen (2000), 49 Peirce (1870b), 264 Peirce (1870c), 264 Peirce (1870a), 138, 264 Peirce (1880b), 46, 77 Peirce (1880a), 124 Peirce (1883b), 264 Peirce (1883c), 264 Peirce (1883a), 237, 241, 264 Peirce (1902), 124 Peirce (1903), 112 Peirce (1904), 112 Peirce, 112 Pigozzi (1975), 76 Pitas and Venetsanopoulos (1991), 277, 278, 280–285 Plato (circa 360 B.C.), 157 Poincaré et al. (1904), 23 Poincaré (1908b), 8 Poincaré (1908a), 8 Potts (1860), 7 de la Vallée-Poussin (1915), 258 Pratt (1992), 264 Pudlák and Tůma (1977), 230 Pudlák and Tůma (1980), 230, 234 Pugh (2002), 23–25, 58 Quine (1979), 128, 185 Rao (2004), 234 Rayburn (1969), 234 Recorde (1557), 7 Renedo et al. (2003), 132, 138, 149, 185 Restall (2000), 168, 169, 182, 183 Restall (2001), 185 Restall (2004), 185 Rickart (1960), 272–274 Riečan (1957), 84 Riesz (1909), 211 Riesz (1913), 259 Rockafellar (1970), 267 Roman (2008), 111, 112, 217 Rota (1964), 61, 216, 218, 234 Rota (1997), 61 Roth (2006), 129 Rudeanu (1961), 138 Rudin (1976), 24, 25, 27, 31, 36, 40, 41, 58 Russell (1951), 14, 187 Saliǐ (1988), 84, 93, 105, 110– 113, 131, 214, 230 Sasaki (1954), 171, 193, 196 Schnare (1968), 227, 234, 235 Schröder (1890), 90, 120, 185 Schröder (1895), 264 Schröder (2003), 45, 254 Serra (1982), 285 Sheffer (1913), 124, 137


version 0.30

page 324

REFERENCE INDEX

Sheffer (1920), 49, 66, 107 Shen and Vereshchagin (2002), 45, 50, 53, 55 Shiva (1998), 185, 186 Sholander (1951), 100, 136 Shramko and Wansing (2005), 184 Sikorski (1969), 116, 119, 131, 137, 138 Simmons (2007), 287 Simon (2011), 267 Smets (2006), 182, 183 Sobociński (1952), 166, 181 Sobociński (1979), 76 Stanley (1997), 55–57, 67 Steen and Seebach (1978), 211 Steiner (1966), 226, 234, 235 Stern (1999), 79, 84, 109, 140, 149, 191 Stillwell (2002), 22 Stone (1935), 134, 137 Stone (1936), 214, 234 Straßburger (2005), 179 Strichartz (1995), 23 Stroock (1999), 214 Suppes (1972), 6, 8, 240, 241,

version 0.30

244, 248, 249, 254, 264 Svozil (1994), 49, 109 Tait (2000), 4 Takeuti (1975), 10 Tamura (1975), 70, 76 Tao (2010), 201 Tao (2011), 201 Tarski (1941), 264 Tarski (1966), 68, 76 Taylor (1920), 124 Taylor (1979), 76 Taylor (2008), 76 Thakare et al. (2002), 107 Thurston (1956), 14, 16, 31, 42 Thurston (2007), 42 Tietze (1923), 211 Trillas et al. (2004), 138 Troelstra and van Dalen (1988), 158 Udriste (1994), 267 Vaidyanathaswamy (1947), 227, 234 Vaidyanathaswamy (1960), 221, 223, 227, 234, 251, 255, 256 van Rooij (1968), 226, 234,


Daniel J. Greenhoe

235 Varadarajan (1985), 161 van de Vel (1993), 266, 269 Vereščagin and Shen (2002), 10 Vladimirov (2002), 138 von Neumann (1929), 12 von Neumann (1960), 89 Vretblad (2003), 262–264 Watson (1994), 227, 234 Whitehead (1898), 120, 185 Whiteman (1937), 138 Whitesitt (1995), 126, 185, 206 Whitman (1946), 230 Wiener (1914), 6 Wilker (1982), 223, 234 Wolf (1998), 4 Xu (1999), 174 Xu et al. (2003), 174, 182, 184 Yager (1979), 163 Yager (1980), 163 Zermelo (1908b), 4 Zermelo (1908a), 4 Żyliński (1925), 128


SUBJECT INDEX

𝐶 ∗ algebra, 275 𝐿1 lattice, 141 𝐿2 lattice, 141 𝐿22 lattice, 141 𝐿32 lattice, 141 𝐿42 lattice, 141 𝐿52 lattice, 141 𝑀2 lattice, 183 𝑀4 lattice, 141 𝑀6 lattice, 141 𝑂6 lattice, 140, 142, 147, 148, 172, 192, 197 𝑂8 lattice, 140 ℝ3 Euclidean space, 193 𝛼𝖿 , 259 ∗-algebra, 40 𝖿 ⨢ 𝗀, 259 𝖿 𝗀, 259 ∗-algebra, 38, 272, 272–274 𝑥 commutes 𝑦, 132 Łukasiewicz 3-valued logic, 166, 174, 181, 181 Łukasiewicz 5-valued logic, 182, 182 LATEX, vi TEX-Gyre Project, vi XƎLATEX, vi GLB, 58 LUB, 58 σ-algebra, 202, 214 σ-ring, 202, 215 attention markers, 29 problem, 125 3-dimensional Euclidean space, 172 Abel, Niels Henrik, 287 absolute value, x, 29, 35, 153 absorbtive, 97 absorption, 121, 178 absorptive, 61, 62, 65, 68, 70, 71, 91, 92, 98–100, 104, 117–

120, 130, 132, 136, 137, 143, 145, 147, 174, 185, 194, 196, 212, 214, 220, 221 absorptive property, 83, 84 Addition, 31 addition, 17 complex, 31 Minkowski, 278 multiplication, 31 addition operator, 23 additive, 255 additive identity element, 32 additive inverse, 32 adjoint, 274 adjunction, 124, 124, 126, 128, 187, 206, 206 Adobe Systems Incorporated, vi affine, 267 algebra, 271, 271, 272 algebra of sets, xi, 111, 186, 202, 207, 211, 214, 214, 215, 220–222, 231, 232 algebraic ring, 215, 223 algebraic ring properties of rings of sets, 223 algebraic structure, 100, 136, 142 algebraically isomorphic, 72 algebras 𝐶 ∗ -algebra, 275 ∗-algebra, 272 algebras of sets, 93, 214, 220, 226, 228, 229, 232, 233 alphabetic, 55 alphabetic order relation, 50 alternate denial, 187 alternative denial, 186, 186 AND, xi, 187 and, 187 anti-chain, 57 anti-symmetric, 46, 119, 120,

224, 248, 248, 249 anti-symmetry, 45 antiautomorphic, 38, 272 antichain, 54, 54, 57 antilinear, 39, 273 antisymmetric, 25, 59, 60 antitone, 140, 146, 157–161, 163–169, 175–178, 192, 195 antitonic, 157 Archimedean Property, 27 Aristotelian logic, 179 arithmetic axiom, 90 arity, 250 associative, 16, 19, 32, 33, 56, 59, 61, 62, 68, 70, 77, 91, 104, 117, 120, 121, 129, 130, 132– 136, 143, 145, 146, 185, 203, 212, 214, 220, 221, 242, 260, 261, 280 associative property, 83, 84 asymmetric, 248 atomic, 84, 112 Avant-Garde, vi axiom Achimedean property, 27 Eudoxus, 27 induction, 14 Peano, 14 axiom of extension, 4, 5, 257 axiom of induction, 14–21 axiom of specification, 4, 5 axiom of the excluded middle, 4 axioms excluded middle, 4 ball, 266 closed, 154 open, 154 Banach algebra, 274 Banach space, 12

page 326

SUBJECT INDEX

Barber of Seville, 4 base, 20 base set, 46 Bell numbers, 218, 218 Benzene ring, 140 Bernstein-Cantor-Schröder Theorem, 254 biconditional, 124, 124, 187 bijection, 14 bijective, xi, 53, 72, 161, 252, 253 binary, 137, 250 binary operation, 202 Birkhoff distributivity criterion, 90, 94, 94, 213 Birkhoff's Theorem, 105 BN4 , 169 BN4 logic, 174, 183, 183 Boolean, 107, 108, 115, 115, 137, 141, 146, 147, 150, 155, 170, 171, 173, 174, 179, 180, 183, 196, 197, 199, 214, 222 boolean, 139, 197 Boolean 4-valued logic, 182, 183 Boolean addition, 124, 124, 126, 187 Boolean algebra, 57, 71, 90, 111, 115, 115, 116, 120–122, 125–131, 132, 133, 134, 136, 139, 146, 147, 149, 185, 214, 215, 221 boolean algebra, 146 Boolean algebras, 9, 71, 124, 186, 228, 257 boolean algebras, 214 Boolean lattice, 115, 115, 171, 175–179, 192 Boolean logic, 179, 179, 180 Borel set, 232, 232 both, 183 bottom, 187 bound greatest lower bound, 58 infimum, 58 least upper bound, 58 supremum, 58 boundary, 171 boundary condition, 142, 158, 161–163, 193, 194 boundary condition (Theorem 11.5 page 162), 172 boundary conditions, 161, 168 bounded, xi, 29, 71, 71, 115, 117–121, 131, 132, 137, 139, 146, 177, 185, 221 bounded lattice, 72, 109, 115, 116, 140, 146, 150, 154, 157– 159, 161–164, 171, 174, 192– 194 version 0.30

bounded lattices, 191, 192 Byrne's FORMULATION A, 134 Byrne's FORMULATION B, 136 Byrne's Formulation A, 131 Byrne's Formulation B, 131 C star algebra, 275 cancellation, 97, 98 cancellation criterion, 97 cancellation hypothesis, 99 Cancellation property, 90 Cantor, Georg, 4 cardinal approach, 13 cardinal arithmetic, 56 cardinality, 202 Carl Spitzweg, 287 Cartesian product, x, 3, 6, 201, 210 cartesian product, 55 center, 146, 147, 196, 196– 199 chain, 47, 54, 57, 61, 72 characteristic function, x, 258 characterization, 154 characterizations, 90 Boolean algebra, 131 distributive lattices, 90 characterize, 68 Chinese lantern, 149 classic 10, 71, 116 classic 10 Boolean properties, 120 classic logic, 179 classical 2-value logic, 184, 185 classical 2-valued logic, 185 classical bi-variate logic, 180 classical implication, 168, 174, 175, 180, 182–184 classical logic, 179, 180 closed, 224 closed ball, 154 closed interval, 265, 266, 266 closed set, 211 closed unit ball, 154, 155 closing, 285 closure, 142 commutative, 16, 18, 32–34, 56, 59, 61, 62, 68, 70, 77, 79, 81, 82, 91, 92, 97–100, 104, 117–121, 132–137, 143, 145, 146, 163, 176, 178, 185, 192, 194–196, 212, 214, 215, 220, 221, 261, 277, 280 commutative group, 32, 33 commutative property, 83, 84, 95 commutes, 149, 193, 193, 194, 196 comparable, 25, 45, 46, 47, 54, 57, 80, 84, 265


Daniel J. Greenhoe

complement, x, 3, 10, 109, 109, 124–126, 128, 129, 137, 147, 165, 186, 203, 206 lattice, 109 set, 109 complement 𝑥, 125, 186, 187, 206 complement 𝑥), 125 complement 𝑦, 125, 186, 187, 206 complement 𝑦), 125 complemented, 84, 107, 108, 109, 110, 111, 115, 117–121, 131–134, 137, 139, 142, 143, 146, 185, 214, 215, 221 complemented lattice, 109, 142 complements, 119, 142, 143, 227 complete, 10, 58 complete disjunction, 186, 186, 187 completeness axiom, 58 complex distributive, 34 complex number, 31 complex numbers, 13, 31 conjugate, 37 field, 35 group, 32, 33 composition function, 260 concave, 266, 266, 267, 267 conjuction, 187 conjugate, 37, 39, 262 conjugate function, 261 conjugate linear, 272, 273 conjunction, 124, 186, 187 conjunctive de Morgan, 159, 160, 162 conjunctive de morgan, 142 conjunctive de Morgan ineq., 159 conjunctive de Morgan inequality, 162 conjunctive distributive, 89, 91, 101–103 connected, 248 continuous, xi, 159, 262, 275 contrapositive, 157 converges, 29 converse, 241 convex, 266, 267, 267, 268 functional, 267 strictly, 267 convex function, 267 convex set, 266, 266, 267 coordinate wise, 55 Coordinatewise order relation, 50 coordinatewise order relation, 50 2016 December 27 Tuesday UTC

page 327

SUBJECT INDEX correspondence, 249 cotinuous, 263 counting measure, xi covering relation, 48 covers, 47 cut, 24 de Morgan, 112, 119–121, 143, 145, 147, 160, 162, 163, 167, 177, 183, 195, 196, 221, 222 de Morgan logic, 179, 179 de Morgan negation, 158, 159, 159, 166–169, 179, 183 de Morgan's Law, 221 de Morgan's law, 124, 125, 128–130 de Morgan's Laws, 121, 221 de Morgan's laws, 111, 117, 185 de Morgan, Augustus, 115 Dedekind cuts, 24 Dedekind, Richard, 14, 24 definitions 𝐶 ∗ algebra, 275 ∗-algebra, 272 σ-algebra, 214 σ-ring, 215 absolute value, 29, 35 addition, 17 addition operator, 23 algebra, 271 algebra of sets, 214 algebraically isomorphic, 72 alphabetic, 55 alphabetic order relation, 50 alternative denial, 186 antichain, 54 Banach algebra, 274 base, 20 base set, 46 Bell numbers, 218 Benzene ring, 140 bijective, 252 Boolean algebra, 115, 116, 132 Boolean lattice, 115 Boolean logic, 179 Borel set, 232 bounded, 71 cardinality, 202 Cartesian product, 6 cartesian product, 55 center, 196 chain, 47 characteristic function, 258 Chinese lantern, 149 classical 2-value logic, 184 2016 December 27 Tuesday UTC

closed ball, 154 closed interval, 265, 266 closed set, 211 closed unit ball, 154 closing, 285 commutes, 193 comparable, 46 complement, 10, 203 complemented lattice,

imaginary number, 31 imaginary part, 273 implies, 8 incomparable, 46 indicator function, 258 inhibit x, 186 inhibit y, 186 injective, 252 intersection, 10, 203 intuitionalistic logic,

109 complete, 58 complete disjunction, 186 complex number, 31 complex numbers, 31 conjugate, 37, 262 convex, 267 convex set, 266 coordinate wise, 55 coordinatewise order relation, 50 correspondence, 249 covers, 47 cut, 24 de Morgan logic, 179 diamond, 93 dictionary, 55 dictionary order relation, 50 difference, 10, 203 dilation, 285 direct product, 55 direct sum, 55 disjoint union, 55 distance function, 153 dual, 56, 61 empty set, 5 emptyset, 203 epigraph, 267 equal, 251, 262 equivalence class, 7 equivalence relation, 7 equivalent, 23 erosion, 285 exponent, 20 exponential numbers, 218 false, 4 fully ordered set, 47 function, 8, 249 functionally complete, 126 fuzzy logic, 179 half-open interval, 265, 266 Hasse diagram, 48 hermitian, 272 hexagon, 140 hypograph, 267 identity function, 248 image, 247 Daniel J. Greenhoe

179 inverse, 241 involution, 272 isomorphic, 72 isomorphism, 72 isotone, 75 join semilattice, 59 joint denial, 186 Kleene logic, 179 Kronecker delta function, 259 labeled, 51 lattice, 61 lattice with negation, 159 least upper bound property, 58 length, 54 lexicographical, 55 lexicographical order relation, 50 linear order relation, 47 linearly ordered set, 47 logic, 179 lower bounded, 71 M3 lattice, 93 map, 249 meet semilattice, 59 metric, 153 metric space, 153 Minkowski addition, 277 Minkowski subtraction, 277 MO2 lattice, 149 modular orthocomplemeted lattice, 150 multiplication operator, 23 multiplicative condition, 274 N5 lattice, 82 natural numbers, 14 negative, 22 normal, 272 normed algebra, 274 number of topologies, 218 O6 lattice, 140 one, 14 one-to-one, 252


version 0.30

page 328

SUBJECT INDEX

only if, 8 onto, 252 open ball, 154 open interval, 265, 266 open set, 211 opening, 285 order, 202 order preserving, 51 order relation, 46 ordered pair, 6 ordered set, 46 ordinal product, 55, 56 ordinal sum, 55 ortho logic, 179 orthocomplemented lattice, 140 orthogonal, 191 orthogonality, 191 orthomodular lattice, 148 partial order relation, 46 partially ordered set, 46 partition, 216 paving, 201 pentagon, 82 permutable, 171 poset, 46 positive, 75 power set, 5, 201 preorder relation, 45 preordered set, 45 product, 17 projection, 272 proper subset, 10 rational number, 23 real number, 24 real part, 273 relation, 7, 238 relative complement, 109 resolvent, 271 ring of sets, 215 Sasaki projection, 171 Schwartz class, 262 self-adjoint, 272 set, 4 set equality, 10 set function, 201 set non-equality, 10 set structure, 201 space, 11 spectral radius, 271 spectrum, 271 strictly convex, 267 subposet, 54 subset, 10 subvaluation, 75 successor, 14 sum, 15 supremum, 58 surjective, 252 version 0.30

symmetric difference, 10, 203 symmetric set, 285 tempered distribution, 262 test function, 262 topological space, 211 topology, 211 topology on a finite set, 211 totally ordered set, 47 transformation, 249 translation operator, 277 translation space, 277 true, 4 union, 10, 203 unit ball, 154 unital, 271 universal set, 203 unlabeled, 51 upper bounded, 71 valuation, 75 whole numbers, 22 width, 54 Descartes, René, ix diamond, 93 dictionary, 55 dictionary order relation, 50 difference, x, 10, 187, 203, 206 dilation, 285 Dilworth's theorem, 54, 57, 111 Dirac delta distribution, 263 direct product, 55 direct sum, 55 Discrete lattice, 66 discrete negation, 163, 167 Discrete Time Fourier Series, xii Discrete Time Fourier Transform, xii discrete topology, 211, 227 Dishkant implication, 175, 176 disjoint union, 55 disjunction, 124, 186, 187 disjunctive de Morgan, 160, 162, 163 disjunctive de morgan, 142 disjunctive de Morgan ineq., 159 disjunctive de Morgan inequality, 162 disjunctive distributive, 89, 91, 92, 103, 104 distance function, 153 distributes, 261 distributions, 262 distributive, 34, 38, 56, 64, 86,


Daniel J. Greenhoe

89, 89, 90, 90, 91, 94, 97, 104– 108, 111, 112, 115–121, 124, 130–133, 136, 137, 139, 146, 147, 171, 175, 177, 179, 183, 185, 212–215, 220–222, 272, 277 left, 19 right, 19 distributive inequalities, 65, 89 distributive lattice, 100, 212, 221 distributive lattices, 89 distributive laws, 105, 121, 221 distributivity, 89, 100 domain, x, 243, 244 dual, 46, 56, 61 dual discrete negation, 164, 165 dual distributive, 89 dual distributivity, 89 dual modular, 79 dual modularity, 79, 149 duals, 63 Elkan's law, 71, 131, 138, 149 ellipse, 241 empty set, xi, 4, 5, 187, 206 emptyset, 203 entailment, 174 epigraph, 267 equal, 251, 262 equality, 10 functions, 251 equality by definition, x equality relation, x equational bases, 90 distributive lattices, 90 equational basis, 68 equivalence, 179, 186, 187, 206 equivalence class, 7 equivalence relation, 3, 7 equivalence relations, 3, 248 equivalent, 23 erosion, 285 Euclidean space, 142, 266 Eudoxus axiom, 27 Euler numbers, 51, 67 Euler, Leonhard, 31 examples Łukasiewicz 3-valued logic, 166, 174, 181 Łukasiewicz 5-valued logic, 182 Aristotelian logic, 179 BN4 , 169 BN4 logic, 174, 183 Boolean 4-valued logic, 182 classical logic, 179 2016 December 27 Tuesday UTC

page 329

SUBJECT INDEX Coordinatewise order relation, 50 Discrete lattice, 66 discrete negation, 163, 167 dual discrete negation, 164, 165 factors of 12, 110 Heyting 3-valued logic, 167, 174, 182 Jaśkowski's first matrix, 167, 182 Kleene 3-valued logic, 166, 174, 180 lattices on 1–3 element sets, 67 lattices on 8 element sets, 68 lattices on a 4 element set, 67 lattices on a 5 element set, 67 lattices on a 6 element set, 67 lattices on a 7 element set, 68 Lexicographical order relation, 50 quantum implication, 183 RM3 logic, 166, 174, 181 Sasaki hook, 183 Sasaki hook logic, 174 exception, 124, 124–126, 129, 187 excluded middle, 109, 116, 121, 132, 142, 159, 162, 163, 175–179, 191, 192, 194 exclusive OR, xi exclusive-or, 187 existensial quantifier, 9 existential quantifier, xi explosion, 185 exponent, 20 exponential numbers, 51, 67, 218, 218 factors of 12, 110 false, xi, 3, 4, 8, 9, 180, 183– 185, 187 field, 35 field of sets, 109 field properties, 13 finite, 55, 71, 154, 202, 211, 230 finite orthomodular, 150 finite width, 112 FontLab Studio, vi for each, xi Fourier Series, xii Fourier Transform, xii Fréchet, Marice René, 11 2016 December 27 Tuesday UTC

fraction equivalence, 23 order, 23 Free Software Foundation, vi Frege, Gottlob, 15 fully ordered set, 47 function, 3, 6, 8, 153, 157, 158, 165, 237, 249, 249–251 +,×, 259 arithmetic, 259 characteristic, 258 conjugate, 261 domain, 244 equality, 251 identity, 248 indicator, 258 inverse, 238, 241–243 null space, 244 range, 244 function addition, 259 function multiplication, 259 function subtraction, 259 functional, 249, 267 functionally complete, 126, 126–130, 206, 206, 207 functions, xi, 3, 8, 238 absolute value, 153 adjoint, 274 bijective, 252 classical implication, 174, 175, 180, 183, 184 complement, 109, 109 de Morgan negation, 159, 159, 166–169, 179 Dishkant implication, 176 equivalence, 179 function, 153 fuzzy negation, 158, 162, 163, 166–168, 179 height, 72, 154, 155 implication, 173–178, 180–182, 184 indicator function, 258, 259 injective, 252 intuitionalistic negation, 179 intuitionistic negation, 158, 163–168 Kalmbach implication, 176 Kleene negation, 159, 165, 166, 168, 169, 179–182, 184 Kronecker delta function, 259 length, 72 logical AND, 185 logical equivalence, 185 logical OR, 185 Daniel J. Greenhoe

metric, 154, 155 minimal negation, 158, 158, 159, 163–168, 179 negation, 158, 165, 169, 170, 192 non-tollens implication, 177 one-to-one, 252 one-to-one and onto, 252 onto, 252 ortho negation, 142, 159, 163, 164, 166–170, 179, 180, 183, 191, 194 orthomodular identity, 178 orthomodular negation, 159 relevance implication, 178 Sasaki hook, 121 Sasaki implication, 175 strict negation, 159, 159 strong negation, 159 subminimal negation, 157, 158, 164, 165 surjective, 252 unique complement, 109 valuation, 153, 153, 154 fuzzy, 165, 167, 168, 182 fuzzy logic, 179, 179 fuzzy negation, 158, 158, 162, 163, 166–168, 179 Gelfand-Mazur Theorem, 275 geodesically between, 266 glb, 58 Golden Hind, vi greatest common divisor, 66, 107 greatest lower bound, xi, 58, 58, 60, 61, 120, 137, 143, 146, 212, 224 group, 203, 260, 261, 278 additive, 32 multiplicative, 33 Gutenberg Press, vi half-open interval, 265, 266 Halmos, Paul R., 8 Handbook of Algebras, 271 Hasse diagram, 47, 48, 49, 204, 225 Hasse diagrams, 48 height, 72, 154, 155 hermitian, 272, 272 hermitian components, 274 Heuristica, vi hexagon, 140


version 0.30

page 330

SUBJECT INDEX

Heyting 3-valued logic, 167, 174, 182, 182 Hilbert space, 12, 142 homogeneous, 30, 36, 260 homogenous, 260 horseshoe, 175 Housman, Alfred Edward, vii Huntington properties, 90, 111, 112 Huntington's FIRST SET, 116 Huntington's axiom, 132, 133, 136, 147 Huntington's fifth set, 131, 133 Huntington's first set, 131, 131 Huntington's Fourth Set, 136, 146 Huntington's fourth set, 131, 132, 133 Huntington's problem, 113 Husimi's conjecture, 150 hypograph, 267 idempotency, 172 idempotent, 59, 61–63, 65, 70, 73, 74, 77, 92, 100, 104, 117–120, 130, 132–134, 136, 146, 177, 185, 194, 212, 214, 220, 221, 242, 260 idempotent property, 95 identity, 117–121, 132, 133, 137, 146, 185, 186, 194, 203, 214, 215, 221, 277 identity element, 248 identity function, 248, 248 if, xi if and only if, xi image, x, 247 image set, 243 imaginary number, 31 imaginary part, xi, 273 imaginary quantities, 31 implication, 124, 124, 173, 174, 174–182, 184, 186, 187, 206 implied by, xi, 186, 187 implies, xi, 8 implies and is implied by, xi impossible quantity, 31 inclusive OR, xi incomparable, 45, 46, 47, 54 increasing, 282 independent, 61, 68 indicator function, x, 258, 258, 259 indiscrete topology, 211, 227 inequalities distributive, 65 Jensen's, 267 median, 65 minimax, 64 version 0.30

modular, 65 infimum, 58 infinitely differentiable, 262 inhibit 𝑥, 124, 124, 125, 186, 187, 206 inhibit 𝑦, 186, 187 inhibit x, 186 inhibit y, 186 injective, xi, 252, 253 inner-product, xi inner-product space, 12 integers, 13 intersection, x, 3, 10, 142, 187, 203, 206, 206, 224 interval, 109, 265, 266 into, 252, 253 intuitionalistic logic, 179, 179 intuitionalistic negation, 158, 179 intuitionistic, 165, 167, 168, 182 intuitionistic negation, 158, 162–168 inverse, 72, 203, 241, 241 inverse function, 238, 241– 243 invertible, 272 involutary, 38 involution, 38, 147, 194, 272 involutory, 120, 121, 125, 126, 130, 135, 140, 143, 144, 146, 147, 159, 160, 162–164, 166–169, 176–178, 195, 196, 272 irreflexive, 248 irreflexive ordering relation, xi isomorphic, 53, 72, 72–74 isomorphism, 53, 72, 72 isotone, 71, 72, 75, 75, 154, 155, 251, 255 Jaśkowski's first matrix, 167, 182 Jensen's Inequality, 267 join, xi, 58, 90, 117, 124–126, 137, 186, 187 join absorptive, 101–103 join associatiave, 103 join associative, 102 join commutative, 102–104, 134 join idempotent, 101 join identity, 134, 144 join semilattice, 59, 59 join super-distributive, 65, 89 join-associative, 144 join-commutative, 144 join-distributive, 146 join-identity, 71, 136


Daniel J. Greenhoe

join-meet-absorptive, 145 joint denial, 186, 186, 187 Kalmbach implication, 175, 176 Kaneyoshi, Urabe, 288 Kenko, Yoshida, 288 Kleene, 167 Kleene 3-valued logic, 166, 174, 180, 180 Kleene condition, 159, 162, 163, 166, 169, 170 Kleene logic, 179 Kleene negation, 158, 159, 165, 166, 168–170, 179–182, 184 Kronecker delta function, 259, 259 Kronecker, Leopold, 23 Kuratowski, Kazimierz, 6 labeled, 51 largest algebra, 214 lattice, 5, 58, 61, 61, 62, 65, 68, 70, 82, 89, 90, 94, 97, 104, 115, 132, 142, 146, 153, 154, 160, 161, 163, 174, 176, 185, 191, 192, 194, 212, 214, 221, 265, 266 complemented, 109 distributive, 90 isomorphic, 72 M3, 93 N5, 82, 84, 92 product, 76 relatively complemented, 109 Lattice characterization in 2 equations and 5 variables, 70 Lattice characterizations in 1 equation, 70 lattice complement, 109 lattice of partitions, 230 lattice of topologies, 226, 227 lattice subvaluation metric, 75 lattice valuation metric, 75 lattice with negation, 159, 175, 179, 191–194, 196 lattice with ortho negation, 184 lattices, 9, 178, 192, 214 lattices on 1–3 element sets, 67 lattices on 8 element sets, 68 lattices on a 4 element set, 67 lattices on a 5 element set, 67 lattices on a 6 element set, 67 lattices on a 7 element set, 68 Law of Simplicity, 77 law of the excluded middle, 185 2016 December 27 Tuesday UTC

page 331

SUBJECT INDEX Law of Unity, 77 least common multiple, 66, 107 least upper bound, xi, 58, 58, 59, 61, 72, 137, 145, 146, 174, 212, 224 least upper bound , 120, 224 least upper bound property, 25, 58, 58 left distributive, 19, 34, 134, 261, 282, 283 Leibniz, Gottfried, ix, 173 length, 54, 57, 72 lexicographical, 55 Lexicographical order relation, 50 lexicographical order relation, 50 linear, 61, 153, 262, 263 linear bounded, xi linear order relation, 47 linear space, 12, 265–267, 271 linearly ordered, 47, 106, 182 linearly ordered lattice, 180– 182 linearly ordered set, 47, 265 Liquid Crystal, vi logic, 173, 179, 179 propositional, 9 logic operators, 3 logical AND, 185, 185 logical and, 3, 6, 9, 185, 202 logical equivalence, 185 logical exclusive or, 9 logical exclusive-or, 9, 202 logical if and only if, 185 logical implies, 185 logical not, 9, 202 logical operation, 9 AND, 9 NOT, 9 OR, 9 XOR, 9 logical operations if and only if, 9 implied by, 9 implies, 8 logical OR, 185, 185 logical or, 3, 9, 185, 202 lower bound, 58, 58, 70, 192– 194 lower bounded, 71, 71, 191 lub, 58 M2 lattice, 183 M-symmetric, 80, 85 M3 lattice, 93, 93, 111 Machiavelli, Niccolò, 287 map, 249 maps to, x material implication, 175 2016 December 27 Tuesday UTC

maximin, 64 measure space, 12 median, 65, 90 median inequality, 65, 89 Median property, 90 median property, 91, 92 meet, xi, 58, 63, 90, 117, 124– 126, 137, 186, 187 meet associative, 103 meet commutative, 101–104 meet idempotent, 101, 102 meet semilattice, 59, 59, 60 meet sub distributive, 89 meet sub-distributive, 65 meet-associative, 145 meet-commutative, 145 meet-distributive, 146 meet-idempotent, 143 meet-identity, 71, 136 metric, xi, 153, 154, 155 metric lattice, 154 metric space, 11, 12, 153, 154, 265, 266 metrics lattice subvaluation, 75 lattice valuation, 75 minimal negation, 158, 158, 159, 163–168, 179 minimax, 64 minimax inequality, 64, 65 Minkowski addition, 277, 277 Minkowski subtraction, 277, 277 Minkowski sum, 142 MO2 lattice, 149 modular, 65, 79, 80, 81, 82, 84–86, 92, 94, 98, 105, 108, 112, 139, 150, 154, 155 Modular inequality, 65 modular inequality, 65 modular lattice, 84, 86 modular orthocomplemented, 139, 141 modular orthocomplemeted lattice, 150 modularity, 79, 99, 100, 149 modularity inequality, 65 modus ponens, 174 monotone, 29, 191, 192 Monotony laws, 63 multiplication, 31 multiplication operator, 23 multiplicative condition, 30, 36, 274 multiplicative identity element, 33 multiplicative inverse, 33 multiply complemented, 109, 110, 142 multipy complemented, 165 Daniel J. Greenhoe

mutually exclusive, 216 N5 lattice, 82, 82, 84, 92, 111 Naive set, 4 Naive set theory, 3 naive set theory, 4 nand, 187 natural numbers, 13, 14 negation, 124, 158, 165, 169, 170, 186, 192, 259 negation 𝑥, 187 negation 𝑦, 187 negative, 22 neither, 183 neutral, 180 non-associative, 192 non-Boolean, 140, 141, 147, 148, 173, 183 non-complemented, 110 non-contradiction, 109, 116, 132, 140, 158, 159, 161, 163, 164, 166–172, 177, 185, 191, 192, 194 non-distributive, 92, 105, 107, 111, 183, 213 non-empty, 216 non-equality, 10 non-join-distributive, 147 non-meet-distributive, 148 non-modular, 82, 84–87, 148, 155 non-negative, 30, 36, 153 non-orthocomplemented, 142 non-orthomodular, 140, 148 non-self dual, 68 non-semimodular, 84–87 non-tollens implication, 175, 177 nondegenerate, 30, 36, 153 nor, 187 normal, 272 normed algebra, 30, 36, 274 normed vector space, 12 NOT, xi not, 3 not 𝑥, 187 not 𝑦, 187 not antitone, 164, 165, 169, 170 not bijective, 253 not injective, 253 not modular orthocomplemented, 140 not strong modus ponens, 175 not surjective, 253 null space, x, 243, 244 nullary, 202, 250 number complex, 31 natural, 14


version 0.30

page 332

SUBJECT INDEX

number of lattices, 67, 86, 107 number of posets, 51 number of topologies, 218 O6 lattice, 140, 140, 165, 175 O6 lattice with ortho negation, 183 O6 orthocomplemented lattice, 183 one, 14, 187 one-to-one, 252, 253 one-to-one and onto, 252, 253 only if, xi, 8 onto, 252, 253 open ball, 154, 154 open interval, 265, 266 open set, 211 opening, 285 operations adjunction, 124, 124, 187, 206 alternate denial, 187 and, 187 biconditional, 124, 124, 187 Boolean addition, 124, 124, 187 bottom, 187 closure, 142 complement, 3, 125, 137 complement 𝑥, 125, 187, 206 complement 𝑥), 125 complement 𝑦, 125, 187, 206 complement 𝑦), 125 complete disjunction, 187 conjuction, 187 difference, 187, 206 Discrete Time Fourier Series, xii Discrete Time Fourier Transform, xii disjunction, 187 empty set, 187, 206 equivalence, 187, 206 exception, 124, 124, 125, 187 exclusive-or, 187 false, 187 Fourier Series, xii Fourier Transform, xii function, 6 greatest lower bound, 212 implication, 124, 124, 187, 206 implied by, 187 inhibit 𝑥, 124, 124, 125, version 0.30

187, 206 inhibit 𝑦, 187 intersection, 3, 142, 187, 206, 224 join, 58, 90, 117, 124, 125, 137, 187 joint denial, 187 least upper bound, 212 logical AND, 185 logical and, 3, 6, 9, 185 logical exclusive-or, 9 logical if and only if, 185 logical not, 9 logical OR, 185 logical or, 3, 9, 185 meet, 58, 63, 90, 117, 124, 125, 137, 187 Minkowski sum, 142 nand, 187 negation 𝑥, 187 negation 𝑦, 187 nor, 187 not 𝑥, 187 not 𝑦, 187 one, 187 or, 187 projection 𝑥, 187, 206 projection 𝑦, 187, 206 rejection, 124, 124, 187, 206 Sasaki projection, 171, 172, 193, 195 sasaki projection, 172 Sasaki projection of 𝑦 onto 𝑥, 171 set difference, 3, 46 set intersection operator, 6 Sheffer stroke, 124, 124, 187, 206 summation operator, 268 symmetric difference, 3, 187, 206 ternary rejection, 138 top, 187 transfer 𝑥, 125, 187 transfer 𝑦, 125, 187 translation operator, 278 true, 187 union, 3, 187, 206, 224 universal set, 187, 206 Z-Transform, xii zero, 187 operator adjoint, 273 operator norm, xi OR, 187 or, 187 order, x, xi, 9, 13, 202


Daniel J. Greenhoe

metric, 50 order preserving, 51, 53, 56, 72–74 order relation, 45, 46, 46–49, 224, 265 order relations, 248 alphabetic, 50 coordinatewise, 50 dictionary, 50 lexicographical, 50 order structures, 72 order-reversing, 157 ordered pair, x, 3, 6, 6, 10 ordered set, 45, 46, 46, 49, 57, 58, 61, 72, 119, 212, 224, 265, 266 linearly, 47 totally, 47 ordinal approach, 13 ordinal product, 55, 56 ordinal sum, 55 ortho lattice, 174–178, 191, 192 ortho logic, 179, 179 ortho negation, 142, 158, 159, 163, 164, 166–170, 176, 179, 180, 183, 191–194 ortho-complemented, 112 orthocomplement, 140, 140 orthocomplemented, 139, 140, 140–142, 146–148, 150, 155, 172, 180 Orthocomplemented lattice, 139 orthocomplemented lattice, 140, 140, 142, 143, 146, 148, 149, 171, 172, 191–194, 196, 197 orthocomplemented lattices, 140 orthocomplemented O6 lattice, 175 orthogonal, 191, 192 orthogonality, 149, 191, 192 orthomodular, 132, 139–141, 148–150, 159, 196 orthomodular identity, 148, 178, 195 orthomodular lattice, 148, 149, 175, 178 orthomodular negation, 158, 159 partial order relation, 46 partially ordered set, 45, 46, 59, 60 partition, 7, 202, 216, 216, 220 partitions, 93, 211, 220 paving, 201, 201, 202 Peano axioms, 13 Peano's axioms, 14 2016 December 27 Tuesday UTC

page 333

SUBJECT INDEX Peirce's Theorem, 112, 113 pentagon, 82, 82 permutable, 171 Poincaré, Jules Henri, 8 pointwise order relation, 56 Pointwise ordering relation, 53 poset, 45, 46 order preserving, 51 poset product, 47, 76 posets number, 51, 67 positive, 75, 154, 155 positive integers, 45 power set, xi, 5, 5, 10, 58, 105, 201, 201, 202, 211, 214, 215 pre-topology, 220 preorder relation, 45, 45 preordered set, 45 preserves joins, 72, 73 preserves meets, 72, 74 Principle of duality, 63, 116, 146 principle of duality, 63, 106, 116, 117 Proclus Lycaeus, 13 product, 17, 18, 19, 47 lattice, 76 poset, 47 projection, 272 projection 𝑥, 187, 206 projection 𝑦, 187, 206 proper subset, x, 10, 10 proper superset, x properties 𝑥 commutes 𝑦, 132 absolute value, x absorbtive, 97 absorption, 121, 178 absorptive, 61, 62, 65, 68, 70, 71, 91, 92, 98–100, 104, 117–120, 130, 132, 136, 137, 143, 145, 147, 174, 185, 194, 196, 212, 214, 220, 221 absorptive property, 83, 84 affine, 267 algebra of sets, xi algebraic ring, 223 AND, xi anti-symmetric, 46, 119, 120, 224, 248, 248, 249 anti-symmetry, 45 antiautomorphic, 38, 272 antisymmetric, 25, 59, 60 antitone, 140, 146, 157– 161, 163–169, 175–178, 192, 195 antitonic, 157 2016 December 27 Tuesday UTC

Archimedean, 27 arity, 250 associative, 56, 59, 61, 62, 68, 70, 77, 91, 104, 117, 120, 121, 129, 130, 132–136, 143, 145, 146, 185, 203, 212, 214, 220, 221, 242, 260, 261, 280 associative property, 83, 84 asymmetric, 248 atomic, 84, 112 bijective, 53, 72, 161, 253 binary, 137, 250 Boolean, 107, 108, 115, 115, 137, 141, 146, 147, 150, 155, 170, 171, 173, 174, 179, 180, 183, 196, 197, 199, 214, 222 boolean, 139, 197 Boolean algebra, 90, 111, 121, 122, 125, 133, 136 boundary, 171 boundary condition, 142, 158, 161–163, 193, 194 boundary condition (Theorem 11.5 page 162), 172 boundary conditions, 161, 168 bounded, 29, 71, 115, 117–121, 131, 132, 137, 139, 146, 177, 185, 221 cancellation, 97, 98 Cartesian product, x characteristic function, x closed, 224 commutative, 56, 59, 61, 62, 68, 70, 77, 79, 81, 82, 91, 92, 97–100, 104, 117–121, 132–137, 143, 145, 146, 163, 176, 178, 185, 192, 194–196, 212, 214, 215, 220, 221, 261, 277, 280 commutative property, 83, 84, 95 comparable, 25, 45, 47, 54, 57, 80, 84, 265 complement, x, 128, 129, 147 complemented, 84, 107, 108, 109, 111, 115, 117–121, 131–134, 137, 139, 142, 143, 146, 185, 214, 215, 221 complete, 10 concave, 266, 266, 267, 267 conjugate linear, 272 conjunctive de Morgan, 159, 160, 162 conjunctive de morgan, Daniel J. Greenhoe

142 conjunctive de Morgan ineq., 159 conjunctive de Morgan inequality, 162 conjunctive distributive, 89, 91, 101–103 connected, 248 continuous, 159, 262 contrapositive, 157 converges, 29 convex, 266–268 cotinuous, 263 counting measure, xi de Morgan, 112, 119– 121, 143, 145, 147, 160, 162, 163, 167, 177, 183, 195, 196, 221, 222 de Morgan negation, 169, 183 de Morgan's law, 124, 125, 128–130 de Morgan's laws, 111, 117, 185 difference, x disjunctive de Morgan, 160, 162, 163 disjunctive de morgan, 142 disjunctive de Morgan ineq., 159 disjunctive de Morgan inequality, 162 disjunctive distributive, 89, 91, 92, 103, 104 distributes, 261 distributive, 38, 56, 64, 86, 89, 89, 90, 90, 91, 94, 97, 104–108, 111, 112, 115–121, 124, 130–133, 136, 137, 139, 146, 147, 171, 175, 177, 179, 183, 185, 212–215, 220–222, 272, 277 distributivity, 100 domain, x dual distributive, 89 dual modular, 79 Elkan's law, 131, 138, 149 empty set, xi entailment, 174 equality by definition, x equality relation, x excluded middle, 109, 116, 121, 132, 142, 159, 162, 163, 175–179, 191, 192, 194 exclusive OR, xi existensial quantifier, 9 existential quantifier, xi explosion, 185 false, xi, 3, 8, 9, 185


version 0.30

page 334

SUBJECT INDEX

finite, 55, 71, 154, 202, 211, 230 finite orthomodular, 150 finite width, 112 for each, xi functionally complete, 126–130, 206, 206, 207 fuzzy, 165, 167, 168, 182 geodesically between, 266 glb, 58 greatest common divisor, 66 greatest lower bound, xi, 61 hermitian, 272 homogeneous, 30, 36, 260 homogenous, 260 Huntington properties, 90, 112 Huntington's axiom, 132, 133, 136, 147 idempotency, 172 idempotent, 59, 61–63, 65, 70, 73, 74, 77, 92, 100, 104, 117–120, 130, 132–134, 136, 146, 177, 185, 194, 212, 214, 220, 221, 242, 260 idempotent property, 95 identity, 117–121, 132, 133, 137, 146, 185, 194, 203, 214, 215, 221, 277 if, xi if and only if, xi image, x imaginary part, xi implied by, xi implies, xi implies and is implied by, xi impossible quantity, 31 inclusive OR, xi incomparable, 45, 47, 54 increasing, 282 independent, 61, 68 indicator function, x infinitely differentiable, 262 injective, 253 inner-product, xi intersection, x into, 253 intuitionistic, 165, 167, 168, 182 intuitionistic negation, 162 inverse, 203 invertible, 272 involutary, 38 involution, 147, 194 version 0.30

involutory, 120, 121, 125, 126, 130, 135, 140, 143, 144, 146, 147, 159, 160, 162– 164, 166–169, 176–178, 195, 196, 272 irreflexive, 248 irreflexive ordering relation, xi isomorphic, 53, 72, 74 isomorphism, 53 isotone, 71, 72, 75, 154, 155, 251 join, xi join absorptive, 101–103 join associatiave, 103 join associative, 102 join commutative, 102– 104, 134 join idempotent, 101 join identity, 134, 144 join super-distributive, 65, 89 join-associative, 144 join-commutative, 144 join-distributive, 146 join-identity, 71, 136 join-meet-absorptive, 145 Kleene, 167 Kleene condition, 159, 162, 163, 166, 169, 170 Kleene negation, 169, 170, 180, 181 lattice, 70, 97 lattice complement, 109 law of the excluded middle, 185 least common multiple, 66 least upper bound, xi, 61 left distributive, 134, 261, 282, 283 length, 57 linear, 61, 153, 262, 263 linearly ordered, 47, 106, 182 linearly ordered lattice, 180–182 logical and, 9 logical exclusive or, 9 logical not, 9 logical or, 9 lower bound, 192–194 lower bounded, 71, 191 M2 lattice, 183 M-symmetric, 80, 85 maps to, x median, 90 median inequality, 65, 89 median property, 91, 92


Daniel J. Greenhoe

meet, xi meet associative, 103 meet commutative, 101–104 meet idempotent, 101, 102 meet sub distributive, 89 meet sub-distributive, 65 meet-associative, 145 meet-commutative, 145 meet-distributive, 146 meet-idempotent, 143 meet-identity, 71, 136 metric, xi Minkowski addition, 277 Minkowski subtraction, 277 modular, 65, 79, 80, 81, 82, 85, 86, 92, 94, 98, 105, 108, 112, 139, 150, 154, 155 modular orthocomplemented, 139, 141 modus ponens, 174 monotone, 29, 191, 192 multiplicative condition, 30, 36 multiply complemented, 109, 110, 142 multipy complemented, 165 mutually exclusive, 216 naive set theory, 4 non-associative, 192 non-Boolean, 140, 141, 147, 148, 173, 183 non-complemented, 110 non-contradiction, 109, 116, 132, 140, 158, 159, 161, 163, 164, 166–172, 177, 185, 191, 192, 194 non-distributive, 92, 105, 107, 111, 183, 213 non-empty, 216 non-join-distributive, 147 non-meet-distributive, 148 non-modular, 82, 84–87, 148, 155 non-negative, 30, 36, 153 non-orthocomplemented, 142 non-orthomodular, 140, 148 non-self dual, 68 non-semimodular, 84– 87 nondegenerate, 30, 36, 2016 December 27 Tuesday UTC

page 335

SUBJECT INDEX 153 NOT, xi not antitone, 164, 165, 169, 170 not bijective, 253 not injective, 253 not modular orthocomplemented, 140 not strong modus ponens, 175 not surjective, 253 null space, x nullary, 202, 250 one-to-one, 253 one-to-one and onto, 253 only if, xi onto, 253 open ball, 154 operator norm, xi order, x, xi order preserving, 53, 56, 72–74 order-reversing, 157 ordered pair, x ortho negation, 170 ortho-complemented, 112 orthocomplemented, 139, 140, 140–142, 146–148, 150, 155, 172, 180 orthogonal, 192 orthomodular, 132, 139– 141, 148–150, 159, 196 orthomodular identity, 148, 195 positive, 154, 155 power set, xi preserves joins, 72, 73 preserves meets, 72, 74 principle of duality, 63, 116 proper subset, x proper superset, x range, x real part, xi reflexive, 3, 7, 25, 45, 46, 59, 60, 80, 81, 84, 119, 224, 248, 248, 249 reflexive ordering relation, xi relation, x, 8 relational and, x, 9 relatively complemented, 109 right distributive, 134, 260, 261, 282, 283 ring of sets, xi self-dual, 63, 68 semimodular, 80, 86, 87 set complement, 109 2016 December 27 Tuesday UTC

set of algebras of sets, xi set of rings of sets, xi set of topologies, xi span, xi strict, 159 strictly antitone, 159 strong, 159 strong entailment, 174– 178, 180–182, 184 strong modus ponens, 174, 175, 180–182, 184 strongly connected, 248 subadditive, 30, 36, 153 subminimal, 167 subset, x subspace lattice, 49 subvaluation, 75 super set, x surjective, 238, 251, 253, 255 symmetric, 3, 7, 79, 149, 153, 192, 194, 194, 195, 248, 248, 249 symmetric difference, x symmetry, 45, 196 ternary, 138, 250 there exists, xi topology of sets, xi totally ordered, 47, 50 transitive, 3, 7, 25, 45, 46, 48, 59, 60, 119, 120, 224, 248, 248, 249 translation invariant, 280, 282 triangle inequality, 30, 36, 153, 265 true, x, 8, 9, 185 unary, 202, 250 unbounded, 71 union, x uniquely comp., 185 uniquely complemented, 90, 109, 110–112, 147, 221 universal quantifier, xi, 9 upper bound, 161, 192, 193 upper bounded, 71, 191 valuation, 75, 155 vector norm, xi weak double negation, 158, 161, 163–168, 192 weak entailment, 174, 178, 181 weak modus ponens, 174–178, 181, 182, 184 width, 55, 57 propositional logic, 9 pstricks, vi quantum implication, 175, Daniel J. Greenhoe

183 quotes Abel, Niels Henrik, 287 Cantor, Georg, 4 de Morgan, Augustus, 115 Dedekind, Richard, 14, 24 Descartes, René, ix Euler, Leonhard, 31 Fréchet, Marice René, 11 Frege, Gottlob, 15 Halmos, Paul R., 8 Housman, Alfred Edward, vii Jevons, William Stanley, 173 Kaneyoshi, Urabe, 288 Kenko, Yoshida, 288 Kronecker, Leopold, 23 Leibniz, Gottfried, ix, 173 Machiavelli, Niccolò, 287 Poincaré, Jules Henri, 8 Proclus Lycaeus, 13 Russull, Bertrand, vii, 14, 187 Stravinsky, Igor, vii quotient structures, 265 range, x, 243, 244 rational number, 23, 23 rational numbers, 13 real number, 24 real numbers, 13, 24 real part, xi, 273 real valued lattice, 153 rectangular coordinates, 32 reflexive, 3, 7, 25, 45, 46, 59, 60, 80, 81, 84, 119, 224, 248, 248, 249 reflexive ordering relation, xi rejection, 124, 124, 126–128, 187, 206, 206 relation, x, 3, 7, 7, 8, 10, 79, 89, 165, 201, 238, 250 anti-symmetric, 249 equivalence, 7 inverse, 241 reflexive, 249 symmetric, 249 transitive, 249 relational and, x, 9 relations, xi, 3, 260 classical implication, 168, 175, 182 commutes, 149, 193, 193, 194, 196 complement, 165 converse, 241 covering relation, 48


version 0.30

page 336 Dishkant

SUBJECT INDEX implication,

175 distributivity, 89 domain, 243 dual, 46 dual distributivity, 89 dual modularity, 79, 149 function, 165 horseshoe, 175 identity element, 248 image set, 243 implication, 174, 179 inverse, 241 Kalmbach implication, 175 logical implies, 185 material implication, 175 modularity, 79, 149 non-tollens implication, 175 null space, 243 order relation, 45, 48, 49, 265 orthogonality, 149, 192 pointwise order relation, 56 preorder relation, 45 quantum implication, 175 range, 243 relation, 165 relevance implication, 175 Sasaki hook, 175 relative complement, 109, 109 relatively complemented, 109 relevance implication, 175, 178 resolvent, 271 right distributive, 19, 34, 134, 260, 261, 282, 283 ring of sets, xi, 202, 207, 211, 215, 215, 216, 220, 222, 223, 231, 232, 234 rings of sets, 93, 215, 216, 220, 232 RM3 logic, 166, 174, 181, 181 Russell's Paradox, 4 Russull, Bertrand, vii, 14, 187 Sasaki hook, 121, 175, 183 Sasaki hook logic, 174, 183 Sasaki implication, 175 Sasaki projection, 171, 171, 172, 193, 195 sasaki projection, 172 Sasaki projection of 𝑦 onto 𝑥, 171 scalar multiplication, 259 version 0.30

Schwartz class, 262, 262 self-adjoint, 272 self-dual, 63, 68 semilattice join, 59 meet, 59, 60 semilinear, 39, 273 semimodular, 80, 86, 87 Serpiński spaces, 212 set, 3, 4, 4, 266 power, 5, 201 rational numbers, 23 ring, 215 symmetric, 285 set complement, 109 set difference, 3, 46, 206 set equality, 10 set function, 201 set inclusion, 10, 224 set intersection operator, 6 set non-equality, 10 set of algebras of sets, xi set of rings of sets, xi set of topologies, xi set operations, 3 set structure, 201, 201, 202, 216, 220–224 set structures algebra of sets, 221 pre-topology, 220 set symmetric difference operator, 6 sets, 3 operations, 10, 202, 204 ordered set, 49 positive integers, 45 real numbers, 24 Sheffer stroke, 124, 124, 126– 128, 187, 206, 206 Sheffer stroke functions, 185 size, 13 smallest algebra, 214 Sophist, 157 space, 11 metric, 153, 154 topological, 211 span, xi spectral radius, 271 spectrum, 271 star algebra, 38 star-algebra, 40, 272 Stone, 131 Stone Representation Theorem, 214, 221 Stravinsky, Igor, vii strict, 159 strict negation, 159, 159 strictly antitone, 159 strictly convex, 267, 267 strong, 159 strong entailment, 174–178,


Daniel J. Greenhoe

180–182, 184 strong modus ponens, 174, 175, 180–182, 184 strong negation, 159 strongly connected, 248 structures 𝐿1 lattice, 141 𝐿2 lattice, 141 𝐿22 lattice, 141 𝐿32 lattice, 141 𝐿42 lattice, 141 𝐿52 lattice, 141 𝑀2 lattice, 183 𝑀4 lattice, 141 𝑀6 lattice, 141 𝑂6 lattice, 140, 142, 147, 148, 172, 192, 197 𝑂8 lattice, 140 ℝ3 Euclidean space, 193 ∗-algebra, 272–274 Łukasiewicz 3-valued logic, 181 Łukasiewicz 5-valued logic, 182 σ-algebra, 202, 214 σ-ring, 202, 215 3-dimensional Euclidean space, 172 algebra, 271, 272 algebra of sets, 202, 207, 211, 214, 214, 220–222, 231, 232 algebraic structure, 142 algebras of sets, 214, 229, 232 anti-chain, 57 antichain, 54, 57 ball, 266 Banach space, 12 binary operation, 202 BN4 logic, 183 Boolean 4-valued logic, 183 Boolean algebra, 57, 71, 115, 115, 116, 120, 121, 131, 132, 133, 134, 136, 139, 146, 147, 149, 185, 215, 221 boolean algebra, 146 Boolean algebras, 9 boolean algebras, 214 Boolean lattice, 115, 115, 171, 175–179, 192 Boolean logic, 179, 179, 180 bounded lattice, 72, 109, 115, 116, 140, 146, 150, 154, 157–159, 161–164, 171, 174, 192–194 bounded lattices, 191, 192 center, 146, 147, 196, 2016 December 27 Tuesday UTC

page 337

SUBJECT INDEX 196–199 chain, 47, 54, 57, 61, 72 classic logic, 179 classical 2-value logic, 184, 185 classical 2-valued logic, 185 classical bi-variate logic, 180 classical logic, 180 closed ball, 154 closed interval, 266, 266 closed unit ball, 154, 155 complement, 165 complemented lattice, 109, 142 complements, 142, 143, 227 convex function, 267 convex set, 266, 266, 267 de Morgan logic, 179, 179 de Morgan negation, 158, 167 diamond, 93 discrete topology, 211, 227 distributive lattice, 100, 212, 221 distributive lattices, 89 epigraph, 267 Euclidean space, 142, 266 fully ordered set, 47 function, 157, 158, 250, 251 functional, 267 functions, 8 fuzzy logic, 179, 179 fuzzy negation, 158 greatest lower bound, 146, 224 group, 278 half-open interval, 266 Hasse diagram, 49, 204, 225 Heyting 3-valued logic, 182 Hilbert space, 12, 142 hypograph, 267 indiscrete topology, 211, 227 inner-product space, 12 interval, 265, 266 intuitionalistic logic, 179, 179 intuitionalistic negation, 158 intuitionistic negation, 166 join semilattice, 59 2016 December 27 Tuesday UTC

Kleene 3-valued logic, 180 Kleene logic, 179 Kleene negation, 158, 170 largest algebra, 214 lattice, 5, 58, 61, 62, 65, 68, 70, 82, 89, 115, 142, 146, 153, 154, 160, 161, 163, 174, 176, 185, 191, 192, 194, 265, 266 lattice of partitions, 230 lattice of topologies, 226, 227 lattice with negation, 159, 175, 179, 191–194, 196 lattice with ortho negation, 184 lattices, 9, 178, 192, 214 least upper bound, 146, 224 linear space, 12, 265– 267, 271 linearly ordered set, 47, 265 logic, 173, 179, 179 logical and, 202 logical exclusive-or, 202 logical not, 202 logical or, 202 lower bound, 70 M3 lattice, 93, 111 measure space, 12 meet semilattice, 60 metric lattice, 154 metric space, 11, 12, 154, 265, 266 minimal negation, 158, 159, 165 MO2 lattice, 149 modular lattice, 84, 86 modular orthocomplemeted lattice, 150 N5 lattice, 82, 111 negation, 169 normed vector space, 12 O6 lattice, 140, 140, 165, 175 O6 lattice with ortho negation, 183 O6 orthocomplemented lattice, 183 open ball, 154, 154 open interval, 266 order, 9 order relation, 47 order structures, 72 ordered pair, 6, 10 ordered set, 45, 57, 58, 72, 265, 266 ortho lattice, 174–178, Daniel J. Greenhoe

191, 192 ortho logic, 179, 179 ortho negation, 158, 166, 168–170, 176, 192–194 Orthocomplemented lattice, 139 orthocomplemented lattice, 140, 140, 142, 143, 146, 148, 149, 171, 172, 191– 194, 196, 197 orthocomplemented lattices, 140 orthocomplemented O6 lattice, 175 orthomodular lattice, 148, 149, 175, 178 orthomodular negation, 158 partially ordered set, 45, 59 partition, 7, 202, 216, 216, 220 partitions, 211 paving, 201, 202 pentagon, 82 poset, 45 power set, 5, 10, 58, 201, 202, 211, 214, 215 real valued lattice, 153 relation, 7, 8, 10, 79, 89, 250 relations, 260 ring of sets, 202, 207, 211, 215, 215, 216, 220, 222, 223, 231, 232 rings of sets, 215, 216, 232 RM3 logic, 181 Sasaki hook logic, 183 Serpiński spaces, 212 set, 266 set structure, 201, 202, 216, 220–224 smallest algebra, 214 subminimal negation, 158 topologies, 211, 212, 226 topology, 202, 211, 212, 215, 220, 227, 231 topology on finite set, 202 totally ordered set, 47 trivial topology, 211 unit ball, 154 unital ∗-algebra, 272 unital algebra, 271 unlabeled lattices, 67 upper bound, 70 vectors, 265 weights, 268 subadditive, 30, 36, 153


version 0.30

page 338

SUBJECT INDEX

subminimal, 167 subminimal negation, 157, 158, 164, 165 subposet, 54 subset, x, 10 subspace lattice, 49 subvaluation, 75, 75 successor, 13, 14, 14, 15, 17, 20 sum, 15, 15, 16, 19 summation operator, 268 super set, x supremum, 58 surjective, xi, 238, 251, 252, 253–255 symmetric, 3, 7, 79, 149, 153, 192, 194, 194, 195, 248, 248, 249 symmetric difference, x, 3, 10, 187, 203, 206, 206 symmetric set, 285 symmetry, 45, 196 tempered distribution, 262 ternary, 138, 250 ternary rejection, 138 test function, 262 The Book Worm, 287 the law of the excluded middle, 257 theorems algebraic ring properties of rings of sets, 223 Archimedean Property, 27 Bernstein-CantorSchröder, 254 Birkhoff distributivity criterion, 90, 94, 94, 213 Birkhoff's Theorem, 105 Byrne's FORMULATION A, 134 Byrne's FORMULATION B, 136 Byrne's Formulation A, 131 Byrne's Formulation B, 131 cancellation criterion, 97 Cancellation property, 90 classic 10, 116 classic 10 Boolean properties, 120 de Morgan's Law, 221 Dilworth, 54 Dilworth's theorem, 57, 111 distributive inequalities, 65 Elkan's law, 71 version 0.30

Eudoxus axiom, 27 Gelfand-Mazur, 275 Huntington properties, 111 Huntington's FIRST SET, 116 Huntington's fifth set, 131, 133 Huntington's first set, 131, 131 Huntington's Fourth Set, 136, 146 Huntington's fourth set, 131, 132, 133 Jensen's Inequality, 267 Lattice characterization in 2 equations and 5 variables, 70 Lattice characterizations in 1 equation, 70 Median property, 90 minimax inequality, 64, 65 Modular inequality, 65 Monotony laws, 63 Peirce's Theorem, 112, 113 Pointwise ordering relation, 53 Principle of duality, 63, 116, 146 Stone, 131 Stone Representation Theorem, 214, 221 there exists, xi top, 187 topological space, 211 topologies, 211, 212, 226, 233 discrete, 211 indiscrete, 211 number of, 218 trivial, 211 topology, 202, 211, 211, 212, 215, 220, 227, 231 discrete, 225 indiscrete, 225 trivial, 225 topology of sets, xi topology on a finite set, 211 topology on finite set, 202 totally ordered, 47, 50 totally ordered set, 21, 25, 47 transfer 𝑥, 125, 187 transfer 𝑦, 125, 187 transfer x, 186 transfer y, 186 transformation, 249 transitive, 3, 7, 25, 45, 46, 48, 59, 60, 119, 120, 224, 248, 248, 249 translation invariant, 280,


Daniel J. Greenhoe

282 translation operator, 277, 278 translation space, 277 triangle inequality, 30, 36, 153, 265 trivial topology, 211 true, x, 3, 4, 8, 9, 180, 183– 185, 187 truth table, 6 unary, 202, 250 unbounded, 71 undecided, 180 union, x, 3, 10, 187, 203, 206, 206, 224 unique complement, 109 uniquely comp., 185 uniquely complemented, 90, 109, 110–112, 147, 221 unit ball, 154 unital, 271 unital ∗-algebra, 272 unital algebra, 271 universal quantifier, xi, 9 universal set, 187, 203, 206 unlabeled, 51 unlabeled lattices, 67 upper bound, 57, 58, 58, 70, 161, 192, 193 upper bounded, 71, 71, 191 Utopia, vi valuation, 75, 75, 153, 153– 155 values GLB, 58 LUB, 58 both, 183 false, 180, 183, 184 greatest lower bound, 58 infimum, 58 least upper bound, 58, 72, 174 lower bound, 58, 58 neither, 183 neutral, 180 orthocomplement, 140, 140 true, 180, 183, 184 undecided, 180 upper bound, 58, 58 vector norm, xi vector spaces, 249 vectors, 265 weak double negation, 158, 161, 163–168, 192 weak entailment, 174, 178, 181 weak modus ponens, 174– 178, 181, 182, 184 2016 December 27 Tuesday UTC

page 339

SUBJECT INDEX weights, 268 whole numbers, 13, 22 width, 54, 55, 57 Wáng Hàn Zōng Zhōng Mińg


Tǐ Fán, vi Wáng Hàn Zōng Zhōng Fǎng Sòng Fán, vi Z-Transform, xii

Daniel J. Greenhoe

zahlen, 22 Zermelo-Fraenkel (ZF) set theory, 4 zero, 186, 187


version 0.30

page 340

LAST PAGE

…last page …please stop reading … version 0.30


Daniel J. Greenhoe


Sets, Relations, and Order Structures [version 0.30]

Sets, Relations, and Order Structures [version 0.30]

Suggest Documents

&030#Buy; 'MSN Webcam Recorder' Full Version

Elsevier Order Sets

Basic Structures: Sets, Functions, Sequences, and Sums

EVALUATION SETS AND MAPPINGS: THE ORDER

EVALUATION SETS AND MAPPINGS: THE ORDER ... - CiteSeerX

+030> Online Read Baby Bargains (Version 12.0- released 2017 ...

interpolative relations and interpolative preference structures - doiSerbia

T-030

030#Get; 'Kingdom Tales 2' Cracked Version - Google URL Shortener

and second-order relations - Jennifer Vonk

grammatical relations and word order in

12-030

Relations on Intuitionistic Fuzzy Soft Sets

SAP2000 Version 16.0.0 - Computers and Structures, Inc.

Discrete Mathematics, Chapters 2 and 9: Sets, Relations and ...

Discrete Mathematics, Chapters 2 and 9: Sets, Relations and ...

Order Structures and Generalisations of Szpilrajn's Theorem

Synopsis Data Structures for Massive Data Sets

STRUCTURES OF GENERALIZED FUZZY SETS IN NON

Order Sets in Computerized Physician Order Entry ... - Semantic Scholar

nested intervals and sets: concepts, relations to ... - Semantic Scholar

Observability don't care sets and Boolean relations - Computer ... - EPFL

RELATIONS BETWEEN (Îº, Ï)-REGULAR SETS AND STAR ...

Regular sets of higher-order pushdown stacks

Sets, Relations, and Order Structures [version 0.30]