MRA-Wavelet subspace architecture for logic

ve l e a MRA-W t subspace architecture for logic, probability, and symbolic sequence processing Daniel J. Greenhoe Tuesday 14th February, 2017 Abstract The linear subspaces of a multiresolution analysis (MRA) and the linear subspaces of the wavelet analysis induced by this MRA, together with the set inclusion relation ⊆, form a very special lattice of subspaces which herein is called a primorial lattice. This paper introduces an operator 𝐑 that extracts a set of 2𝘕 −1 element Boolean lattices from a 2𝘕 element Boolean lattice. Used recursively, a sequence of Boolean lattices with decreasing order is generated—a structure that is similar to an MRA. A second operator, which is a special case of a “difference operator”, is introduced that operates on consecutive Boolean lattices 𝙇𝑛2 and 𝙇𝑛−1 to produce a sequence 2 of orthocomplemented lattices. These two sequences, together with the subset ordering relation ⊆, form a primorial lattice ℙ. A logic or probability constructed on a Boolean lattice 𝙇𝘕2 likewise induces a primorial lattice ℙ. Such a logic or probability can then be rendered at 𝘕 different “resolutions” by selecting any one of the 𝘕 Boolean lattices in ℙ and at 𝘕 different “frequencies” by selecting any of the 𝘕 different orthocomplemented lattices in ℙ. Furthermore, ℙ can be used for symbolic sequence analysis by projecting sequences of symbols onto the sublattices in ℙ using one of three lattice projectors introduced. ℙ can be used for symbolic sequence processing by judicious rejection and selection of projected sequences. Examples of symbolic sequences include sequences of logic values, sequences of probabilistic events, and genomic sequences (as used in “genomic signal processing ”).

Contents 1 Introduction 1.1 Analyses for linear spaces . . . 1.2 Multiresolution analyses . . . . 1.3 Wavelet analyses . . . . . . . . . 1.4 Primoral lattices . . . . . . . . . 1.5 Applications of primoral lattices

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2 Primoral lattice subspace structure 2.1 Primorial Lattices . . . . . . . . . . . . . 2.2 Reduction operator on boolean lattices . 2.3 Difference operator on bounded lattices 2.4 Projections on primorial lattices . . . . .

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3 3 3 5 6 6

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6 7 9 10 12

3 A generalized probability function 17 3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1

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4 Applications 4.1 Logic analysis . . . . . . . . . . . . . 4.2 Fuzzy logic analysis . . . . . . . . . . 4.3 Probability analysis . . . . . . . . . . 4.4 Symbolic sequence analysis . . . . . 4.5 Symbolic sequence processing (SSP) 4.6 Genomic Signal Processing (GSP) . .

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20 20 21 22 22 24 25

A Order structures A.1 Order . . . . . . . . . . . . . . . . . . . . . . . . . A.1.1 Order relations . . . . . . . . . . . . . . . A.1.2 Representation . . . . . . . . . . . . . . . A.1.3 Decomposition . . . . . . . . . . . . . . . A.1.4 Decomposition examples . . . . . . . . . A.2 Lattices . . . . . . . . . . . . . . . . . . . . . . . . A.2.1 Definition . . . . . . . . . . . . . . . . . . A.2.2 Bounded lattices . . . . . . . . . . . . . . A.2.3 Atomic lattices . . . . . . . . . . . . . . . . A.2.4 Modular Lattices . . . . . . . . . . . . . . A.2.5 Distributive Lattices . . . . . . . . . . . . A.2.6 Complemented lattices . . . . . . . . . . . A.2.7 Boolean lattices . . . . . . . . . . . . . . . A.3 Orthocomplemented Lattices . . . . . . . . . . . A.3.1 Definition . . . . . . . . . . . . . . . . . . A.3.2 Properties . . . . . . . . . . . . . . . . . . A.3.3 Restrictions resulting in Boolean algebras A.3.4 Orthomodular lattices . . . . . . . . . . .

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27 27 27 28 28 30 32 32 34 35 35 36 38 39 40 40 42 43 44

B Functions on lattices B.1 Valuations . . . . . . . . . . . B.2 Negation . . . . . . . . . . . . B.2.1 Definitions . . . . . . . B.2.2 Properties of negations B.3 Projections . . . . . . . . . . . B.4 Logics . . . . . . . . . . . . . .

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C Relations on lattices C.1 Orthogonality . C.2 Commutativity . C.3 Center . . . . . . C.4 D-Posets . . . .

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D MRA-wavelet subspace structure D.1 Transversal Operators . . . . . D.2 The Structure of Wavelets . . . D.3 Multiresolution analysis . . . D.4 Wavelet analysis . . . . . . . . D.5 Fast Wavelet Transform (FWT)

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References

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Reference Index

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vel MRA-Wa et subspace architecture …

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Table of Contents

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Subject Index

77

License

84

1 Introduction 1.1 Analyses for linear spaces An analysis of a linear space 𝙓 is any sequence ⦅𝙑𝑗 ⦆𝑗∈ℤ of linear subspaces of 𝙓 . The partial or

complete reconstruction of 𝙓 from ⦅𝙑𝑗 ⦆𝑗∈ℤ is a synthesis.1 Some analyses are completely characterized by a transform. For example, a Fourier analysis is a sequence of subspaces with sinusoidal bases. Examples of subspaces in a Fourier analysis include 𝙑1 = 𝗌𝗉𝖺𝗇{𝑒𝑖𝑥 }, 𝙑2.3 = 𝗌𝗉𝖺𝗇{𝑒𝑖2.3𝑥 }, 𝙑√2 = 𝗌𝗉𝖺𝗇{𝑒𝑖√2𝑥 }, etc. A transform is loosely defined as a function that maps a family of functions into an analysis. A very useful transform (a “Fourier transform”) for Fourier Analysis is 1 𝖿(𝑥)𝑒−𝑖𝜔𝑥 𝖽𝑥 [𝐅𝖿̃ ](𝜔) ≜ ∫ √2𝜋 ℝ An analysis can be partially characterized by its order structure scaling subspace with respect to an order relation such as the set inclusion relation ⊆. Most transforms have a very simple M-𝑛 order structure, as illustrated to the right. The M-𝑛 lattices for 𝑛 ≥ 3 are modular 𝙑0 𝙑1 but non-distributive. Analyses typically have one subspace that is a scaling subspace; and this subspace is often simply a family of constants (as is the case with Fourier Analysis).2

analysis of 𝙓 𝙓 𝙑2

⋯

𝙑𝑛−1

𝟬

Some examples of the order structures of some analyses are illustrated in Figure 1 (page 4) and Figure 4 (page 8).

1.2 Multiresolution analyses In 1989, Stéphane G. Mallat introduced the Multiresolution Analysis (MRA). 3 1

The word analysis comes from the Greek word ἀvάλυσις, meaning “dissolution” (📘 Perschbacher (1990), page 23 ⟨entry 359⟩), which in turn means “the resolution or separation into component parts” (📘 Black et al. (2009), http: //dictionary.reference.com/browse/dissolution) 2

transform structure reference: 📘 Greenhoe (2013) page 29 ⟨§2.2⟩. The M-𝑛 lattices for 𝑛 ≥ 3 are modular: Lemma A.56 (page 37). The M-𝑛 lattices for 𝑛 ≥ 3 are non-distributive: Theorem A.57 (page 37). 3 📘 Mallat (1999) page 240, Definition D.12 (page 58)



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1 INTRODUCTION

Cosine analysis (even Fourier series)

Cosine polynomial analysis

𝙓 = 𝗌𝗉𝖺𝗇{cos(2𝜋𝑛𝑥)|𝑛 = 0, 1, 2, 3}

𝙓 = 𝗌𝗉𝖺𝗇{cos𝑛 (2𝜋𝑥)|𝑛 = 0, 1, 2, 3}

scaling subspace

𝟬

𝟬

scaling subspace

Chebyshev polynomial analysis

Hadamard-3 analysis

𝙓 = 𝗌𝗉𝖺𝗇{𝑇 𝑛 (𝑥)|𝑛 = 0, 1, 2, 3}

𝙓 = 𝗌𝗉𝖺𝗇𝐻3

scaling subspace

𝟬

𝟬

scaling subspace

Figure 1: Examples of order structures for selected analyses A multiresolution analysis together with the set inclusion relation ⊆ form the linearly ordered set (⦅𝙑𝑗 ⦆ , ⊆), illustrated to the right by a Hasse diagram. Subspaces 𝙑𝑗 increase in “size” with increasing 𝑗. That is, they contain more and more vectors (functions) for larger and larger 𝑗—with the upper limit of this sequence being 𝙇𝟤ℝ . Alternatively, we can say that approximation within a subspace 𝙑𝑗 yields greater “resolution” for increasing 𝑗.4

𝙇𝟤ℝ

entire linear space

⋮

𝙑2 𝙑1 𝙑0 𝙑−1

larger subspaces smaller subspaces

⋮ 𝟬

smallest subspace

A multiresolution analysis provides “coarse” approximations of a function in a linear space 𝙇𝟤ℝ at multiple “scales” or “resolutions”. Key to this process is a sequence of scaling functions. Most traditional transforms feature a single scaling function 𝜙(𝑥) set equal to one (𝜙(𝑥) = 1). This allows for convenient representation of the most basic functions, such as constants.5 A multiresolution system, on the other hand, uses a generalized form of the scaling concept:6 1. Instead of the scaling function simply being set equal to unity (𝜙(𝑥) = 1), a multiresolution analysis is often constructed in such a way that the scaling function 𝜙(𝑥) forms a partition of unity such that ∑𝑛∈ℤ 𝜙(𝑥 − 𝑛) = 1. 2. Instead of there being just one scaling function, there is an entire sequence of scaling functions ⦅2𝑗/2 𝜙(2𝑗 𝑥)⦆𝑗∈ℤ , each corresponding to a different “resolution”. 4

References: 📘 Michel and Herget (1993) page 83 ⟨Theorem 3.2.12⟩, 📘 Kubrusly (2001) page 67 ⟨Theorem 2.14⟩, 📘 Greenhoe (2013) page 38 ⟨§2.3.2 Order structure⟩, 📘 Greenhoe (2014a) ⟨Theorem 7.1⟩. multiresolution analysis: Definition D.12 (page 58). multiresolution analysis: Definition D.12 (page 58). linearly ordered set: Definition A.4 (page 27). Hasse diagram: Definition A.6 (page 28). 5 📃 Jawerth and Sweldens (1994) page 8 6 The concept of a scaling space was perhaps first introduced by Taizo Iijima in 1959 in Japan, and later as the Gaussian Pyramid by Burt and Adelson in the 1980s in the West. 📘 Mallat (1989) page 70, 📘 Iijima (1959), 📘 Burt and Adelson (1983), 📘 Adelson and Burt (1981), 📘 Lindeberg (1993), 📘 Alvarez et al. (1993), 📘 Guichard et al. (2012), 📘 Weickert (1999) ⟨historical survey⟩ CC BY 4.0


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1

square pulse: −5

−4

−3

−2

−1

0

1

2

3

4

5

0

1

2

3

4

5

1

3 2

2

1

tent function: −5

−4

−3

−2

−1

1 1 2

raised cosine: −2

− 32

−1

1+𝛽 −2

− 12

1−𝛽 2

1−𝛽 −2

1 2

1+𝛽 2

1

sine squared: −5

−4

−3

−2

−1

0

1

2

3

4

5

0

1

2

3

4

5

1

centered B-spline: −5

−4

−3

−2

−1

Figure 2: Some illustrations of partition of unity See Figure 2 (page 5) for some illustrations of functions with the partition of unity property.7 See Figure 4 (page 8) for some illustrations of scaling functions.

1.3 Wavelet analyses The term “wavelet” comes from the French word “ondelette”, meaning “small wave”. And in essence, wavelets are “small waves” (as opposed to the “long waves” of Fourier analysis) that form a basis for 𝙇𝟤ℝ .8 One method for wavelet construction is by use of the Multiresolution Analysis (MRA). And since its introduction in 1989 by Stéphane G. Mallat, the MRA has become the dominate method for wavelet construction. Furthermore, P.G. Lemarié has proved that all wavelets with compact support are generated by an MRA.9 7

For further information about partition of unity and splines, see 📃 Greenhoe (2014c). 📘 Strang and Nguyen (1996) page ix, 📘 Atkinson and Han (2009) page 191 9 📃 Lemarié (1990), 📘 Mallat (1999) page 240 8



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2 PRIMORAL LATTICE SUBSPACE STRUCTURE

Primoral lattices

𝙑0

𝙑2

𝙇𝟤ℝ

⋯

su g in al sc

📘 Greenhoe (2013) has demonstrated that an MRA together with the wavelet system that it generates forms a very special subspace lattice structure, as illustrated to the right. In this paper, this type of lattice is called the primorial lattice.10 Some specific MRA-wavelet subspace primoral lattices are illustrated in Figure 4 (page 8).

bs pa c

es

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Daniel J. Greenhoe

𝙑1

wavelet subspaces 𝙒0

𝙒1

⋯

𝟬

1.5

Applications of primoral lattices

This paper seeks to further exploit the primoral lattice structure by applying it to other fields including logic, fuzzy logic, probability, symbolic sequence analysis, and symbolic sequence processing.11 In particular, this paper introduces a reduction operator 𝐑 that extracts a set of 2𝘕 −1 element Boolean lattices from a 2𝘕 element Boolean lattice. Used recursively, a sequence of Boolean lattices with decreasing order is generated—a structure that is similar to an MRA. A second operator⦸, which is a special case of a “difference operator”, is introduced that operates on consecutive Boolean lattices 𝙇𝑛2 and 𝙇𝑛−1 to produce a sequence of orthocomplemented lattices. These two sequences, together 2 with the subset ordering relation ⊆, form a primorial lattice ℙ.12 A logic or probability constructed on a Boolean lattice 𝙇𝘕 2 likewise induces a primorial lattice ℙ. Such a logic or probability can then be rendered at 𝘕 different “resolutions” by selecting any one of the 𝘕 Boolean lattices in ℙ and at 𝘕 different “frequencies” by selecting any of the 𝘕 different orthocomplemented lattices in ℙ. Furthermore, ℙ can be used for symbolic sequence analysis by projecting sequences of symbols onto the sublattices in ℙ using one of three lattice projectors introduced. ℙ can be used for symbolic sequence processing by judicious rejection and selection of projected sequences. Examples of symbolic sequences include sequences of logic values, sequences of probabilistic events, and genomic sequences (as used in “genomic signal processing ”).

2

Primoral lattice subspace structure

This section formally defines the primoral lattice structure (next definition), provides some examples (Example 2.2–Example 2.5), lists some simple properties of primoral lattices (Proposition 2.6), and introduces two operators (Section 2.2 and Section 2.3) on Boolean lattices that can be used to produce a primoral lattice. 10

Reference: 📘 Greenhoe (2013) page 72 ⟨Section 2.4.3 Order structure⟩. primorial lattice: Definition 2.1 (page 7). logic: Section 4.1 (page 20)., fuzzy logic: Section 4.2 (page 21). probability: Section 4.3 (page 22). symbolic sequence analysis: Section 4.4 (page 22). symbolic sequence processing: Section 4.5 (page 24). 12 reduction operator 𝐑: Definition 2.7 (page 9). bounded lattice difference operator ⦸: Definition 2.13 (page 10). difference operator: Definition C.18 (page 54). orthocomplemented lattice: Definition A.72 (page 40). 11

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um be

rs

2.1 Primorial Lattices

im

or

ia

𝑦3 = 𝑦 2 ∨ 𝑥 2

pr

𝑦2 = 𝑦 1 ∨ 𝑥 1 𝑦1 = 𝑦 0 ∨ 𝑥 0 𝑦0

𝑥0

𝑥1

𝑥𝘕 −1

𝑥2

2310

ln

𝑦𝘕 = 𝑦𝘕 −1 ∨ 𝑥𝘕 −1

210 30

6 3

2

7

5

atoms

11

prime numbers 1

0

(B) p-lattice of primorial and prime numbers ce

s

(A) general p-lattice

𝙇𝟤ℝ

in

g

su

𝑦3 ≜ 𝑦2 ∪ (3 + 5ℤ)

sc

al

𝑦2 ≜ 𝑦1 ∪ (2 + 5ℤ) 𝑦1 ≜ (0 + 5ℤ) ∪ (1 + 5ℤ) cosets/partition of ℤ

𝙑0

𝙑2

⋯

bs

pa

ℤ = 𝑦 3 ∪ 𝑦4

𝙑1

wavelet subspaces 𝙒0

𝙒1

⋯

4

3

2

1

0

+

+

+

+

+

5ℤ

5ℤ

5ℤ

5ℤ

5ℤ

∅

𝟬

(C) p-lattice of cosets

(D) p-lattice of MRA and wavelet subspaces

Figure 3: Some selected primorial lattices (see Example 2.2 page 7–Example 2.5 page 7)

2.1 Primorial Lattices Definition 2.1. Let 𝑋 ≜ {0, 𝑥0 , 𝑥1 , … , 𝑥𝘕 , 𝑦0 , 𝑦1 , … , 𝑦𝘕 } be a set. A lattice 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) is primorial if 1. 0 is the least element of 𝙇 2. 𝙇 is atomic (Definition A.44 page 35) and {𝑦0 , 𝑥0 , 𝑥1 , … , 𝑥𝘕 } are atoms of 𝙇 3. 𝑦𝑛+1 = 𝑦𝑛 ∨ 𝑥𝑛 . A lattice that is primorial is a primorial lattice, or simply a p-lattice.

and and

Example 2.2. A general primorial lattice is illustrated to in Figure 3 page 7 (A). Example 2.3. 13 The set of primorial numbers and prime numbers ordered by the divides (“|”) relation forms a primorial lattice, as illustrated in Figure 3 page 7 (B). Example 2.4. Any partition, along with successive unions of the partition elements, generates a primorial lattice. One example of this is the cosets of ℤ, which generate a finite primorial lattice, as illustrated in Figure 3 page 7 (C). Example 2.5. A special characteristic of MRA-wavelet analysis is that its order structure with respect to the ⊆ relation is not a simple M𝑛 lattice (as is with the case of Fourier and several other analyses). Rather, it is a primorial lattice as illustrated in Figure 3 page 7 (D) and in Figure 4 page 8. Proposition 2.6.

14

Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice.

13

📘 Greenhoe (2013) page 30, 💻 Sloane (2014) ⟨http://oeis.org/A002110⟩ 📘 Greenhoe (2013) page 72 ⟨Section 2.4.3 Order structure⟩ 14 📘 Greenhoe (2013) page 52 ⟨Proposition 2.6⟩

13



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Haar/Daubechies-𝑝1 wavelet analysis

Daubechies-𝑝2 wavelet analysis

scaling subspaces

scaling subspaces

𝟬

𝟬

Figure 4: some MRA-wavelet systems (Example 2.5 page 7) ⎧ ⎪ ⎪ 𝙇 is ⎪ ⟹ ⎨ { primorial } ⎪ ⎪ ⎪ ⎩

1. 2. 3. 4. 5. 6.

𝙇 is nondistributive 𝙇 is nonmodular 𝙇 is complemented ⟺ 𝙇 is finite 𝙇 is not uniquely complemented 𝙇 is nonorthocomplemented 𝙇 is nonBoolean

(Definition A.53 page 36)

and


and


and


and


and


.

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

✎PROOF: 1. Proof that 𝙇 is nondistributive: (a) 𝙇 contains the N5 lattice (Definition A.49 page 36). (b) Because 𝙇 contains the 𝑁5 lattice, 𝙇 is nondistributive (Theorem A.57 page 37). 2. Proof that 𝙇 is nonmodular and nondistributive: (a) 𝙇 contains the 𝑁5 lattice (Definition A.49 page 36). (b) Because 𝙇 contains the 𝑁5 lattice, 𝙇 is nonmodular

(Theorem A.50 page 36).

3. Proof that 𝙇 is noncomplemented: ′

′

𝑣

′

𝑥 =𝑦 =𝑣 =𝑧 𝑧′ = {𝑥, 𝑦, 𝑣} 𝑥″ = (𝑥′ )′ = 𝑧′ = {𝑥, 𝑦, 𝑣} ≠ 𝑥

1 𝑦

𝑥

𝑧 0

4. Proof that 𝙇 is nonBoolean: (a) 𝙇 is nondistributive (item (1) page 8). (b) Because 𝙇 is nondistributive, it is nonBoolean (Definition A.69 page 39).

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2.2 Reduction operator on boolean lattices

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Daniel J. Greenhoe

2.2 Reduction operator on boolean lattices Definition 2.7. Let 𝔹 be the set of all bounded lattices (Definition A.39 page 34). Let 𝙇𝘕 2 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) 𝘕 be a Boolean lattice (Definition A.69 page 39) with 2 elements and 𝘕 ∈ ℕ (𝘕 is a positive integer). 𝘕 𝘕 The operator 𝐑 is the lattice reduction operator of 𝙇𝘕 2 and 𝐑𝙇2 is the reduction of 𝙇2 if ⎧ ⎪ ⎪ 𝐑𝙇𝘕 2 ≜ ⎨𝙇 ∈ 𝔹| ⎪ ⎪ ⎩

1. 2. 3. 4.

𝙇 is a 2𝘕 −1 element Boolean lattice 𝙇 ⊆ 𝙇𝘕 2 {0, 1} ∈ 𝙇 {𝑥, 𝑦} is an orthocomplemented pair in 𝙇 ⟹ {𝑥, 𝑦} is an orthocomplemented pair in 𝙇𝘕 2

and and and

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

𝘕 Note that in Definition 2.7, the order relation ≤ is the same for both 𝙇𝘕 2 and any 𝙇 in 𝐑𝙇2 . That is, if 𝑥 ≤ 𝑦 in 𝙇𝘕 2 , then 𝑥 ≤ 𝑦 in 𝙇 as well.

Example 2.8. Let 𝙇22 be a Boolean lattice (Definition A.69 page 39) of order 2. Let 𝐑 be the lattice reduction operator 𝐑 and 𝐑𝙇22 be the reduction of 𝙇22 (Definition 2.7 page 9). Then 𝐑𝙇22 yields a set of exactly one 22−1 value Boolean lattice, as illustrated next:

1 ⎛ 𝑝 ⎜ 𝐑⎜ 𝑝⟂ ⎜ 0 ⎝

⎞ ⎟ ⎧ ⎪ ⎟=⎨ ⎟ ⎪ ⎩ ⎠

⎫ ⎪ ⎬ 0 ⎪ ⎭ 1

Example 2.9. Let 𝙇32 be a Boolean lattice (Definition A.69 page 39) of order 3. Let 𝐑 be the lattice reduction operator 𝐑 and 𝐑𝙇32 be the reduction of 𝙇32 (Definition 2.7 page 9). The operation 𝐑𝙇32 yields a set of three 22 value Boolean lattices, as illustrated next:

⎛ ⟂ ⟂ 1 ⎜ 𝑟 𝑞 ⎜ 𝐑⎜ 𝑞 ⎜ 𝑝 ⎜ 0 ⎝

⎞ 1 ⎟ ⎧ 𝑝 ⎟ ⎪ ⎟=⎨ 𝑝⟂ 𝑟 ⎟ ⎪ 0 ⎟ ⎩ ⎠

𝑝⟂

1

1

𝑞

,

𝑟 𝑞⟂ 0

,

0

⎫ ⎪ ⟂ ⎬ 𝑟 ⎪ ⎭

Example 2.10. Let 𝙇42 be a Boolean lattice (Definition A.69 page 39) of order 4. Let 𝐑 be the lattice reduction operator 𝐑 and 𝐑𝙇42 be the reduction of 𝙇42 (Definition 2.7 page 9). The operation 𝐑𝙇42 yields a set of ten 23 value Boolean lattices, as illustrated in Figure 5 (page 10). Remark 2.11. In a boolean lattice 𝙇𝘕 2 (Definition A.69 page 39), besides the pair {0, 1}, there are a total of 𝘕 −1 2 −1 orthocomplemented (Definition A.72 page 40) pairs of elements. But note that any arbitrary 2𝘕 −1 −2 pairs of orthocomplemented pairs does not in general generate a boolean lattice. The lattice 𝙇42 , for example, has 24−1 − 1 = 7 orthocomplemented pairs besides {0, 1}. To generate an 𝙇32 lattice, we 7! = 35 ways of selecting 3 pairs from 𝙇4 , but need 3 orthocomplemented pairs. There are (73) = 3!4! 2



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⎛ ⎜ ⎜ ⎜ ⎜ 𝐑⎜ ⎜ ⎜ ⎜ ⎜ ⎝

Daniel J. Greenhoe


1 𝑠⟂ 𝑑

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𝑏⟂

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0

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

Figure 5: reduction of 𝙇42 (Example 2.10 page 9) only 10 of these ways generate a boolean lattice (Example 2.10 page 9). All other ways fail. For example, if we were to select the pairs {0, 𝑤, 𝑤⟂ , 𝑎, 𝑎⟂ , 𝑏, 𝑏⟂ , 1}, we would get the orthocomplemented, but non-boolean (Definition A.69 page 39) lattice illustrated to the right; In particular, it is complemented, but non-distributive. For example, 𝑎 𝑤⟂ ∧ (𝑎 ∨ 𝑏) = 𝑤⟂ ≠ 0 = 0 ∨ 0 = (𝑤⟂ ∧ 𝑎) ∨ (𝑤⟂ ∧ 𝑏). Alternatively, note that the set ⟂ ⟂ {1, 𝑎, 𝑤, 0, 𝑏 , 𝑤 } together with the ordering relation ≤ form an O6 sublattice (Defini- 𝑏⟂ tion A.73 page 41), which contains an N5 sublattice, which implies that the lattice to the right is non-distributive (by the Birkhoff distributivity criterion Theorem A.57 page 37).

1 𝑤⟂ 𝑤

𝑏

𝑎⟂

0

Example 2.12. Let 𝙇52 be a Boolean lattice (Definition A.69 page 39) of order 5. Let 𝐑 be the lattice reduction operator 𝐑 and 𝐑𝙇52 be the reduction of 𝙇52 (Definition 2.7 page 9). The result of the operation 𝐑𝙇52 is partially illustrated in Figure 6 (page 11).

2.3

Difference operator on bounded lattices

Definition 2.13. Let 𝑋 ⧵𝑌 be the standard set difference of a set 𝑋 and a set 𝑌 . Let 𝙇𝑥 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) and 𝙇𝑦 ≜ ( 𝑌 , ∨, ∧, 0, 1 ; ≤) be bounded lattices (Definition A.39 page 34). The bounded lattice difference 𝙇𝑥 ⦸𝙇𝑦 of 𝙇𝑥 and 𝙇𝑦 is the lattice 𝙇 such that 𝙇 ≜ ( (𝑋 ⧵𝑌 ) ∪ {0, 1}, ∨, ∧, 0, 1 ; ≤) Example 2.14. Let⦸be the bounded lattice difference operator (Definition 2.13 page 10). 1 𝑏

𝑎

⦸ 𝑝

𝑞 0

CC BY 4.0

1

1

𝑐

𝑞 𝑏

𝑟

𝑐

𝑎

= 𝑟

𝑝

0 vel MRA-Wa et subspace architecture …

0

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2.3 Difference operator on bounded lattices

⎛ ⎜ ⎜ 𝑡⟂ ⎜ ⎜ ⟂ ⎜ 𝑗 𝑖⟂ ⎜ 𝐑⎜ 𝑏 ⎜ 𝑎 ⎜ ⎜ 𝑝 ⎜ ⎜ ⎜ ⎝ ⎧ ⎪ ⎪ ⎪ ⎪ 𝑎 ⎪ ⎪ ⎪ ⎪ 𝑝 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⟂ ⎪ 𝑗 =⎨ ⎪ ⎪ 𝑞 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 𝑎 ⎪ ⎪ ⎪ 𝑝 ⎪ ⎪ ⎩

1 𝑟⟂

𝑠⟂ 𝑔⟂

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0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

1 ⟂

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1 𝑓⟂

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1 𝑠⟂

𝑝⟂

⟂

𝑐⟂

⟂

0

𝑡⟂

page 11

Daniel J. Greenhoe

…

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

Figure 6: reduction of 𝙇52 (Example 2.12 page 10)



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Proposition 2.15. Let 𝔹 be the set of all bounded lattices (Definition A.39 page 34). Let ⦸ be the bounded lattice difference operator (Definition 2.13 page 10). (𝔹,⦸, ⊆) is a D-poset (Definition C.18 page 54). 𝘕 −1 Theorem 2.16. Let 𝙇 ≜ 𝙇𝘕 be the bounded lattice difference (Definition 2.13 page 10) of a Boolean 2 ⦸ 𝙇2 𝘕 −1 lattice 𝙇2 (Definition A.69 page 39) and a Boolean lattice 𝙇𝘕 selected from the set 𝐑𝙇𝘕 2 2 (Definition 2.7 page 9). 𝑛 𝑛 𝑛−1 Let 𝑋 ≜ {𝙇2 |𝑛 = 1, 2, …} ∪ {𝙇2 ⦸𝙇2 |𝑛 = 2, 3, …}. 𝘕 𝘕 −1 1. 𝙇 ⦸𝙇 is an orthocomplemented lattice (Definition A.72 page 40) and 2 2 2. The structure ℙ ≜ ( 𝑋, ∨, ∧ ; ⊆) is a primorial lattice (Definition 2.1 page 7).

✎PROOF: 1. Proof that 𝙇𝘕2 ⦸ 𝙇𝘕2 −1 is an orthocomplemented lattice: (a) 𝙇𝘕2 is a Boolean lattice by definition. (b) 𝙇𝘕2 −1 is also a Boolean lattice (Definition 2.7 page 9). (c) Every lattice that is Boolean is also orthocomplemented (Proposition A.80 page 43). (d) By definition of 𝙇𝘕2 ⦸𝙇𝘕2 −1 , orthocomplemented pairs are removed from 𝙇𝘕2 and the orthocomplemented pair {0, 1} is put back in. (e) What remains in 𝙇𝘕2 ⦸ 𝙇𝘕2 −1 is a set of orthocomplemented pairs, ordered with the same ordering relation ≤ that orders 𝙇𝘕2 . (f) All remaining orthocomplemented pairs are still involutory: 𝑥 = 𝑥⟂⟂

∀𝑥∈𝑋

(g) All remaining orthocomplemented pairs are still antitone because the ordering relation ≤ in 𝙇𝘕2 and 𝙇𝘕2 ⦸ 𝙇𝘕2 −1 is the same. (h) All remaining orthocomplemented pairs still have the non-contradiction property because suppose that in 𝙇𝘕2 ⦸ 𝙇𝘕2 −1 , there is an element 𝑥 such that 𝑥 ∧ 𝑥⟂ = 𝑚 ≠ 0. Then in 𝙇𝘕2 , it would also be true that 𝑥 ∧ 𝑥⟂ ≠ 0. This cannot be true (is a contradiction); so therefore for all 𝑥 in 𝙇𝘕2 ⦸𝙇𝘕2 −1 , 𝑥 ∧ 𝑥⟂ = 0 (non-contradiction property). (i) So 𝙇𝘕2 ⦸ 𝙇𝘕2 −1 is an orthocomplemented lattice (Definition A.72 page 40). 2. Proof that ( 𝑋 ≜ {𝙇𝑛2 |𝑛 = 1, 2, …} ∪ {𝙇𝑛2 ⦸ 𝙇𝑛−1 2 |𝑛 = 2, 3, …} , ⊆) is a primorial lattice: This follows directly from the construction of the bounded lattice difference (Definition 2.13 page 10) and the definition of primorial lattices (Definition 2.1 page 7).

✏ 𝘕 Definition 2.17. Let 𝙇𝘕 2 be a 2 element Boolean lattice (Definition A.69 page 39). The lattice ℙ as described in Theorem 2.16 is a primorial lattice generated by 𝙇𝘕 2.

Example 2.18. Figure 7 (page 13) illustrates a primorial lattice generated by 𝙇52 .

2.4

Projections on primorial lattices

This section introduces three lattice projections. When performing analysis in a primorial lattice (Definition 2.1 page 7), it is necessary to project a point that exists in a lattice of “high resolution” onto a lattice 𝙇 of lower resolution that may or may not contain this point. The three projections introduced here are the CC BY 4.0


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2.4 Projections on primorial lattices

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Daniel J. Greenhoe

1 𝑠⟂

𝑡⟂

𝑗⟂

𝑖⟂

ℎ⟂

𝑎

𝑏

𝑐

𝑔⟂

𝑓⟂

𝑑

𝑝

𝑞

𝑞⟂

𝑟⟂

𝑑⟂

𝑒⟂

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𝑠

𝑝⟂

𝑔

𝑐⟂

𝑏⟂

𝑎⟂

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𝑖

𝑗

𝑡 𝙇52

0 1 𝑡⟂

n

→

𝑠

io ut

𝑒

gr es ol

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𝑓⟂ 𝑟

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cr ea sin

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in

𝙇42

1 𝑡

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⟂

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orthocomplemented 𝑓

𝑞

𝑡

0 𝙇32

𝑞

0

⟂

𝑡⟂

𝑓⟂

𝑓

𝑡

𝑎 𝑑

𝑟

⟂

𝑠 0

𝙇32 ⦸𝙇22

𝙇42 ⦸𝙇32

1

Boolean

𝑟⟂

𝑒

𝑒⟂

𝑑

0 𝙇22

1

𝑠⟂

1

1

𝑞

1

0

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𝑒⟂

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𝑐 𝑝

𝑝⟂

𝑑⟂ 𝑔⟂ 𝑔

𝑏⟂ 𝑐

𝑑

𝙇52 ⦸𝙇42

⟂

ℎ 𝑒

𝑎⟂

0

𝙇2

Figure 7: a primorial lattice generated by 𝙇52



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Daniel J. Greenhoe


1. zero primorial projection (Definition 2.19 page 14) which assigns to 0 any point that does not exist in 𝙇 2. Sasaki primorial projection (Definition 2.20 page 14) which assigns a projection value using the Sasaki projection (Definition B.22 page 49) 3. metric primorial projection (Definition 2.22 page 14) which assigns a projection value based on a lattice metric (Definition B.7 page 46). Definition 2.19. Let ℙ be a primorial lattice (Definition 2.17 page 12) generated by a Boolean lattice 𝙇𝘕 2 (Definition A.69 page 39). Let 𝙇 ≜ ( 𝑌 , ∨, ∧, 0, 1 ; ≤) be a lattice in ℙ. Let 𝕩 ≜ ⦅𝑥𝑛 ⦆ be a sequence over the set 𝑋. The zero primorial projection Φ𝙇 (𝑥) of 𝑥 onto 𝙇 is defined as ∀𝑥∈𝑋 Φ𝑧𝙇 (𝑥) ≜ [{𝑥, 0} ∩ 𝑌 ] ⋁ 𝙇

The zero primorial projection Φ𝑧𝙇 (𝕩) of 𝕩 onto 𝙇 is defined as Φ𝑧𝙇 (𝕩) ≜ ⦅𝑦𝑛 ⦆ where 𝑦𝑛 ≜ Φ𝑧𝙇 (𝑥𝑛 ) ∀𝑥𝑛 ∈⦅𝑥𝑛 ⦆, 𝑦𝑛 ∈⦅𝑦𝑛 ⦆. Definition 2.20. Let ℙ and 𝕩 be defined as in Definition 2.19 (page 14). Let ℙ be a primorial lattice 𝘕 (Definition 2.17 page 12) generated by a Boolean lattice 𝙇 (Definition A.69 page 39). Let 𝙇 ≜ ( 𝑌 , ∨, ∧, 0, 1 ; ≤) be 2 a lattice in ℙ. Let 𝕩 ≜ ⦅𝑥𝑛 ⦆ be a sequence over the set 𝑋. The Sasaki primorial projection Φ𝑠𝙇 (𝑥) of 𝑥 onto 𝙇 is defined as ∀𝑥∈𝙇 𝜙 (𝑥)|𝑦 ∈ 𝑌 } ∩ 𝑌 ] Φ𝑠𝙇 (𝑥) ≜ ⋁ [{ 𝑦 𝙇

where 𝜙𝑦 (𝑥) is the Sasaki projection of 𝑥 onto 𝑦 (Definition B.22 page 49) in the smallest Boolean lattice 𝙇𝘔 2 that contains both 𝑥 and 𝙇. The Sasaki primorial projection Φ𝑠𝙇 (𝕩) of 𝕩 onto 𝙇 is defined as Φ𝑠𝙇 (𝕩) ≜ ⦅𝑦𝑛 ⦆ where 𝑦𝑛 ≜ Φ𝑠𝙇 (𝑥𝑛 ) ∀𝑥𝑛 ∈⦅𝑥𝑛 ⦆. The Sasaki primorial projection yields a kind of maxmini (Theorem A.35 page 33) result: Proposition 2.21. Let Φ𝙇 (𝑥) be the Sasaki primorial projection of 𝑥 onto 𝙇 in a primorial lattice ℙ. Φ𝑠𝙇 (𝑥) = ∀𝑥∈𝑋 [{𝑥 ∧ 𝑦|𝑦 ∈ 𝑌 } ∩ 𝑌 ] ⋁ 𝙇

✎PROOF: Φ𝑠𝙇 (𝑥) ≜

𝜙 (𝑥)|𝑦 ∈ 𝑌 } ∩ 𝑌 ] ⋁ [{ 𝑦 ≜ (𝑥 ∨ 𝑦⟂ ) ∧ 𝑦|𝑦 ∈ 𝑌 } ∩ 𝑌 ] ⋁ [{ (𝑥 ∧ 𝑦) ∨ (𝑦⟂ ∧ 𝑦)|𝑦 ∈ 𝑌 } ∩ 𝑌 ] ⋁ [{ = [{(𝑥 ∧ 𝑦) ∨ (0)|𝑦 ∈ 𝑌 } ∩ 𝑌 ] ⋁ = [{𝑥 ∧ 𝑦|𝑦 ∈ 𝑌 } ∩ 𝑌 ] ⋁ =

by def. of Sasaki primorial projection (Definition 2.20 page 14) by definition of Sasaki projection (Definition B.22 page 49) by distributive prop. (Theorem A.70 page 39) by noncontradiction property (Theorem A.70 page 39) by bounded property (Theorem A.70 page 39)

✏ Definition 2.22. Let ℙ and 𝕩 be defined as in Definition 2.19. The metric primorial projection Φ𝑚 𝙇 (𝑥) of 𝑥 onto 𝙇 is defined as Φ𝑚 𝙇 (𝑥) ≜

𝖡 (𝑥, 𝑟) ∩ 𝑌 ] ⋀[

where

𝙇

𝖡 (𝑥, 𝑟) is the closed ball in (𝙇𝘔 2 , 𝖽) with the smallest radius 𝑟 that contains 𝑥 𝑎𝑛𝑑 𝘔 2. (𝙇 , 𝖽) is a metric lattice (Definition B.7 page 46) 𝑎𝑛𝑑 2 𝘔 3. 𝙇 is the smallest Boolean lattice (Definition A.69 page 39) containing 𝑥 𝑎𝑛𝑑 2 𝘔 4. the valuation function defining 𝖽 is the height function on 𝙇 . 2 The metric primorial projection Φ𝙇 (𝕩) of 𝕩 onto 𝙇 is defined as Φ𝙇 (𝕩) ≜ ⦅𝑦𝑛 ⦆ such that 𝑦𝑛 ≜ Φ𝙇 (𝑥𝑛 ). 1.

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2.4 Projections on primorial lattices

page 15

Daniel J. Greenhoe

Example 2.23. Here are examples of the primorial projections Φ𝑧𝙊 (𝑥) (Definition 2.19 page 14), Φ𝑠𝙊 (𝑥) 6 6 𝑚 (Definition 2.20 page 14), and Φ (𝑥) (Definition 2.22 page 14) in the primorial lattice (Definition 2.1 page 7) generated 𝙊 6

by the Boolean lattice (Definition A.69 page 39) 𝙇52 ≜ ( 𝑋, the lattice 𝙊6 ≜ 𝙇32 ⦸𝙇22 ≜ ( 𝑌 , ∨, ∧, 0, 1 ; ≤). projection 𝑥 in 𝙊6 ≜ 𝙇32 ⦸𝙇22 𝑥 in 𝙇32 𝑥 = 0 𝑓 𝑡 𝑡⟂ 𝑓 ⟂ 1 𝑞 𝑞 ⟂ 𝑟 Φ𝑧𝙊 (𝑥) = 0 𝑓 𝑡 𝑡⟂ 𝑓 ⟂ 1 0 0 0 6 Φ𝑠𝙊 (𝑥) = 0 𝑓 𝑡 𝑡⟂ 𝑓 ⟂ 1 0 1 0 6 (𝑥) = 0 𝑓 𝑡 𝑡⟂ 𝑓 ⟂ 1 0 0 0 Φ𝑚 𝙊 6

∨, ∧, 0, 1 ; ≤) as illustrated in Figure 7 page 13 onto 𝑥 in 𝙇42 𝑟⟂ 𝑠 0 0 𝑓⟂ 0 𝑓⟂ 0

𝑥 in 𝙇52 𝑔 ⟂ 𝑝 𝑝⟂ 0 0 0 𝑓 0 1 𝑓 0 1

𝑠⟂ 𝑔 0 0 𝑓⟂ 𝑡 𝑓⟂ 𝑡

𝑑 𝑑⟂ 0 0 0 𝑡 0 𝑡

✎PROOF: 1. Proof for zero primorial projection values: Φ𝑧𝙊 (0) =

({0} ∪ {0}) ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] ⋁[ ({𝑓 } ∪ {0}) ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] Φ𝑧𝙊 (𝑓 ) = ⋁[ 6 ({𝑡} ∪ {0}) ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] Φ𝑧𝙊 (𝑡) = ⋁[ 6 Φ𝑧𝙊 (𝑡⟂ ) = 𝑡⟂ ∪ {0}) ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] ⋁ [({ } 6 Φ𝑧𝙊 (𝑓 ⟂ ) = 𝑓 ⟂ ∪ {0}) ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] ⋁ [({ } 6 ({1} ∪ {0}) ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] Φ𝑧𝙊 (1) = ⋁[ 6 ({𝑞} ∪ {0}) ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] Φ𝑧𝙊 (𝑞) = ⋁[ 6 Φ𝑧𝙊 (𝑞 ⟂ ) = 𝑞 ⟂ ∪ {0}) ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑞 ⟂ , 1}] ⋁ [({ } 6 ({𝑟} ∪ {0}) ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] Φ𝑧𝙊 (𝑟) = ⋁[ 6 Φ𝑧𝙊 (𝑟⟂ ) = 𝑟⟂ ∪ {0}) ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑟⟂ , 1}] ⋁ [({ } 6 ({𝑠} ∪ {0}) ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] Φ𝑧𝙊 (𝑠) = ⋁[ 6 Φ𝑧𝙊 (𝑠⟂ ) = 𝑠⟂ ∪ {0}) ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑟⟂ , 1}] ⋁ [({ } 6 ({𝑔} ∪ {0}) ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] Φ𝑧𝙊 (𝑔) = ⋁[ 6 Φ𝑧𝙊 (𝑔 ⟂ ) = 𝑔 ⟂ ∪ {0}) ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑟⟂ , 1}] ⋁ [({ } 6 ({𝑝} ∪ {0}) ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] Φ𝑧𝙊 (𝑝) = ⋁[ 6 𝑝⟂ ∪ {0}) ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑟⟂ , 1}] Φ𝑧𝙊 (𝑝⟂ ) = ⋁ [({ } 6 ({𝑑} ∪ {0}) ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] Φ𝑧𝙊 (𝑑) = ⋁[ 6 Φ𝑧𝙊 (𝑑 ⟂ ) = 𝑑 ⟂ ∪ {0}) ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑟⟂ , 1}] ⋁ [({ } 6 6

= = = =

⋁ ⋁ ⋁

[{0}]

=0

[{0, 𝑓 }]

=𝑓

[{0, 𝑡}]

=𝑡

⋁ [{

0, 𝑡⟂ }]

= 𝑡⟂

0, 𝑓 ⟂ }] ⋁ [{ = [{1, 0}] ⋁ = [{0}] ⋁ = 0, 𝑞 ⟂ }] ⋁ [{ = [{0}] ⋁ = 0, 𝑟⟂ }] ⋁ [{ = [{0}] ⋁ = [{0}] ⋁ = [{0}] ⋁ = [{0}] ⋁ = [{0}] ⋁ = [{0}] ⋁ = [{0}] ⋁ = [{0}] ⋁

= 𝑓⟂

=

=1 =0 =0 =0 =0 =0 =0 =0 =0 =0 =0 =0 =0

2. Proof for Sasaki primorial projection (Definition 2.20 page 14): Φ𝑠𝙊 (0) =

⋁

Φ𝑠𝙊 (𝑓 ) =

⋁

Φ𝑠𝙊 (𝑡) =

⋁

6

6

6

[{0 ∧ 𝑦|𝑦 ∈ 𝑌 } ∩ 𝑌 ]

=

[{𝑓 ∧ 𝑦|𝑦 ∈ 𝑌 } ∩ 𝑌 ]

=

[{𝑡 ∧ 𝑦|𝑦 ∈ 𝑌 } ∩ 𝑌 ]

=

⋁ ⋁ ⋁

=𝑓

{0, 𝑡}

=𝑡

0, 𝑓 , 𝑡⟂ } ⋁{ = 0, 𝑡, 𝑓 ⟂ } ⋁{

= 𝑡⟂

=

[{0, 0, 𝑡, 0, 𝑡, 𝑡} ∩ 𝑌 ]

=

𝑡⟂ ∧ 𝑦|𝑦 ∈ 𝑌 } ∩ 𝑌 ]

=

⋁ [{

0, 𝑓 , 0, 𝑡⟂ , 𝑞, 𝑡⟂ } ∩ 𝑌 ]

Φ𝑠𝙊 (𝑓 ⟂ ) =

⋁ [{

𝑓 ⟂ ∧ 𝑦|𝑦 ∈ 𝑌 } ∩ 𝑌 ]

=

⋁ [{

0, 0, 𝑡, 𝑞, 𝑓 ⟂ , 𝑓 ⟂ } ∩ 𝑌 ]

Tuesday 14 February, 2017 9:12pm UTC Copyright © Daniel J. Greenhoe

{0, 𝑓 }

[{0, 𝑓 , 0, 𝑓 , 0, 𝑓 } ∩ 𝑌 ]

⋁ [{

6 th

=0

=

Φ𝑠𝙊 (𝑡⟂ ) = 6

{0}

[{0, 0, 0, 0, 0, 0} ∩ 𝑌 ]


⋁ ⋁ ⋁

=

version 0.80

= 𝑓⟂ CC BY 4.0

page 16

Daniel J. Greenhoe

Φ𝑠𝙊 (1) =

⋁

Φ𝑠𝙊 (𝑞) =

⋁

6

6


0, 𝑓 , 𝑡, 𝑓 ⟂ , 𝑡⟂ , 1} ∩ 𝑌 ] ⋁ [{ = [{0, 0, 0, 𝑞, 0, 𝑞} ∩ 𝑌 ] ⋁ = 0, 𝑓 , 𝑡, 𝑓 , 𝑡, 𝑞 ⟂ } ∩ 𝑌 ] ⋁ [{ = [{0, 𝑟, 0, 𝑟, 0, 𝑟} ∩ 𝑌 ] ⋁ = 0, 𝑠, 𝑡, 𝑒, 𝑓 ⟂ , 𝑟⟂ } ∩ 𝑌 ] ⋁ [{ = [{0, 𝑠, 0, 𝑠, 0, 𝑠} ∩ 𝑌 ] ⋁ = 0, 0, 𝑡, 𝑑, 𝑓 ⟂ , 𝑠⟂ } ∩ 𝑌 ] ⋁ [{ = [{0, 0, 𝑡, 𝑝, 𝑔, 𝑔} ∩ 𝑌 ] ⋁ = 0, 𝑓 , 0, 𝑔 ⟂ , 0, 𝑔 ⟂ } ∩ 𝑌 ] ⋁ [{ = [{0, 0, 0, 𝑝, 𝑝, 𝑝} ∩ 𝑌 ] ⋁ = 0, 𝑓 , 𝑡, 𝑔 ⟂ , 𝑡, 𝑝⟂ } ∩ 𝑌 ] ⋁ [{ = [{0, 𝑟, 0, 𝑑, 0, 𝑑} ∩ 𝑌 ] ⋁ = 0, 𝑠, 𝑡, 0, 𝑔, 𝑑 ⟂ } ∩ 𝑌 ] ⋁ [{

[{1 ∧ 𝑦|𝑦 ∈ 𝑌 } ∩ 𝑌 ]

=

[{𝑞 ∧ 𝑦|𝑦 ∈ 𝑌 } ∩ 𝑌 ]

Φ𝑠𝙊 (𝑞 ⟂ ) =

𝑞 ⟂ ∧ 𝑦|𝑦 ∈ 𝑌 } ∩ 𝑌 ] ⋁ [{ Φ𝑠𝙊 (𝑟) = [{𝑟 ∧ 𝑦|𝑦 ∈ 𝑌 } ∩ 𝑌 ] ⋁ 6 Φ𝑠𝙊 (𝑟⟂ ) = 𝑟⟂ ∧ 𝑦|𝑦 ∈ 𝑌 } ∩ 𝑌 ] ⋁ [{ 6 Φ𝑠𝙊 (𝑠) = [{𝑠 ∧ 𝑦|𝑦 ∈ 𝑌 } ∩ 𝑌 ] ⋁ 6 𝑠⟂ ∧ 𝑦|𝑦 ∈ 𝑌 } ∩ 𝑌 ] Φ𝑠𝙊 (𝑠⟂ ) = ⋁ [{ 6 Φ𝑠𝙊 (𝑔) = [{𝑔 ∧ 𝑦|𝑦 ∈ 𝑌 } ∩ 𝑌 ] ⋁ 6 Φ𝑠𝙊 (𝑔 ⟂ ) = 𝑔 ⟂ ∧ 𝑦|𝑦 ∈ 𝑌 } ∩ 𝑌 ] ⋁ [{ 6 Φ𝑠𝙊 (𝑝) = [{𝑝 ∧ 𝑦|𝑦 ∈ 𝑌 } ∩ 𝑌 ] ⋁ 6 𝑝⟂ ∧ 𝑦|𝑦 ∈ 𝑌 } ∩ 𝑌 ] Φ𝑠𝙊 (𝑝⟂ ) = ⋁ [{ 6 Φ𝑠𝙊 (𝑑) = [{𝑑 ∧ 𝑦|𝑦 ∈ 𝑌 } ∩ 𝑌 ] ⋁ 6 Φ𝑠𝙊 (𝑑 ⟂ ) = 𝑑 ⟂ ∧ 𝑦|𝑦 ∈ 𝑌 } ∩ 𝑌 ] ⋁ [{ 6 6

= = = =

⋁ ⋁ ⋁ ⋁

𝑌

=1

{0}

=0

{0, 𝑓 , 𝑡}

=1

{0}

=0

0, 𝑡, 𝑓 ⟂ } ⋁{ {0} = ⋁ = 0, 𝑡, 𝑓 ⟂ } ⋁{ {0, 𝑡} = ⋁ {0, 𝑓 } = ⋁ {0} = ⋁ {0, 𝑓 , 𝑡} = ⋁ {0} = ⋁ {0, 𝑡} = ⋁

=

= 𝑓⟂ =0 = 𝑓⟂ =𝑡 =𝑓 =0 =1 =0 =𝑡

3. Proof for metric primorial projection (Definition 2.22 page 14): Φ𝑚 (0) = 𝙊

⋀[

𝖡 (0, 0) ∩ 𝑌 ]

=

{0} ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] ⋀[

=

Φ𝑚 (𝑓 ) = 𝙊

𝖡 (𝑓 , 0) ∩ 𝑌 ] ⋀[

=

{𝑓 } ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] ⋀[

=

=

{𝑡} ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] ⋀[

=

6

6

Φ𝑚 (𝑡) = 𝙊 6

⋀[

𝖡 (𝑡, 0) ∩ 𝑌 ]

⋀

=0

{𝑓 }

=𝑓

{𝑡}

=𝑡

𝖡 𝑡⟂ , 0 ) ∩ 𝑌 ] ⋀[ (

=

𝑡⟂ ∩ 0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] ⋀ [{ } {

=

𝑡⟂ ⋀{ }

= 𝑡⟂

Φ𝑚 (𝑓 ⟂ ) = 𝙊

𝖡 𝑓 ⟂ , 0) ∩ 𝑌 ] ⋀[ (

=

𝑓 ⟂ ∩ 0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] ⋀ [{ } {

=

𝑓⟂ ⋀{ }

= 𝑓⟂

=

6

Φ𝑚 (1) = 𝙊

⋀[

𝖡 (1, 0) ∩ 𝑌 ]

=

{1} ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] ⋀[

Φ𝑚 (𝑞) = 𝙊

𝖡 (𝑞, 1) ∩ 𝑌 ] ⋀[

=

⋀ [{

𝑞, 0, 𝑡⟂ } ∩ 𝑌 ]

=

𝖡 𝑞 ⟂ , 1) ∩ 𝑌 ] ⋀[ (

=

⋀ [{

𝑞 ⟂ , 𝑡, 1} ∩ 𝑌 ]

=

𝖡 (𝑟, 1) ∩ 𝑌 ] ⋀[

=

[{𝑟, 0, 𝑑, 𝑓 } ∩ 𝑌 ]

=

𝖡 𝑟⟂ , 1 ) ∩ 𝑌 ] ⋀[ (

=

𝑟⟂ , 𝑑 ⟂ , 𝑓 ⟂ , 1 } ∩ 𝑌 ] ⋀ [{

=

=

⋀ [{

𝑠, 0, 𝑒, 𝑓 , 𝑒⟂ } ∩ 𝑌 ]

=

=

⋀ [{

𝑠⟂ , 𝑒⟂ , 𝑓 ⟂ , 𝑑, 1} ∩ 𝑌 ]

=

𝑓 ⟂ , 1} ⋀{

= 𝑓⟂

=

⋀ [{

𝑔, 𝑝, 𝑓 ⟂ , 𝑒⟂ , 𝑑 ⟂ , 𝑡} ∩ 𝑌 ]

=

𝑓 ⟂ , 𝑡} ⋀{

=𝑡

=

⋀ [{

𝑔 ⟂ , 𝑑, 𝑒, 𝑓 , 𝑝⟂ , 𝑡⟂ } ∩ 𝑌 ]

=

𝑓 , 𝑡⟂ } ⋀{

=𝑓

{0}

=0

{1}

=1

6

6

Φ𝑚 (𝑞 ⟂ ) = 𝙊 6

Φ𝑚 (𝑟) = 𝙊 6

Φ𝑚 (𝑟⟂ ) = 𝙊 6

(𝑠) = Φ𝑚 𝙊 6

Φ𝑚 (𝑠⟂ ) = 𝙊 6

Φ𝑚 (𝑔) = 𝙊 6

(𝑔 ⟂ ) = Φ𝑚 𝙊 6

Φ𝑚 (𝑝) = 𝙊 6

Φ𝑚 (𝑝⟂ ) = 𝙊 6

⋀[

𝖡 (𝑠, 1) ∩ 𝑌 ]

𝖡 𝑠⟂ , 1) ∩ 𝑌 ] ⋀[ ( ⋀[

𝖡 (𝑔, 1) ∩ 𝑌 ]

𝖡 𝑔 ⟂ , 1) ∩ 𝑌 ] ⋀[ ( ⋀[

𝖡 (𝑝, 1) ∩ 𝑌 ]

𝖡 𝑝⟂ , 1 ) ∩ 𝑌 ] ⋀[ (

(𝑑) = Φ𝑚 𝙊 6

BY 4.0

⋀

{0}

Φ𝑚 (𝑡⟂ ) = 𝙊 6

CC

⋀

= =

⋀

⋀

[{𝑝, 0, 𝑝, 𝑎, 𝑏, 𝑐, 𝑔} ∩ 𝑌 ]

=

𝑝⟂ , 𝑎 ⟂ , 𝑏 ⟂ , 𝑐 ⟂ , 𝑔 ⟂ , 1 } ∩ 𝑌 ] ⋀ [{

=

{1}

=1

0, 𝑡⟂ } ⋀{

=0

⋀

⋀ ⋀

{𝑡, 1}

=𝑡

{0, 𝑓 }

=0

𝑓 ⟂ , 1} ⋀{ ⋀

⋀ ⋀

{0, 𝑓 }

= 𝑓⟂ =0

𝖡 (𝑑, 2) ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] ⋀[


version 0.80


Daniel J. Greenhoe

page 17

0, 𝑎, 𝑏, 𝑑, 𝑒, 𝑓 , ℎ, 𝑖, 𝑞, 𝑟, 𝑐 ⟂ , 𝑔 ⟂ , 𝑗 ⟂ , 𝑝⟂ , 𝑠⟂ , 𝑡⟂ } ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] ⋀ [{ = 0, 𝑓 , 𝑡⟂ } ⋀{ =0 =

Φ𝑚 (𝑑 ⟂ ) = 𝙊 6

𝖡 𝑑 ⟂ , 2) ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] ⋀[ ( = 𝑐, 𝑔, 𝑗, 𝑝, 𝑠, 𝑡, 𝑎⟂ , 𝑏⟂ , 𝑑 ⟂ , 𝑒⟂ , 𝑓 ⟂ , ℎ⟂ , 𝑖⟂ , 𝑞 ⟂ , 𝑟⟂ , 1} ∩ {0, 𝑓 , 𝑡, 𝑡⟂ , 𝑓 ⟂ , 1}] ⋀ [{ = 𝑡, 𝑓 ⟂ , 1} ⋀{ =𝑡

✏

3 A generalized probability function The traditional probability function 𝖯 is defined on a Boolean lattice. This paper introduces a new definition of 𝖯 which can be defined not only on Boolean lattices, but on some non-Boolean lattices as well. This new freedom is exploited in Section 4.3 (page 22).

3.1 Definitions Definition 3.1. Let 𝙇 ≜ ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤) be a lattice with negation (Definition B.16 page 48). Let Ⓓ be the distributivity relation (Definition A.52 page 36). A function 𝖯 in ℝ𝙓 is a probability on 𝙇 if 1. 𝖯(0) = 0 (nondegenerate) and 2. 𝖯(1) = 1 (normalized) and 3. 𝑥 ≤ 𝑦 ⟹ 𝖯(𝑥) ≤ 𝖯(𝑦) ∀𝑥,𝑦∈𝑋 (monotone) and 𝑥∧𝑦=0 and 4. ⟹ 𝖯(𝑥 ∨ 𝑦) = 𝖯(𝑥) + 𝖯(𝑦) ∀𝑥,𝑦∈𝑋 (additive). { (𝑧, 𝑥, 𝑦) ∈ Ⓓ ∀𝑧 ∈ 𝑋 } If 𝖯 is a probability on a lattice with negation 𝙇, then (𝙇, 𝖯) is a probability space. Remark 3.2. Definition 3.1 page 17 (previous) is not any standard definition of the probability function. On a Boolean lattice, the measure-theoretic probability function, due to A. N. Kolmogorov, is defined as15 (1). 𝖯(1) = 1 (normalized) and (2). 𝖯(𝑥) ≥ 0 ∀𝑥∈𝑋 (nonnegative) and ∞

∞

𝑥𝑛 = 𝖯(𝑥𝑛 ) ∀𝑥𝑛 ∈𝑋 (σ-additive) . ∑ (⋁ ) 𝑛=1 𝑛=1 The advantage of this definition is that 𝖯 is a measure, and hence all the power of measure theory is subsequently at one's disposal in using 𝖯. However, it has often been argued that the requirement of σ-additivity is unnecessary for a probability function. Even as early as 1930, de Finetti argued against it, in what became a kind of polite running debate with Fréchet.16 In fact, Kolmogorov himself provided some argument against σ-additivity when referring to the closely related Axiom of Continuity saying, “Since the new axiom is essential for infinite fields of probability only, it is (3).

⋀

𝑥𝑛 = 0

∞

⟹ 𝖯

𝑛=1

15

📘 Billingsley (1995) pages 22–23 ⟨Probability Measures⟩, 📘 Kolmogorov (1933a), 📘 Kolmogorov (1933b), page 16 ⟨field of probability⟩, 📘 Pap (1995) pages 8–9 ⟨Definition 2.3(13)⟩, 📘 Kalmbach (1986) page 27 16 📃 de Finetti (1930a), 📃 Fréchet (1930a), 📃 de Finetti (1930b), 📃 Fréchet (1930b), 📃 de Finetti (1930c), 📃 Cifarelli and Regazzini (1996) pages 258–260 Tuesday 14th February, 2017 9:12pm UTC Copyright © Daniel J. Greenhoe


version 0.80

CC BY 4.0

page 18

Daniel J. Greenhoe

3 A GENERALIZED PROBABILITY FUNCTION

almost impossible to elucidate its empirical meaning…For, in describing any observable random process we can obtain only finite fields of probability.…” But in its support he added, “This limitation has been found expedient in researches of the most diverse sort.”17 There are several other definitions of probability that only require additivity rather than σ-additivity. On a Boolean lattice, the traditional probability function is defined as18 (1). 𝖯(1) = 1 (normalized) and (2). 𝖯(𝑥) ≥ 0 ∀𝑥∈𝑋 (nonnegative) and (3). 𝑥 ∧ 𝑦 = 0 ⟹ 𝖯(𝑥 ∨ 𝑦) = 𝖯(𝑥) + 𝖯(𝑦) ∀𝑥,𝑦∈𝑋 (additive) . This definition implies (on a Boolean lattice) that (a). 𝖯(0) = 0 (nondegenerate) and (b). 𝖯(𝑥) ≤ 1 ∀𝑥∈𝑋 (upper bounded) and (c). 𝖯(𝑥) = 1 − 𝖯(¬𝑥) ∀𝑥∈𝑋 and (d). 𝖯(𝑥 ∨ 𝑦) ≤ 𝖯(𝑥) + 𝖯(𝑦) ∀𝑥,𝑦∈𝑋 (subadditive) and (e). 𝖯(𝑥 ∨ 𝑦) = 𝖯(𝑥) + 𝖯(𝑦) − 𝖯(𝑥 ∧ 𝑦) ∀𝑥,𝑦∈𝑋 and (f). 𝑥 ≤ 𝑦 ⟹ 𝖯(𝑥) ≤ 𝖯(𝑦) ∀𝑥,𝑦∈𝑋 (monotone) . On a distributive pseudocomplemented lattice, the generalized probability function has been defined as19 (1). 𝖯(0) = 0 (nondegenerate) and (2). 𝖯(1) = 1 (normalized) and (3). 0 ≤ 𝖯(1) ≤ 1 and (4). 𝖯(𝑥 ∨ 𝑦) = 𝖯(𝑥) + 𝖯(𝑦) − 𝖯(𝑥 ∧ 𝑦) ∀𝑥,𝑦∈𝑋 . On an orthomodular lattice, or a finite modular lattice, the quantum probability function is defined as20 (1). 𝖯(0) = 0 (nondegenerate) and (2). 𝖯(1) = 1 (normalized) and (3). 𝑥 ⟂ 𝑦 ⟹ 𝖯(𝑥 ∨ 𝑦) = 𝖯(𝑥) + 𝖯(𝑦) ∀𝑥,𝑦∈𝑋 (additive) . However, for lattices that are not distributive, modular, or orthomodular, none of these definitions work out so well. Take for example the O6 lattice with the “very reasonable” probability function given in Example 3.8 (page 20). This probability space (O6 , 𝖯) fails to be any of the 4 probability functions defined in this Remark. It fails to be a measure-theoretic or traditional probability function because 𝑎∧𝑏=0 but 𝖯(𝑎 ∨ 𝑏) = 𝖯(1) = 1 ≠ 31 + 12 = 𝖯(𝑎) + 𝖯(𝑏) . It fails to be a generalized probability function because 𝖯(𝑎 ∨ 𝑏) = 𝖯(1) = 1 ≠ 31 + 12 − 0 = 𝖯(𝑎) + 𝖯(𝑏) − 𝖯(0) = 𝖯(𝑎) + 𝖯(𝑏) − 𝖯(𝑎 ∧ 𝑏) . It fails to be an quantum probability function because 𝑎⟂𝑏=0 but 𝖯(𝑎 ∨ 𝑏) = 𝖯(1) = 1 ≠ 31 + 21 = 𝖯(𝑎) + 𝖯(𝑏) . In each of these cases, the function 𝖯 fails to be additive. The solution of Definition 3.1 (page 17) is simply to “switch off” additivity when the lattice is not distributive. This method is a little “crude”, but at least it allows us to define probability on a very wide class of lattices, while retaining compatibility with the Boolean case (Proposition 3.3 page 19, Proposition 3.4 page 19, Proposition 3.5 page 19). (end Remark 3.2) 17

📘 Kolmogorov (1933b), page 15 📘 Papoulis (1991) pages 21–22, 📘 Kolmogorov (1933b), page 2 ⟨§1. Axioms I–V⟩ 19 📃 Narens (2014) page 118, 📘 Narens (2007) 20 📃 Greechie (1971) page 126 ⟨DEFINITIONS⟩, 📃 Narens (2014) page 118 18

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3.2 Properties Proposition 3.3. 21 Let (𝙇, 𝖯) be a probability space (Definition 3.1 page 17). 0 ≤ 𝖯(𝑥) ≤ 1 ∀𝑥∈𝑋 ✎PROOF: 0 = 𝖯(0)

by previous result

≤ 𝖯(𝑥)

because 0 ≤ 𝑥 and monotone property (Definition 3.1 page 17)

𝖯(𝑥) ≤ 𝖯(1)

because 𝑥 ≤ 1 and monotone property (Definition 3.1 page 17)

=1

by property of 𝖯 (Definition 3.1 page 17)

✏ Proposition 3.4. 22 Let (𝙇, 𝖯) be a probability space (Definition 3.1 page 17). 𝙇 is ⟹ { 𝖯(𝑥) = 1 − 𝖯(¬𝑥) ∀𝑥∈𝑋 } { orthocomplemented } ✎PROOF: 1 − 𝖯(¬𝑥) = 𝖯(1) − 𝖯(¬𝑥)

by Definition 3.1 page 17

= 𝖯(𝑥 ∨ ¬𝑥) − 𝖯(¬𝑥)

by excluded middle property of ortho negation (Definition B.14 page 48)

= 𝖯(𝑥) + 𝖯(¬𝑥) − 𝖯(¬𝑥)

because (𝑥)(¬𝑥) = 0 and additive property (Definition 3.1 page 17)

= 𝖯(𝑥)

✏ Proposition 3.5. 23 Let (𝙇, 𝖯) be a probability space (Definition 3.1 page 17). 𝙇 is 1. 𝖯(𝑥 ∨ 𝑦) = 𝖯(𝑥) + 𝖯(𝑦) − 𝖯(𝑥 ∧ 𝑦) ∀𝑥,𝑦∈𝑋 ⟹ { Boolean } { 2. 𝖯(𝑥 ∨ 𝑦) ≤ 𝖯(𝑥) + 𝖯(𝑦) ∀𝑥,𝑦∈𝑋

and (Boole's inequality)

✎PROOF: 1. lemma: Proof that 𝖯((¬𝑥) ∧ 𝑦) = 𝖯(𝑦) − 𝖯(𝑥 ∧ 𝑦): 𝖯(𝑦) − 𝖯(𝑥𝑦) = 𝖯(1 ∧ 𝑦) − 𝖯(𝑥𝑦)

by definition of 1 and ∧ (Definition A.28 page 32)

= 𝖯[(𝑥 ∨ ¬𝑥)𝑦] − 𝖯(𝑥𝑦)

by excluded middle property of Boolean lattices

= 𝖯(𝑥𝑦 ∨ ¬𝑥𝑦) − 𝖯(𝑥𝑦)

by distributive property of Boolean lattices

= 𝖯(𝑥𝑦) + 𝖯(¬𝑥𝑦) − 𝖯(𝑥𝑦)

because (𝑥𝑦)(¬𝑥𝑦) = 0 and by additive property

= 𝖯(¬𝑥𝑦) 2. Proof that 𝖯(𝑥 ∨ 𝑦) = 𝖯(𝑥) + 𝖯(𝑦) − 𝖯(𝑥 ∧ 𝑦): 𝖯(𝑥 ∨ 𝑦) = 𝖯(𝑥 ∨ ¬𝑥𝑦)

by property of Boolean lattices

= 𝖯(𝑥) + 𝖯(¬𝑥𝑦)

because (𝑥)(¬𝑥𝑦) = 0 and by additive property

= 𝖯(𝑥) + 𝖯(𝑦) − 𝖯(𝑥 ∧ 𝑦)

by item (1) (page 19)

✏ 21

📘 Papoulis (1991) page 21 ⟨(2-11)⟩ 📘 Papoulis (1991) page 21 ⟨(2-12)⟩ 23 📘 Papoulis (1991) page 21 ⟨(2-13)⟩, 📘 Feller (1970) pages 22–23 ⟨(7.4),(7.6)⟩ 22



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Examples

Example 3.6. The function ¬ on the lattice 𝙇 as illustrated to the right is a Kleene negation (Definition B.14 page 48). Together with the probability function 𝖯, also illustrated to the right, the pair (𝙇, 𝖯) is a probability space (Definition 3.1 page 17).

Example 3.7. The lattice with negation 𝙇 (Definition B.16 page 48) illustrated to the right is a Boolean lattice. Together with the probability function 𝖯, also illustrated to the right, the pair (𝙇, 𝖯) is a probability space (Definition 3.1 page 17).

𝖯(𝑎) =

Example 3.8. The lattice with negation 𝙇 (Definition B.16 page 48) illustrated to the right is an orthocomplemented O6 lattice (Definition A.73 page 41). Together with the probability function 𝖯, also illustrated to the right, the pair (𝙇, 𝖯) is a probability space (Definition 3.1 page 17).

𝖯(𝑐) = 12 𝑐 = ¬𝑏 𝖯(𝑎) = 13 𝑎 = ¬𝑑

4

APPLICATIONS

1 = ¬0 𝖯(1) = 1 𝑎 = ¬𝑎 𝖯(𝑎) = 12 0 = ¬1 𝖯(0) = 0

1 = ¬0 1 3

𝑎 = ¬𝑏

𝖯(1) = 1

𝑏 = ¬𝑎 𝖯(𝑏) = 0 = ¬1

2 3

𝖯(0) = 0

1 = ¬0 𝖯(1) = 1 𝑑 = ¬𝑎 𝖯(𝑑) = 23 𝑏 = ¬𝑐 𝖯(𝑏) = 12 0 = ¬1 𝖯(0) = 0

Applications

This section discusses some possible applications of primorial lattices.

4.1

Logic analysis

𝘕 Let 𝙇𝘕 2 be a 2 -valued Boolean logic (Definition B.27 page 50). Let ℙ be the primorial lattice generated by 𝘕 𝘕 −1 𝙇𝘕 , … , 𝙇22 , 𝙇2 ⦆ in ℙ are Boolean logics with 2 (Definition 2.17 page 12). The sequence of lattices ⦅𝙇2 , 𝙇2 𝑛 decreasing “resolution” (higher values of 𝑛 in 𝙇2 correspond to greater resolution). Thus, we can reduce a very complex logic in 𝙇𝘕 2 to a simpler lower resolution logic.

Moreover, the sequence of ortho logics (Definition B.27 page 50) in ℙ 3 𝘕 −2 𝘕 −1 𝘕 −1 𝘕 2 ⦅𝙇2 ⦸𝙇2 , 𝙇2 ⦸𝙇2 , … , 𝙇2 ⦸𝙇2 , 𝙇2 ⦆ represents the Boolean logic 𝙇𝘕 2 at 𝘕 − 1 progressively lower “frequencies”. Alternatively, we could say that the Boolean logic at resolution 𝘕 is “decomposed” into (or analyzed by) 𝘕 − 1 ortho logics. Moreover, a proposition 𝑝 in a higher resolution space can be projected into a lower resolution space (including the two-value classic logic space) by a projection operator (Section 2.4 page 12).

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4.2 Fuzzy logic analysis

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1 2

1

1 2

1

1 2

1

𝕞1 (𝑥) = ¬𝕞0 (𝑥) 𝑥

𝕞𝑃 (𝑥) = ¬𝕞𝐶 (𝑥) 𝑥

1 2

𝕞𝐴 (𝑥) = ¬𝕞𝑅 (𝑥) 𝑥

1 2

𝕞𝐵 (𝑥) = ¬𝕞𝑄 (𝑥) 𝑥

1

𝕞0 (𝑥) = ¬𝕞1 (𝑥)

1

1 2

𝕞𝑄 (𝑥) = ¬𝕞𝐵 (𝑥) 𝑥

1

1 2

𝕞𝑅 (𝑥) = ¬𝕞𝐴 (𝑥) 𝑥

𝙇32 11 𝕞 (𝑥) = ¬𝕞 (𝑥) 𝐶 𝑃 2

𝑥

𝑥

1 𝕞 (𝑥) = ¬𝕞 (𝑥) = 1 1 0

1 2

1 𝕞 (𝑥) = ¬𝕞 (𝑥) = 1 1 0

1 2

1

1 2

𝕞𝐵 (𝑥) = ¬𝕞𝑄 (𝑥) 𝑥

𝑥 1

1 2

1

1 2

𝕞𝑄 (𝑥) = ¬𝕞𝐵 (𝑥) 𝑥

1 𝕞 (𝑥) = ¬𝕞 (𝑥) = 02 𝙇 0 1

1 2

𝑥

1

1 2

𝑥 1

𝕞𝑃 (𝑥) = ¬𝕞𝐶 (𝑥) 𝑥

1 2

𝕞𝐴 (𝑥) = ¬𝕞𝑅 (𝑥) 𝑥

1 2

2

1

𝕞𝑅 (𝑥) = ¬𝕞𝐴 (𝑥) 3 2𝑥

𝙇2 ⦸𝙇2

𝕞𝐶 (𝑥) = ¬𝕞𝑃 (𝑥) 𝑥

1 𝕞 (𝑥) = ¬𝕞 (𝑥) = 0 0 1

1 2

𝑥

11 𝕞 (𝑥) = ¬𝕞 (𝑥) = 1 1 0 2 𝑥 𝙇2 11 𝕞 (𝑥) = ¬𝕞 (𝑥) = 0 0 1 2 𝑥

Figure 8: primorial lattice for fuzzy subset logic (Example 4.1 page 21)

4.2 Fuzzy logic analysis Fuzzy logics (Definition B.27 page 50) can be constructed on Boolean and orthocomplemented lattices24 such that together with the subset ordering relation ⊆, form of a primorial lattice ℙ (Definition 2.1 page 7). A Boolean fuzzy logic 𝙇𝘕 2 can then be rendered at 𝘕 − 1 different “resolutions” using the Boolean lattices of ℙ and analyzed at 𝘕 − 1 “frequencies” using the orthocomplemented lattices of ℙ, as described in Section 4.1 (page 20). Example 4.1. Figure 8 (page 21) illustrates a fuzzy subset logic 25 on a primorial lattice. The lattice 𝙇32 contains both monotonic and non-monotonic membership functions. These are separated into lower resolution spaces 𝙇22 containing the non-monotonic membership functions (neglecting 1 and 0), 𝙇32 ⦸ 𝙇22 containing the monotonic membership functions, and 𝙇2 containing crisp set logic. A projection operator (Section 2.4 page 12) can be used to project a membership function onto any of these spaces as perhaps called for by a given application. 24

📃

25

📃

Greenhoe (2014b) ⟨§2.2⟩ Greenhoe (2014b) ⟨§3.2⟩



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APPLICATIONS

𝖯(1) = 1 𝖯(𝑠⟂ ) = 3 4

𝖯(𝑝) =

1 2

𝖯(𝑞) =

n tio lu gr es o in cr ea s in

𝖯(𝑞 ⟂ ) =

𝑐⟂

3 4

𝑏⟂

3 𝖯(𝑟) = 16 𝖯(𝑠) =

𝖯(𝑝⟂ ) =

1 2

𝖯(𝑎⟂ ) =

1 4

1 16

𝙇42

𝖯(1) = 1

𝖯(𝑎) =

3 4

𝖯(𝑝) =

1 2

𝖯(𝑞 ⟂ ) =

𝖯(𝑞) =

𝖯(1) = 1 𝖯(𝑝⟂ ) =

1 2

1 4

13 16

𝖯(0) = 0 3 4

1 4

𝖯(0) = 0

𝖯(𝑝) =

𝑐

𝑏

→

𝖯(𝑎) =

𝖯(𝑟⟂ ) =

15 16

𝖯(0) = 0 𝙇22

𝑞⟂ 1 2𝑎⟂

𝖯(𝑝⟂ ) =

1 2

𝖯(𝑎⟂ ) =

1 4

𝙇32

← increasing uncertainty 1 1 𝖯(𝑠⟂ ) = 15 𝖯(𝑟⟂ ) = 13 16 16 𝑎 ⟂ 𝑐 𝖯(𝑏⟂ ) = 𝑐 𝑞 𝑏3 1 𝖯(𝑟) = 16 𝖯(𝑠) = 16 0 0

5 16

𝙇42 ⦸𝙇32

𝙇32 ⦸𝙇22 𝖯(1) = 1

Boolean/ classic probabilities

𝙇2 𝖯(0) = 0

Figure 9: primorial lattice with probability function (Example 4.2 page 22)

4.3

Probability analysis

A logic is a lattice with negation (Definition B.16 page 48) and with an implication function defined on it. In this paper, a probability is a lattice with negation and with a probability function (Definition 3.1 page 17) defined on it. 𝘕 Let 𝙇𝘕 2 be the 2 -element Boolean lattice generated by an 𝘕 -event Boolean probability space (Def𝘕 inition 3.1 page 17). Let ℙ be the primorial lattice (Definition 2.1 page 7) generated by 𝙇 . Then in ℙ, the 2 probability space can be rendered at progressively lower resolutions using the Boolean lattices of ℙ, and can be analyzed at assorted “frequencies” using the orthocomplemented lattices of ℙ.

Example 4.2. A primorial lattice with a probability function is illustrated in Figure 9 (page 22).

4.4

Symbolic sequence analysis

Definitions. Finding some properties of a sequence 𝕩 that is constructed over a field 𝔽 may be referred to as sequence analysis or discrete-time signal analysis. If we somehow mathematically alter 𝕩 with an operator 𝐀 to produce a new sequence 𝕪 ≜ 𝐀𝕩, then this may be referred to as sequence processing, or more commonly as discrete-time signal processing or digital signal processing (DSP). CC BY 4.0


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Basis theory. Sequence analysis and sequence processing typically make use of basis theory. In basis theory in general (of which Fourier analysis and wavelet analysis are special cases), we represent some point 𝕩 (𝕩 is a sequence) in a Banach space (a complete normed linear space) by a linear combination of a basis sequence ⦅𝑥𝑛 ⦆ such that 𝕩≛ 𝑎 𝑥 ∑ 𝑛 𝑛 𝑛∈𝑍

where ≛ represents strong convergence with respect to the norm ‖⋅‖ of the Banach space. Each element 𝑎𝑛 is a member of the field 𝔽 of the Banach space and the sequence ⦅𝑎𝑛 ⦆ is often referred to as a “transform” (Fourier transform, discrete-time Fourier transform, wavelet transform, etc.) In order to be able to successfully compute any transform (such as a Fourier transform or wavelet transform) in a Banach space or even a finite linear space, the sequence 𝕩 needs to be somehow related to the field 𝔽 over which the Banach space is constructed.

The problem. Let 𝐅̃ be the discrete-time Fourier transform operator and 𝐖 be a discrete-time ̃ or 𝐖𝕩. This is a problem in symbolic sequence wavelet transform. Suppose we want to compute 𝐅𝕩 analysis and symbolic signal processing in general because of the following reasons: 1. The symbols in 𝕩 have no field structure; so we can't even add them. 𝘉 and ◻ 𝘊 are symbols, we can't say, for 𝘈 ◻, 2. The symbols in 𝕩 have no order structure; so if ◻, 𝘉 𝘉 𝘊 𝘈 < ◻ or ◻ < ◻, etc. example, ◻ 3. The symbols in 𝕩 have no topology except for some arguably trivial topologies;26 so we 𝘉 than it is to ◻, 𝘊 etc. 𝘈 is “closer” to ◻ can't say, for example, that ◻ In fact, symbol sequence analysis does not just cause problems for Fourier or wavelet analysis only— it causes problems for basis theory in general because a basis is constructed in a Banach space, and symbolic sequences are in general not constructed in Banach spaces. A kind of “hack” solution may be to map the symbols to points ⦅𝑝1 , 𝑝2 , … , 𝑝𝘕 ⦆ in the complex plane ℂ. If these points are chosen such that they are distinct, not on either the real or imaginary axes, and |𝑝1 | = |𝑝2 | = … = |𝑝𝘕 |, then that would seem to be a good start, because now the mapped symbols have a field structure, and they are arguably unordered (arguably we can't say any one of them is greater or less than any other, just as in the original symbol sequence). But we still have the topology problem. If we map, say, 4 symbols to 4 points in ℂ as 𝑝1 = 1, 𝑝2 = −1, 𝑝3 = 𝑖, and 𝑝4 = −𝑖, then “𝑝1 ” is closer (with respect to the metric induced by the norm |⋅|) to “𝑝3 ” then it is to “𝑝2 ”: 1/2 1/2 1/2 𝖽(𝑝1 , 𝑝3 ) = |𝑝1 − 𝑝3 | = (𝑝21 − 𝑝23 ) = (12 − 𝑖2 ) = √2 ⪇ 2 = (22 − 02 ) = 𝖽(𝑝1 , 𝑝2 ) This unwanted topological property is introduced by the mapping, will affect the transform, but yet is not a property of the original symbolic sequence. “Frequency” properties may be useful in symbolic sequence analysis and symbolic sequence processing. But the point here is that any kind of basis theory technique (including Fourier or wavelet techniques) may result in a kind of imperfect “hack” solution. 26

𝘉 ◻}, 𝘊 𝘈 ◻, These topologies include the indiscrete topology {∅, 𝑋} where 𝑋 ≜ {◻, discrete topology 𝟚𝑋 (references: 📘 Munkres (2000), page 77, 📘 Kubrusly (2011) page 107 ⟨Example 3.J⟩, 📘 Steen and Seebach (1978) pages 42–43 ⟨II.4⟩, 📘 DiBenedetto (2002) page 18 ), and the topology induced by the discrete metric 𝖽(𝑥, 𝑦) ≜ {1 for 𝑥 ≠ 𝑦, 0 for 𝑥 = 𝑦} (references: 📘 Giles (1987), page 13, 📘 Copson (1968), page 24, 📘 Khamsi and Kirk (2001) page 19 ⟨Example 2.1⟩ ).



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APPLICATIONS

Proposed solution. The solution proposed here is to perform symbolic sequence analysis using primorial lattices. Suppose we have a sequence 𝕩 over a set of 𝘕 symbols (each element in the sequence can be any one of 𝘕 different symbols). Let ℙ be the primorial lattice generated by 𝙇𝘕 2. 𝘕 𝘕 𝘉 in 𝙇 , where ◻ 𝘉 𝘈 ∨◻ 𝘈 and ◻ The orthogonal 𝘕 atoms of 𝙇2 represent the 𝘕 symbols. The element ◻ 2 𝘉 (it is not 𝘈 OR ◻ are 2 symbols, represents the event of a particular position in the sequence being ◻ 𝘉 𝘈 AND ◻). possible for a particular position to be both ◻ Any symbol in 𝙇𝘕 2 can be projected onto any other Boolean or orthocomplemented lattice in ℙ by use of a lattice projection (Section 2.4 page 12). The result of projecting an entire sequence onto a lattice in ℙ is another sequence (Definition 2.19 page 14). So after projection, a sequence on 𝙇𝘕 2 results in 𝘕 − 1 sequences of lower resolution and 𝘕 − 1 sequences of assorted frequencies. This is similar in form to the Fast Wavelet Transform, as illustrated in Figure 20 (page 64).

4.5

Symbolic sequence processing (SSP)

Introduction. The previous section discusses symbolic sequence analysis—meaning we are not trying to change the properties of the sequence, we are only trying to understand its properties. This section discusses symbolic sequence processing (or symbolic signal processing )—meaning we are trying to change the properties of the sequence. Digital signal processing (DSP) or discrete-time signal processing operates on a sequence constructed over a field 𝔽 , where 𝔽 is typically either ℝ or ℂ. Often by use of simple multiplication and addition operations on elements of the sequence, one can change the properties of the sequence. Often when the properties are related to Fourier analysis, the DSP operations are called “filtering”.

The problem. Multiplication and addition operations commonly used in DSP require field properties. In symbolic sequence processing, we don't in general have a field.

Proposed solution. Sequence processing of, or “filtering” on, a symbolic sequence 𝕩 can be performed by judicious selection and/or rejection of the various projections onto the logics in the primorial lattice ℙ. For example, if one wants 𝕩 at a lower “resolutions”, then simply select the sequence from a projection onto the Boolean logic at resolution lower than 𝘕 . If one wants to “filter out” the “high frequency” components of 𝕩, then simply discard the projections onto the higher frequency orthocomplemented lattices before synthesizing a new sequence from the “low frequency” component sequences. Synthesis of two projection sequences 𝕪 and 𝕫 into a new sequence 𝕩′ can be performed, for example, by pointwise join such that 𝕪 ⊕ 𝕫 ≜ ⦅𝑦𝑛 ⦆𝑛∈ℤ

⦅𝑧 ⦆ ⋁ 𝑛 𝑛∈ℤ ≜ ⦅𝑦𝑛 ∨ 𝑧𝑛 ⦆𝑛∈ℤ

≜ ⦅𝑥𝑛 ⦆𝑛∈ℤ ≜𝕩

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4.6 Genomic Signal Processing (GSP)

Daniel J. Greenhoe

𝘛 (◻ 𝘈 ∨ ◻)

re as in gr es

ol

ut

io n

→

𝘎⟂ ◻

◻ 𝘈

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1 𝘛⟂ ◻ ◻ 𝘈⟂

𝘊⟂ ◻

𝑏

𝑐⟂ 𝑐

⟂ 𝘊 ∨◻ 𝘎 ) = (◻ 𝘛 𝘈 ∨ ◻) 𝑏⟂ (◻

𝘛 ◻

𝘊 ◻

𝘎 ◻

0

in c

𝙇42 1 𝘛 ◻ 𝘈 ∨◻ ◻ 𝘈⟂

𝘛⟂ ◻

◻ 𝘈

orthocomplemented

𝘛 ◻ 𝘊 ∨◻ 𝘎 ◻ 0

𝙇32

1 𝘛 ◻ 𝘈 ∨◻

𝘊 ∨◻ 𝘎 ◻

𝘛⟂ ◻

𝘎⟂ ◻

◻ 𝘈⟂

◻ 𝘈

0

1

1 𝘛 ◻

𝑏

0 𝙇22

1 0

𝑐

𝑐

𝘊 ◻

𝘎 ◻

𝑏⟂

0

𝙇32 ⦸𝙇22

Boolean

𝘊⟂ ◻ ⟂

𝙇42 ⦸𝙇32

𝙇2

Figure 10: primorial lattice for genomic signal processing (GSP) with 𝐴 ∨ 𝑇 and 𝐶 ∨ 𝐺 analysis features (Example 4.3 page 25)

4.6 Genomic Signal Processing (GSP) Genomic Signal Processing (GSP) is simply a special case of Symbolic Sequence Processing with 𝘕 = 𝘊 ◻, 𝘛 and ◻, 𝘎 each of which corresponds 𝘈 ◻, 4. In GSP, the 4 symbols are commonly referred to as ◻, to a nucleobase (adenine, thymine, cytosine, and guanine, respectively).27 The sequence itself is called a genome. A typical genome sequence contains a large number of symbols (about 3 billion for humans, 29751 for the SARS virus).28 𝘛 and (◻∨ 𝘊 ◻) 𝘎 are of special interest. Portions of 𝘈 ◻) Example 4.3. Traditionally in GSP, the symbols (◻∨ 𝘛 content separate at lower temperatures than do those with high 𝘈 ∨ ◻) a genome sequence high in (◻ 29 𝘊 ∨ ◻) 𝘎 content. (◻ Therefore, one could construct a primorial lattice induced by 𝙇42 that allows for 𝘛 and/or ◻ 𝘊 ∨◻ 𝘎 in some lower resolution space. An example is illustrated 𝘈 ∨◻ convenient analysis of ◻ in Figure 10 (page 25).

Example 4.4. In some cases, genomic sequences with more than 4 symbols (𝘕 > 4) have been 27

📃 Mendel (1853) ⟨Mendel (1853): gene coding uses discrete symbols⟩, 📃 Watson and Crick (1953a) page 737 ⟨Watson and Crick (1953): gene coding symbols are adenine, thymine, cytosine, and guanine⟩, 📃 Watson and Crick (1953b) page 965, 📘 Pommerville (2013) page 52 28 💻 GenBank (2014) ⟨http://www.ncbi.nlm.nih.gov/genome/guide/human/⟩⟨Homo sapiens, NC_000001– NC_000022 (22 chromosome pairs), NC_000023 (X chromosome), NC_000024 (Y chromosome), NC_012920 (mitochondria)⟩, 💻 GenBank (2014) ⟨http://www.ncbi.nlm.nih.gov/nuccore/30271926⟩⟨SARS coronavirus, NC_004718.3⟩ 📃 S. G. Gregory (2006) ⟨homo sapien chromosome 1⟩, 📃 Runtao He (2004) ⟨SARS coronavirus⟩ 29 📘 Cristianini and Hahn (2007) page 13 ⟨Remark 1.2⟩ Tuesday 14th February, 2017 9:12pm UTC Copyright © Daniel J. Greenhoe


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4

APPLICATIONS

1 𝘊⟂ ◻

𝘎⟂ ◻

𝑗⟂

𝑖⟂

ℎ⟂

𝑎

𝑏

𝑐

𝘛⟂ ◻ ◻ 𝘈⟂

𝑔⟂ 𝑓 ⟂ 𝑑

𝘟 ◻

◻ 𝘈

𝘟⟂ ◻

𝑒⟂

𝑑⟂

𝑐⟂

𝑏⟂

𝑎⟂

𝑒

𝑓

𝑔

ℎ

𝑖

𝑗

𝘛 ◻

𝘊 ◻

𝘎 ◻

𝙇52

0

𝘛 (◻ 𝘈 ∨ ◻)

gr es o

lu

tio

n

→

𝘎⟂ ◻

in

◻ 𝘈

1 𝘛⟂ ◻ ◻ 𝘈⟂

𝘊⟂ ◻

𝑏

𝑐⟂ 𝑐

⟂ 𝘊 ∨◻ 𝘎 ) = (◻ 𝘛 𝘈 ∨ ◻) 𝑏⟂ (◻

𝘛 ◻

𝘊 ◻

𝘎 ◻

cr ea s

0

in

𝙇42 1 𝘛 ◻ 𝘈 ∨◻ ◻ 𝘈⟂

𝘛⟂ ◻

orthocomplemented

𝘛 ◻ 𝘊 ∨◻ 𝘎 ◻ 0

◻ 𝘈

𝙇32

1 𝘛 ◻ 𝘈 ∨◻

1

1 𝘊 ∨◻ 𝘎 ◻

𝘛⟂ ◻

𝘎⟂ ◻

◻ 𝘈⟂ 𝘛 ◻

◻ 𝘈

0

1

𝑏

0 𝙇22

𝑐

𝑐

𝘊 ◻

𝘎 ◻

0

𝙇32 ⦸𝙇22

𝙇42 ⦸𝙇32

1

Boolean

𝘊⟂ ◻ ⟂

0

𝑑⟂

ℎ⟂ 𝑒⟂ 𝑎 𝑏

⟂

𝑐

𝑏

𝘟 ◻

𝘟⟂ ◻

𝑔⟂ 𝑔

𝑏⟂ 𝑐

𝑑

⟂

𝑒

ℎ 𝑎⟂ 𝙇52 ⦸𝙇42

0

𝙇2

Figure 11: primorial lattice for genomic signal processing (GSP) with extra symbol 𝑋

(Example 4.4

page 25)

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𝘟 in the higher studied.30 Figure 11 (page 26) illustrates a primorial lattice with an extra symbol ◻ 4 𝘊 ◻, 𝘎 and ◻ 𝘛 in the lower resolution 𝙇 𝘈 ◻, resolution 𝙇52 Boolean lattice, but with only the symbols ◻, 2 𝘟 can be projected onto any of the lower resolution spaces using a Boolean lattice. The symbol ◻ projection operator (Section 2.4 page 12).

Appendix A

Order structures

A.1 Order A.1.1 Order relations Definition A.1. 31 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) An ordered set is the pair (𝑋, ≤). The set 𝑋 is called the base set of (𝑋, ≤). If 𝑥 ≤ 𝑦 or 𝑦 ≤ 𝑥, then 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. Definition A.2. 32 Let (𝑋, ≤) be an ordered set (Definition A.1 page 27). Let 𝟚𝑋𝑋 be the set of all relations on 𝑋. 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

𝑥⪇𝑦

⟺ def

⟺ def

Example A.3. order relation ≤ (integer less than or equal to) ≥ ⊆ (subset) ⊇ | (divides) ⟸ ⟹ (implies)

dual order relation (integer greater than or equal to) (super set) (divided by) (implied by)

Definition A.4. 33 A relation ≤ is a linear order relation on 𝑋 if 1. ≤ is an order relation (Definition A.1 page 27) and 2. 𝑥 ≤ 𝑦 or 𝑦 ≤ 𝑥 ∀𝑥,𝑦∈𝑋 (comparable). 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. 30

📃 Comnish-Bowden (1985), 📃 Amato (1990), 📃 Elnitski et al. (2003), 📃 Hutter and Benner (2003), 📃 von Krosigk and Benner (2004), 📃 Malyshev et al. (2009), 📃 Jiang and Seela (2010) 31 📘 MacLane and Birkhoff (1999) page 470, 📘 Beran (1985) page 1, 📃 Korselt (1894) page 156 ⟨I, II, (1)⟩, 📃 Dedekind (1900) page 373 ⟨I–III⟩. An order relation is also called a partial order relation. An ordered set is also called a partially ordered set or poset. 32 📃 Peirce (1880) page 2 33 📘 MacLane and Birkhoff (1999) page 470, 📃 Ore (1935) page 410 Tuesday 14th February, 2017 9:12pm UTC Copyright © Daniel J. Greenhoe


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ORDER STRUCTURES

Representation

Definition A.5. 34 𝑦 covers 𝑥 in the ordered set (𝑋, ≤) if 1. 𝑥 ≤ 𝑦 (𝑦 is greater than 𝑥) (𝑧 = 𝑥 or 𝑧 = 𝑦) (there is no element between 𝑥 and 𝑦). 2. (𝑥 ≤ 𝑧 ≤ 𝑦) ⟹ The case in which 𝑦 covers 𝑥 is denoted 𝑥 ≺ 𝑦.

and

An ordered set can be represented in any of three ways: Hasse diagram (Definition A.6 page 28) a set of ordered pairs of order relations (Definition A.1 page 27) a set of ordered pairs of cover relations (Definition A.5 page 28) Definition A.6. Let (𝑋, ≤) be an ordered pair. A diagram is a Hasse diagram of (𝑋, ≤) if it satisfies the following criteria: Each element in 𝑋 is represented by a dot or small circle. For each 𝑥, 𝑦 ∈ 𝑋, if 𝑥 ≺ 𝑦, then 𝑦 appears at a higher position than 𝑥 and a line connects 𝑥 and 𝑦. Example A.7. Here are three ways of representing the ordered set (𝟚{𝑥,𝑦} , ⊆);

Hasse diagrams: If two elements are comparable, then the 1. lesser of the two is drawn lower on the page than the other with a line connecting them.

{𝑥, 𝑦} {𝑦}

{𝑥} ∅

2. Sets of ordered pairs specifying order relations (Definition A.1 page 27): (∅, ∅) , ({𝑥}, {𝑥}) , ({𝑦}, {𝑦}) , ({𝑥, 𝑦}, {𝑥, 𝑦}) , ⊆= { (∅, {𝑥}) , (∅, {𝑦}) , (∅, {𝑥, 𝑦}) , ({𝑥}, {𝑥, 𝑦}) , ({𝑦}, {𝑥, 𝑦}) } 3. Sets of ordered pairs specifying covering relations: ≺= { (∅, {𝑥}) , (∅, {𝑦}) , ({𝑥}, {𝑥, 𝑦}) , ({𝑦}, {𝑥, 𝑦}) }

A.1.3

Decomposition

Definition A.8. 35 The tupple (𝑌 , ⋜) is a subposet of the ordered set (𝑋, ≤) if 1. 𝑌 ⊆ 𝑋 (𝑌 is a subset of 𝑋) and 2 2. ⋜ = (≤ ∩𝑌 ) (⋜ is the relation ≤ restricted to 𝑌 × 𝑌 ) Example A.9.

Subposets of

34 35

include

Birkhoff (1933) page 445 📘 Grätzer (2003) page 2 📃

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Example A.10. Let (𝑋, ≤) ≜ ({0, 𝑎, 𝑏, 𝑐, 𝑝, 1},

1

{ (0, 0) , (𝑎, 𝑎) , (𝑏, 𝑏) , (𝑐, 𝑐) , (𝑝, 𝑝) , (1, 1) , (0, 𝑎) , (0, 𝑏) , (0, 𝑐) , (0, 𝑝) , (0, 1) , (𝑎, 𝑏) , (𝑎, 𝑐) , (𝑎, 1) , (𝑝, 1) ,

𝑐 𝑝

𝑏 𝑎 0

(𝑏, 𝑐) , (𝑏, 1) , (𝑐, 1) , (𝑝, 1) }) (𝑌 , ⋜) ≜ ({0, 𝑎, 𝑐, 𝑝, 1},

1

{ (0, 0) , (𝑎, 𝑎) , (𝑐, 𝑐) , (𝑝, 𝑝) , (1, 1) ,

𝑐 𝑎

(0, 𝑎) , (0, 𝑐) , (0, 𝑝) , (0, 1) ,

𝑝 0

(𝑎, 𝑐) , (𝑎, 1) , (𝑝, 1) , (𝑐, 1) , (𝑝, 1) }). Then (𝑌 , ⋜) is a subposet of (𝑋, ≤) because 𝑌 ⊆ 𝑋 and ⋜= (≤ ∩𝑌 2 ).

A chain is an ordered set in which every pair of elements is comparable (Definition A.4 page 27). An antichain is just the opposite—it is an ordered set in which no pair of elements is comparable (next definition). antichain 36

Definition A.11. The subposet (𝐴, ≤) in the ordered set (𝑋, ≤) is an antichain if all elements in 𝐴 are incomparable (Definition A.1 page 27), such that 𝑥||𝑦 ∀𝑥, 𝑦 ∈ 𝐴

antichain

antichain

Definition A.12. 37 The length 𝓁(𝙇) of a chain (Definition A.4 page 27) 𝙇 with 𝘕 elements is 𝘕 − 1. The length of an ordered set (Definition A.1 page 27) is the length of the longest chain in the ordered set. The width of an ordered set is the number of elements in the largest antichain in the ordered set. Theorem A.13 (Dilworth's theorem). 38 Let (𝑋, ≤) be an ordered set. 1. there exists a partition of (𝑋, ≤) into 𝘕 chains ⎧ width 𝘕 of (𝑋, ≤) ⎪ ⟹ ⎨ 2. there does not exist any partition { is finite } ⎪ of (𝑋, ≤) into less than 𝘕 chains ⎩

and

⎫ ⎪ ⎬ ⎪ ⎭

Definition A.14. 39 Let 𝑋 and 𝑌 be disjoint sets. Let 𝙋 ≜ ( 𝑋, ⋜) and 𝙌 ≜ ( 𝑌 , ⊴) be ordered sets on 𝑋 and 𝑌 . The direct sum of 𝙋 and 𝙌 is defined as 𝙋 + 𝙌 ≜ ( 𝑋 ∪ 𝑌 , ≤) where 𝑥 ≤ 𝑦 if 1. 𝑥, 𝑦 ∈ 𝑋 and 𝑥 ⋜ 𝑦 or 2. 𝑥, 𝑦 ∈ 𝑌 and 𝑥 ⊴ 𝑦 The direct sum operation is also called the disjoint union. The notation 𝑛𝙋 is defined as 𝑛𝙋 ≜ 𝙋 + 𝙋 + ⋯ + 𝙋. ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑛 − 1 “+” operations

Definition A.15. 40 Let 𝑋 and 𝑌 be disjoint sets. Let 𝙋 ≜ ( 𝑋, ⋜) and 𝙌 ≜ ( 𝑌 , ⊴) be ordered sets on 𝑋 and 𝑌 . The direct product of 𝙋 and 𝙌 is defined as 𝙋 × 𝙌 ≜ ( 𝑋 × 𝑌 , ≤) where (𝑥1 , 𝑦1 ) ≤ (𝑥2 , 𝑦2 ) if 𝑥1 ⋜ 𝑥2 and 𝑦1 ⋜ 𝑦2 . 36

📘 Grätzer (2003) page 2 📘 Grätzer (2003) page 2, 📘 Birkhoff (1967) page 5 38 📃 Dilworth (1950a) page 161, 📓 Dilworth (1950b), 📃 Farley (1997) page 4 39 📘 Stanley (1997) page 100 40 📘 Stanley (1997) pages 100–101, 📘 Shen and Vereshchagin (2002) page 43 37



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ORDER STRUCTURES

The direct product operation is also called the cartesian product. The order relation ≤ is called a coordinate wise order relation. The notation 𝙋 𝑛 is defined as 𝙋𝑛 ≜ 𝙋 × 𝙋 × ⋯ × 𝙋. ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑛 − 1 “×” operations

Definition A.16. 41 Let 𝑋 and 𝑌 be disjoint sets. Let 𝙋 ≜ ( 𝑋, ⋜) and 𝙌 ≜ ( 𝑌 , ⊴) be ordered sets on 𝑋 and 𝑌 . The ordinal sum of 𝙋 and 𝙌 is defined as 𝙋 ⊕ 𝙌 ≜ ( 𝑋 ∪ 𝑌 , ≤) where 𝑥 ≤ 𝑦 if 1. 𝑥, 𝑦 ∈ 𝑋 and 𝑥 ⋜ 𝑦 or 2. 𝑥, 𝑦 ∈ 𝑌 and 𝑥 ⊴ 𝑦 or 3. 𝑥 ∈ 𝑋 and 𝑦 ∈ 𝑌 . Definition A.17. 42 Let 𝑋 and 𝑌 be disjoint sets. Let 𝙋 ≜ ( 𝑋, ⋜) and 𝙌 ≜ ( 𝑌 , ⊴) be ordered sets on 𝑋 and 𝑌 . The ordinal product of 𝙋 and 𝙌 is defined as 𝙋 ⊗ 𝙌 ≜ ( 𝑋 × 𝑌 , ≤) 1. 𝑥1 ≠ 𝑥2 and 𝑥1 ⋜ 𝑥2 or where (𝑥1 , 𝑦1 ) ≤ (𝑥2 , 𝑦2 ) if { 2. 𝑥1 = 𝑥2 and 𝑦1 ⊴ 𝑦2 } The order relation ≤ is called a lexicographical order relation, dictionary order relation, or alphabetic order relation. Definition A.18. 43 Let 𝙋 ≜ ( 𝑋, ≤) be an ordered set. Let ≥ be the dual order relation of ≤. The dual of 𝙋 is defined as 𝙋 ∗ ≜ ( 𝑋, ≥) Definition A.19. 44 Let 𝑋 and 𝑌 be disjoint sets. Let 𝙋 ≜ ( 𝑋, ⋜) and 𝙌 ≜ ( 𝑌 , ⊴) be ordered sets on 𝑋 and 𝑌 . 𝙌 𝙋 ≜ ( {𝖿 ∈ 𝑌 𝑋 |𝖿 is order preserving } , ≤) where 𝖿 ≤ 𝗀 if 𝖿(𝑥) ≤ 𝗀(𝑥) ∀𝑥 ∈ 𝑋. The order relation ≤ is called a pointwise order relation. Theorem A.20 (cardinal arithmetic). 45 Let 𝙋 ≜ ( 𝑋, ≤) be an ordered set. 1. 𝙋 + 𝙌 = 𝙌 +𝙋 (commutative) 2. 𝙋 × 𝙌 = 𝙌 ×𝙋 (commutative) 3. (𝙋 + 𝙌 ) + ( ℝ, ≤) = 𝙋 + (𝙌 + ( ℝ, ≤)) (associative) 4. (𝙋 × 𝙌 ) × ( ℝ, ≤) = 𝙋 × (𝙌 × ( ℝ, ≤)) (associative) 5. 𝙋 × (𝙌 + ( ℝ, ≤)) = (𝙋 × 𝙌 ) + (𝙋 × ( ℝ, ≤)) (distributive) 𝙋 +𝙌 6. ( ℝ, ≤) = ( ℝ, ≤)𝙋 × ( ℝ, ≤)𝙌 𝙌 ( ℝ, ≤) 7. (𝙋 ) = 𝙋 𝙌×( ℝ, ≤) Definition A.21. The ordered set 𝙇1 is defined as ({𝑥}, ≤), for some value 𝑥. It is illustrated by the Hasse diagram to the right. Definition A.22. The ordered set 𝙇2 is defined as 𝙇2 ≜ 𝙇21 . It is illustrated by the Hasse diagram to the right.

A.1.4

Decomposition examples

Example A.23. Figure 12 (page 31) illustrates the four ordered set operations +, ×, ⊕, and ⊗. 41

📘 Stanley (1997) page 100 📘 Stanley (1997) page 101, 📘 Shen and Vereshchagin (2002) page 44, 📘 Halmos (1960) page 58, 📘 Hausdorff (1937) page 54 43 📘 Stanley (1997) page 101 44 📘 Stanley (1997) page 101 45 📘 Stanley (1997) page 102 42

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+

=

⊕

=

×

=

⊗

=

page 31

Figure 12: Operations on ordered sets (Example A.23 page 30) longest antichain

1 𝑝

𝑞

𝑟

𝑏 𝑎

𝑐 0

{𝑏, 𝑝}, {𝑐, 𝑞}, {𝑟, 1} } partition 1: { {0, 𝑎}, {𝑏}, {𝑐, 𝑞}, {𝑟, 1} } partition 2: { {0, 𝑎, 𝑝}, {𝑐, 𝑞}, {𝑟} partition 3: { {0, 𝑎, 𝑝, 1}, {𝑏}, } {𝑐, 𝑞}, {𝑟} partition 4: { {0, 𝑏, 𝑝, 1}, {𝑎}, } {𝑞} partition 5: { {0, 𝑐, 𝑟, 1}, {𝑎, 𝑝}, {𝑏}, } {0, {𝑎, {𝑏}, {𝑟} partition 6: 𝑐, 𝑞, 1}, 𝑝}, { } ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ examples of partitions of chains

Figure 13: Lattice of width 4 and examples of minimal order partitions of chains (Example A.26 page 31) Example A.24. 46 The ordered set 𝑛𝙇1 is the anti-chain with 𝑛 elements. The ordered set 4𝙇1 is illustrated to the right. Example A.25. The ordered set 𝟙𝑛 is the chain with 𝑛 elements. The ordered set 𝟙4 is illustrated to the right.

Examples of the Boolean lattices ple A.74 (page 41).


𝙇12 , 𝙇22 , 𝙇32 , 𝙇42 and 𝙇52 are illustrated in Exam-

Example A.26. 47 The longest antichain (Definition A.11 page 29) in the lattice illustrated in Figure 13 (page 31) has 4 elements giving this ordered set a width (Definition A.12 page 29) of 4. The longest chain also has 4 elements, giving the ordered set a length (Definition A.12 page 29) of 3. By Dilworth's theorem (Theorem A.13 page 29), the smallest partition consists of four chains (Definition A.4 page 27). Examples of such minimal order partitions are listed in Figure 13. Definition A.27. Let (𝑋, ≤) be an ordered set and 𝟚𝑋 the power set of 𝑋. For any set 𝐴 ∈ 𝟚𝑋 , 𝑐 is an upper bound of 𝐴 in (𝑋, ≤) if 1. 𝑥 ≤ 𝑐 ∀𝑥 ∈ 𝐴. An element 𝑏 is the least upper bound, or lub, of 𝐴 in (𝑋, ≤) if 2. 𝑏 and 𝑐 are upper bounds of 𝐴 ⟹ 𝑏 ≤ 𝑐. 46 47

📘 Stanley (1997) page 100 Farley (1997) page 4

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The least upper bound of the set 𝐴 is denoted ⋁ 𝐴. It is also called the supremum of 𝐴, which is denoted sup 𝐴. The join 𝑥 ∨ 𝑦 of 𝑥 and 𝑦 is defined as 𝑥 ∨ 𝑦 ≜ ⋁ {𝑥, 𝑦}. Definition A.28. Let (𝑋, ≤) be an ordered set and 𝟚𝑋 the power set of 𝑋. For any set 𝐴 ∈ 𝟚𝑋 , 𝑝 is a lower bound of 𝐴 in (𝑋, ≤) if 1. 𝑝 ≤ 𝑥 ∀𝑥 ∈ 𝐴. An element 𝑎 is the greatest lower bound, or glb, of 𝐴 in (𝑋, ≤) if 2. 𝑎 and 𝑝 are lower bounds of 𝐴 ⟹ 𝑝 ≤ 𝑎. The greatest lower bound of the set 𝐴 is denoted ⋀ 𝐴. It is also called the infimum of 𝐴, which is denoted inf 𝐴. The meet 𝑥 ∧ 𝑦 of 𝑥 and 𝑦 is defined as 𝑥 ∧ 𝑦 ≜ ⋀ {𝑥, 𝑦}. Proposition A.29. Let ( 𝑋, ∨, ∧ ; ≤) be an ordered set (Definition A.1 page 27). 1. 𝑥 ∧ 𝑦 = 𝑥 𝑎𝑛𝑑 𝑥 ≤ 𝑦 ⟺ ∀𝑥,𝑦∈𝑋 { 2. 𝑥 ∨ 𝑦 = 𝑦 } Proposition A.30. Let 𝟚𝑋 be the power set of a set 𝑋. 1. ⋁ 𝐴 ≤ ⋁ 𝐵 and 𝐴⊆𝐵 ⟹ ∀𝐴,𝐵∈𝟚𝑋 { 2. ⋀ 𝐴 ≤ ⋀ 𝐵 }

A.2

Lattices

A.2.1

Definition

The structure available in an ordered set (Definition A.1 page 27) tends to be insufficient to ensure “wellbehaved” mathematical systems. This situation is greatly remedied if every pair of elements in the ordered set has both a least upper bound and a greatest lower bound (Definition A.28 page 32) in the set; in this case, that ordered set is a lattice (next definition). Gian-Carlo Rota (1932–1999) has illustrated 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”.48 Definition A.31. 49 An algebraic structure 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) is a lattice if 1. (𝑋, ≤) is an ordered set ((𝑋, ≤) is a partially or totally ordered set) and 2. ∃𝑥 ∨ 𝑦 ∈ 𝑋 ∀𝑥, 𝑦 ∈ 𝑋 (every pair of elements in 𝑋 has a least upper bound in 𝑋) and 3. ∃𝑥 ∧ 𝑦 ∈ 𝑋 ∀𝑥, 𝑦 ∈ 𝑋 (every pair of elements in 𝑋 has a greatest lower bound in 𝑋). 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 A.4 page 27). Theorem A.32. 50 ( 𝑋, ∨, ∧ ; ≤) is a lattice 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 ∧ 𝑦) = 𝑥 48 49 50

= = = =

⟺ 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).

⎫ ⎪ ⎬ ⎪ ⎭

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 (1933) page 442, 📘 Maeda and Maeda (1970), page 1 📘 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 (1933) page 442, 📃 Dedekind (1900) pages 371–372 ⟨(1)–(4)⟩, 📘 Greenhoe (2016a), page 62 ⟨Theorem 4.3⟩ 📃

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Lemma A.33. 51 Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be lattice (Definition A.31 page 32). 𝑥≤𝑦 ⟺ 𝑥=𝑥∧𝑦 ∀𝑥,𝑦∈𝙇 ✎PROOF: 1. Proof for ⟹ case: by left hypothesis and definition of ∧ (Definition A.28 page 32). 2. Proof for ⟸ case: by right hypothesis and definition of ∧ (Definition A.28 page 32).

✏ Proposition A.34 (Monotony laws). 52 Let ( 𝑋, ∨, ∧ ; ≤) be a lattice. 𝑎 ≤ 𝑏 and 𝑎 ∧ 𝑥 ≤ 𝑏 ∧ 𝑦 and ⟹ { 𝑥 ≤ 𝑦 } { 𝑎∨𝑥 ≤ 𝑏∨𝑦 } Theorem A.35 (Minimax inequality). 𝑚

𝑛

𝑛

53

Let ( 𝑋, ∨, ∧ ; ≤) be a lattice.

𝑚

𝑥 ≤ 𝑥 ⋁ ⋀ 𝑖𝑗 ⋀ ⋁ 𝑖𝑗 𝑖=1 𝑗=1 𝑗=1 𝑖=1 ⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟ maxmini: largest of the smallest

∀𝑥𝑖𝑗 ∈ 𝑋

minimax: smallest of the largest

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 A.54 page 36); and in this case, the lattice is a called a distributive lattice (Definition A.53 page 36). Theorem A.36 (distributive inequalities). 54 ( 𝑋, ∨, ∧ ; ≤) is a lattice ⟹ 𝑥 ∧ (𝑦 ∨ 𝑧) ≥ (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ∀𝑥,𝑦,𝑧∈𝑋 (join super-distributive) ⎧ ⎪ 𝑥 ∨ (𝑦 ∧ 𝑧) ≤ (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) ∀𝑥,𝑦,𝑧∈𝑋 (meet sub-distributive) ⎨ ⎪ (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ∨ (𝑦 ∧ 𝑧) ≤ (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) ∧ (𝑦 ∨ 𝑧) ∀𝑥,𝑦,𝑧∈𝑋 (median inequality). ⎩

and and

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 A.47 page 35). Theorem A.37 (Modular inequality). 55 Let ( 𝑋, ∨, ∧ ; ≤) be a lattice (Definition A.31 page 32). 𝑥≤𝑦 ⟹ 𝑥 ∨ (𝑦 ∧ 𝑧) ≤ 𝑦 ∧ (𝑥 ∨ 𝑧) Theorem A.32 (page 32) gives 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. 51

📘 Holland (1970), 📘 Greenhoe (2016b), pages 62–63 ⟨Lemma 4.1⟩ 📘 Givant and Halmos (2009) page 39, 📃 Doner and Tarski (1969) pages 97–99, 📘 Greenhoe (2016b), page 63 ⟨Proposition 4.1⟩ 53 📘 Birkhoff (1948) pages 19–20, 📘 Greenhoe (2016b), page 64 ⟨Theorem 4.5⟩ 54 📘 Davey and Priestley (2002) page 85, 📘 Grätzer (2003) page 38, 📃 Birkhoff (1933) page 444, 📃 Korselt (1894) page 157, 📘 Müller-Olm (1997) page 13 ⟨terminology⟩, 📘 Greenhoe (2016b), pages 64–65 ⟨Theorem 4.6⟩ 55 📘 Birkhoff (1948) page 19, 📘 Burris and Sankappanavar (1981) page 11, 📃 Dedekind (1900) page 374, 📘 Greenhoe (2016b), page 65 ⟨Theorem 4.7⟩

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Theorem A.38. 56 ( 𝑋, ∨, ∧ ; ≤) is a lattice ⟺ 𝑥 ∨ 𝑦 = 𝑦 ∨ 𝑥 𝑥∧𝑦 = 𝑦∧𝑥 ⎧ ⎪ ⎨ (𝑥 ∨ 𝑦) ∨ 𝑧 = 𝑥 ∨ (𝑦 ∨ 𝑧) (𝑥 ∧ 𝑦) ∧ 𝑧 = 𝑥 ∧ (𝑦 ∧ 𝑧) ⎪ 𝑥 ∧ (𝑥 ∨ 𝑦) = 𝑥 ⎩ 𝑥 ∨ (𝑥 ∧ 𝑦) = 𝑥

A.2.2

ORDER STRUCTURES

∀𝑥,𝑦∈𝑋

(commutative)

and


(associative)

and

∀𝑥,𝑦∈𝑋

(absorptive)

⎫ ⎪ ⎬ ⎪ ⎭

Bounded lattices

Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. By the definition of a lattice (Definition A.31 page 32), 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) (Definition A.27 page 31, Definition A.28 page 32)? If both of these are in 𝑋, then the lattice 𝙇 is said 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. Definition A.39. 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. 𝙇 is lower bounded if (⋀ 𝑋 ) ∈ 𝑋. 𝙇 is bounded if 𝙇 is both upper and lower bounded. A bounded lattice is optionally denoted ( 𝑋, ∨, ∧, 0, 1 ; ≤), where 0 ≜ ⋀ 𝑋 and 1 ≜ ⋁ 𝑋. Proposition A.40. Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. ⟹ {𝙇 is finite} {𝙇 is bounded} 57

Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice with ⋁ 𝑋 ≜ 1 and ⋀ 𝑋 ≜ 0. ⎧ 𝑥 ∨ 1 = 1 ∀𝑥∈𝑋 (upper bounded) and ⎫ ⎪ 𝑥 ∧ 0 = 0 ∀𝑥∈𝑋 (lower bounded) and ⎪ ⟹ {𝙇 is bounded} ⎨ 𝑥 ∨ 0 = 𝑥 ∀𝑥∈𝑋 (join-identity) ⎬ and ⎪ ⎪ ⎩ 𝑥 ∧ 1 = 𝑥 ∀𝑥∈𝑋 (meet-identity) ⎭

Proposition A.41.

Definition A.42. 58 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice (Definition A.39 page 34). The height 𝗁(𝑥) of a point 𝑥 ∈ 𝙇 is the least upper bound of the lengths (Definition A.12 page 29) of all the chains that have 0 and in which 𝑥 is the least upper bound. The height 𝗁(𝙇) of the lattice 𝙇 is defined as 𝗁(𝙇) ≜ 𝗁(1) . Example A.43. The height of the lattice illustrated in Figure 13 (page 31) is 3 because 𝗁(𝙇) ≜ 𝗁(1) 𝓁(𝘾 )|𝘾 is a chain in 𝙇 containing both 0 and 1} ⋁{ = 𝓁 ({0, 𝑎, 𝑝, 1}, ≤) , 𝓁 ({0, 𝑏, 𝑝, 1}, ≤) , 𝓁 ({0, 𝑐, 𝑝, 1}, ≤) , 𝓁 ({0, 𝑐, 𝑞, 1}, ≤) , ⋁{ 𝓁 ({0, 𝑐, 𝑟, 1}, ≤) , } {4 − 1, 4 − 1, 4 − 1, 4 − 1, 4 − 1} = ⋁ {3, 3, 3, 3, 3} = ⋁ =3 ≜

56

📘 Padmanabhan and Rudeanu (2008) pages 7–8, 📘 Beran (1985) page 5, 📃 McKenzie (1970) page 24, Greenhoe (2014b) ⟨Theorem 1.22⟩, 📘 Greenhoe (2016b), pages 68–70 ⟨Theorem 4.8⟩ 📃 Greenhoe (2014b) ⟨§1.2.2⟩, 📘 Greenhoe (2016b), pages 71–72 ⟨Proposition 4.4⟩ 📘 Birkhoff (1967) page 5

📃 57 58

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atomic lattices

anti-atomic

page 35

atomic and anti-atomic

neither atomic nor anti-atomic

Figure 14: Selected atomic, anti-atomic, and neither atomic nor anti-atomic lattices (see Example A.45 page 35) A.2.3 Atomic lattices Definition A.44. 59 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice (Definition A.39 page 34). 𝑥 is an atom of 𝙇 if 𝑥 covers (Definition A.5 page 28) 0. 𝑥 is an anti-atom of 𝙇 if 𝑥 is covered by 1. 𝙇 is atomic if every 𝑥 ∈ 𝑋 ⧵0 can be represented as joins of atoms of 𝙇. 𝙇 is anti-atomic if every 𝑥 ∈ 𝑋 ⧵1 can be represented as meets of anti-atoms of 𝙇. Example A.45. Figure 14 (page 35) illustrates some examples of lattices that are atomic, anti-atomic, both, and neither.

A.2.4 Modular Lattices Definition A.46. 60 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

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. Modular lattices are a generalization of distributive lattices (Definition A.53 page 36) in that all distributive lattices are modular, but not all modular lattices are distributive (Example A.61 page 37, Example A.62 page 38). Definition A.47.

61

A lattice ( 𝑋, ∨, ∧ ; ≤) is modular if

𝑥Ⓜ𝑦

∀𝑥, 𝑦 ∈ 𝑋.

62

Theorem A.48. Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. {𝑥 ≤ 𝑦 𝙇 is modular ⟺ ⟹ 𝑥 ∨ (𝑧 ∧ 𝑦) = (𝑥 ∨ 𝑧) ∧ 𝑦} ⟺ 𝑥 ∨ [(𝑥 ∨ 𝑦) ∧ 𝑧] = (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) ⟺ 𝑥 ∧ [(𝑥 ∧ 𝑦) ∨ 𝑧] = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) 59

∀𝑥,𝑦,𝑧∈𝑋 ∀𝑥,𝑦,𝑧∈𝑋 ∀𝑥,𝑦,𝑧∈𝑋

Larson and Andima (1975) page 178, 📃 Birkhoff (1938) page 800 ⟨see footnote ‡⟩ 📘 Stern (1999) page 11, 📘 Maeda and Maeda (1970), page 1 ⟨Definition (1.1)⟩, 📘 Maeda (1966) page 248 61 📘 Birkhoff (1967) page 82, 📘 Maeda and Maeda (1970), page 3 ⟨Definition (1.7)⟩ 62 📘 Padmanabhan and Rudeanu (2008) page 39, 📘 Ore (1935) page 413 ⟨(2)⟩, 📘 Greenhoe (2016b), pages 81–82 ⟨Theorem 5.1⟩ 📃

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Definition A.49 (N5 lattice/pentagon). 63 ({0, 𝑎, 𝑏, 𝑝, 1}, ≤) with cover relation ≺= {(0, 𝑎) , (𝑎, 𝑏) , (𝑏, 1) , (𝑝, 1) , (0, 𝑝)}. The N5 lattice is also called the pentagon. Hasse diagram to the right.

A

ORDER STRUCTURES

The N5 lattice is the ordered set

The N5 lattice is illustrated by the

1 𝑏 𝑝 𝑎 0

Theorem A.50. 64 Let 𝙇 be a lattice (Definition A.31 page 32). 𝙇 is modular (Definition A.47 page 35) ⟺ 𝙇 does not contain the N5 lattice (Definition A.49 page 36). Theorem A.51. 65 Let 𝘼 ≜ ( 𝑋, ∨, ∧ ; ≤) be an algebraic structure. (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) = [(𝑧 ∧ 𝑥) ∨ 𝑦] ∧ 𝑥 ∀𝑥,𝑦,𝑧∈𝑋 and 𝘼 is a ⟺ { [𝑥 ∨ (𝑦 ∨ 𝑧)] ∧ 𝑧 = 𝑧 } { modular lattice } ∀𝑥,𝑦,𝑧∈𝑋 Examples of modular lattices are provided in Example A.61 (page 37) and Example A.62 (page 38).

A.2.5

Distributive Lattices

Definition A.52. 66 Let ( 𝑋, ∨, ∧ ; ≤) be a lattice (Definition A.31 page 32). 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 Ⓓ∗ ≜ {(𝑥, 𝑦, 𝑧) ∈ 𝑋 3 |𝑥 ∨ (𝑦 ∧ 𝑧) = (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧)} (each (𝑥, 𝑦, 𝑧) is conjunctive distributive). A triple (𝑥, 𝑦, 𝑧) ∈ Ⓓ is alternatively denoted as (𝑥, 𝑦, 𝑧) Ⓓ, and is a distributive triple. A triple (𝑥, 𝑦, 𝑧) ∈ Ⓓ∗ is alternatively denoted as (𝑥, 𝑦, 𝑧) Ⓓ∗ , and is a dual distributive triple. Definition A.53.

67

A lattice ( 𝑋, ∨, ∧ ; ≤) is distributive if

(𝑥, 𝑦, 𝑧) ∈ Ⓓ

∀𝑥, 𝑦, 𝑧 ∈ 𝑋

Not all lattices are distributive. But if a lattice 𝙇 does happen to be distributive (Definition A.53 page 36)— that is all triples in 𝙇 satisfy the distributive property (Definition A.53 page 36)—then all triples in 𝙇 also satisfy the dual distributive property, as well as another property called the median property. The converses also hold (next theorem). Theorem A.54. 68 Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice (Definition A.31 page 32). 𝙇 is distributive (Definition A.53 page 36) ⟺ 𝑥 ∧ (𝑦 ∨ 𝑧) = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ∀𝑥,𝑦,𝑧∈𝑋 (disjunctive distributive) ⟺ 𝑥 ∨ (𝑦 ∧ 𝑧) = (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) ∀𝑥,𝑦,𝑧∈𝑋 (conjunctive distributive) ⟺ (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) ∧ (𝑦 ∨ 𝑧) = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) ∨ (𝑦 ∧ 𝑧) ∀𝑥,𝑦,𝑧∈𝑋 (median property) 63

📘 Beran (1985) pages 12–13, 📃 Dedekind (1900) pages 391–392 ⟨(44) and (45)⟩ 📘 Burris and Sankappanavar (1981) page 11, 📘 Grätzer (1971) page 70, 📃 Dedekind (1900) ⟨cf Stern 1999 page 10⟩, 📘 Greenhoe (2016b), pages 82–84 ⟨Theorem 5.2⟩ 65 📘 Padmanabhan and Rudeanu (2008) pages 42–43, 📃 Riečan (1957) 66 📘 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 functions⟩ 67 📘 Burris and Sankappanavar (1981) page 10, 📘 Birkhoff (1948) page 133, 📃 Ore (1935) page 414 ⟨arithmetic axiom⟩, 📃 Birkhoff (1933) page 453, 📘 Balbes and Dwinger (1975) page 48 ⟨Definition II.5.1⟩ 68 📃 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⟩, 📘 Greenhoe (2016b), pages 90–92 ⟨Theorem 6.1⟩ 64

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Lattices

Definition A.55 (M3 lattice/diamond). 69 The M3 lattice is the ordered set ({0, 𝑝, 𝑞, 𝑟, 1}, ≤) with covering relation ≺= {(𝑝, 1) , (𝑞, 1) , (𝑟, 1) , (0, 𝑝) , (0, 𝑞) , (0, 𝑟)}. The M3 lattice is also called the diamond, and is illustrated by the Hasse diagram to the right. Lemma A.56. 70 𝙇 is an { M3 lattice }

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Daniel J. Greenhoe

⟹

1.

{

2.

𝙇 is not distributive 𝙇 is modular

(Definition A.53 page 36) (Definition A.47 page 35)

1 𝑞 𝑝

𝑟 0

and

}

Theorem A.57 (Birkhoff distributivity criterion). 71 Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. r r r ⎧ ⎪ 𝙇 does not contain N5 as a sublattice r r and 𝙇 is distributive ⟺ r ⎨ rrr ⎪ r ⎩ 𝙇 does not contain M3 as a sublattice 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 is modular, but yet it is not distributive. Theorem A.58. 72 Let ( 𝑋, ∨, ∧ ; ≤) be a lattice. ⟹ {( 𝑋, ∨, ∧ ; ≤) is distributive} ⟸ /

{( 𝑋, ∨, ∧ ; ≤) is modular}

Theorem A.59. 73 Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice (Definition A.31 page 32). 1. 𝙇 is distributive and ⎫ ⎧ ⎪ ⎪ and ⎬ {𝑎 = 𝑏} ⟹ ∀𝑥, 𝑎, 𝑏 ∈ 𝑋 ⎨ 2. 𝑥 ∨ 𝑎 = 𝑥 ∨ 𝑏 ⎪ ⎪ 3. 𝑥 ∧ 𝑎 = 𝑥 ∧ 𝑏 ⎩ ⎭ Proposition A.60. 74 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 𝑙𝑛 1 1 1 1 2 5 15 53 222 1078 5994 37622 262776 2018305 16873364 𝑚𝑛 1 1 1 1 2 4 8 16 34 72 157 343 766 1718 3899 𝑑𝑛 1 1 1 1 2 3 5 8 15 26 47 82 151 269 494 Example A.61. 75 There are a total of 5 unlabeled lattices on a five element set. Of these, 3 are distributive (Proposition A.60 page 37, and thus also modular), one is modular but non-distributive, and one is non-distributive (and non-modular). distributive (and modular) modular non-distributive

69

📘 Beran (1985) pages 12–13, 📘 Korselt (1894) page 157 ⟨𝑝1 ≡ 𝑥, 𝑝2 ≡ 𝑦, 𝑝3 ≡ 𝑧, 𝑔 ≡ 1, 0 ≡ 0⟩ 📘 Birkhoff (1948) page 6, 📘 Burris and Sankappanavar (1981) page 11, 📘 Korselt (1894) page 157 ⟨cf Salii1988 p. 37⟩, 📘 Greenhoe (2016b), pages 93–94 ⟨Lemma 6.2⟩ 71 📘 Burris and Sankappanavar (1981) page 12, 📘 Birkhoff (1948) page 134, 📃 Birkhoff and Hall (1934) ⟨cf Stern 1999 page 9⟩, 📘 Greenhoe (2016b), pages 94–97 ⟨Theorem 6.2⟩ 72 📘 Birkhoff (1948) page 134, 📘 Burris and Sankappanavar (1981) page 11, 📘 Greenhoe (2016b), page 105 ⟨Theorem 6.5⟩ 73 📘 MacLane and Birkhoff (1999) pages 484–485 74 💻 Sloane (2014) ⟨http://oeis.org/A006966⟩, 💻 Sloane (2014) ⟨http://oeis.org/A006982⟩, 💻 Sloane (2014) ⟨http://oeis.org/A006981⟩,📃 Heitzig and Reinhold (2002) ⟨𝑙𝑛 ⟩, 📃 Erné et al. (2002) page 17 ⟨𝑑𝑛 ⟩, 📃 Thakare et al. (2002) 75 📃 Erné et al. (2002) pages 4–5, 📘 Greenhoe (2016b), page 107 ⟨Example 6.2⟩ 70



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A

ORDER STRUCTURES

Example A.62. 76 There are a total of 15 unlabeled lattices on a six element set. Of these, 5 are distributive (Proposition A.60 page 37, and modular), 3 are modular but non-distributive, and 7 are nondistributive (and non-modular). distributive (and modular) modular but non-distributive

non-distributive (and non-modular)

A.2.6

Complemented lattices

Definition A.63. 77 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice (Definition A.39 page 34). 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). An element 𝑥′ in 𝙇 is the unique complement of 𝑥 in 𝙇 if 𝑥′ is a complement of 𝑥 and 𝑦′ is a 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. Example A.64. Here are some examples: non-complemented lattices

uniquely complemented lattices

multiply complemented lattices

Example A.65. Of the 53 unlabeled lattices on a 7 element set, 0 are uniquely complemented, 17 are multiply complemented, and 36 are non-complemented. Theorem A.66 (next) is a landmark theorem in mathematics. Theorem A.66. 78 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. 76

📘 Greenhoe (2016b), page 86 ⟨Example 5.6⟩ 📘 Stern (1999) page 9, 📘 Birkhoff (1948) page 23 78 📃 Dilworth (1945) page 123, 📘 Saliǐ (1988) page 51, 📘 Grätzer (2003) page 378 ⟨Corollary 3.8⟩

77

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Corollary A.67. 79 Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice. 1. 𝙇 is distributive and ⟹ {𝙇 is uniquely complemented} ⟸ / { 2. 𝙇 is complemented } Theorem A.68 (Huntington properties). 80 Let 𝙇 be a lattice. or ⎧ 𝙇 is modular ⎫ ⎪ 𝙇 is 𝙇 is atomic or ⎪ ⎧ ⎫ 𝙇 is ⎪ ⎪ ⎪ ⎪ ⎨ uniquely ⎬ and ⎨ 𝙇 is orthocomplemented or ⎬ ⟹ { distributive } ⎪ ⎪ 𝙇 has finite width or ⎪ ⎩ complemented ⎪ ⎭ ⎪ 𝙇 is de Morgan ⎪ ⎩ ⎭ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ Huntington properties

A.2.7 Boolean lattices Definition A.69. 81 A lattice (Definition A.31 page 32) 𝙇 is Boolean if 1. 𝙇 is bounded (Definition A.39 page 34) and 2. 𝙇 is distributive (Definition A.53 page 36) and 3. 𝙇 is complemented (Definition A.63 page 38). In this case, 𝙇 is a Boolean algebra or a Boolean lattice. In this paper, a Boolean lattice with 2𝘕 elements is sometimes denoted 𝙇𝘕 2. The next theorem presents the classic properties of any Boolean algebra. The first 4 pairs of properties are true for any lattice (Theorem A.32 page 32). The bounded, distributive, and complemented properties are true by definition of a Boolean lattice (Definition A.69 page 39). Theorem A.70 (classic 10 Boolean properties). 82 Let 𝘼 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an algebraic structure. In the event that 𝑨 is a bounded lattice (Definition A.39 page 34), let 𝑥′ represent a complement (Definition A.63 page 38) of an element 𝑥 in 𝘼. 𝘼 is a Boolean algebra ⟺ ∀𝑥, 𝑦, 𝑧 ∈ 𝑋 𝑥∨𝑥 = 𝑥 𝑥∧𝑥 = 𝑥 (idempotent) and 𝑥∨𝑦 = 𝑦∨𝑥 𝑥∧𝑦 = 𝑦∧𝑥 (commutative) and 𝑥 ∨ (𝑦 ∨ 𝑧) = (𝑥 ∨ 𝑦) ∨ 𝑧 𝑥 ∧ (𝑦 ∧ 𝑧) = (𝑥 ∧ 𝑦) ∧ 𝑧 (associative) and 𝑥 ∨ (𝑥 ∧ 𝑦) = 𝑥 𝑥 ∧ (𝑥 ∨ 𝑦) = 𝑥 (absorptive) and 𝑥∨1 = 1 𝑥∧0 = 0 (bounded) and 𝑥∨0 = 𝑥 𝑥∧1 = 𝑥 (identity) and 𝑥 ∨ (𝑦 ∧ 𝑧) = (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) 𝑥 ∧ (𝑦 ∨ 𝑧) = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) (distributive) and ′ ′ 𝑥∨𝑥 = 1 𝑥∧𝑥 = 0 (complemented) and (𝑥 ∨ 𝑦)′ = 𝑥 ′ ∧ 𝑦′ (𝑥 ∧ 𝑦)′ = 𝑥′ ∨ 𝑦′ (de Morgan) and ′ ′ (involutory) (𝑥 ) = 𝑥 disjunctive properties

conjunctive properties

Proposition A.71 (Huntington's fourth set).

83

property name

Let 𝑨 ≜ ( 𝑋, ∨, ∧ ; ≤) be an algebraic structure. 𝑨 is

79

📘 MacLane and Birkhoff (1999) page 488, 📘 Saliǐ (1988) page 30 ⟨Theorem 10⟩ 📘 Roman (2008) page 103, 📘 Adams (1990) page 79, 📘 Saliǐ (1988) page 40, 📃 Dilworth (1945) page 123, 📘 Grätzer (2007), page 698 81 📘 MacLane and Birkhoff (1999) page 488, 📘 Jevons (1864) 82 📃 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 83 📃 Huntington (1933) page 280 ⟨“4th set”⟩ 80



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a Boolean algebra ⟺ 1. 𝑥 ∨ 𝑥 ⎧ ⎪ 2. 𝑥 ∨ 𝑦 ⎨ 3. (𝑥 ∨ 𝑦) ∨ 𝑧 ⎪ ′ ′ ′ ′ ′ ⎩ 4. (𝑥 ∨ 𝑦 ) ∨ (𝑥 ∨ 𝑦)

A.3

= = = =

A

𝑥 𝑦∨𝑥 𝑥 ∨ (𝑦 ∨ 𝑧) 𝑥

ORDER STRUCTURES

∀𝑥∈𝑋

(idempotent)

and

∀𝑥,𝑦∈𝑋

(commutative)

and


(associative)

and

∀𝑥,𝑦∈𝑋.

(Huntington's axiom)

⎫ ⎪ ⎬ ⎪ ⎭

Orthocomplemented Lattices

Orthocomplemented lattices (Definition A.72 page 40) are a kind of generalization of Boolean algebras. The relationship between lattices of several types, including orthocomplemented and Boolean lattices, is stated in Theorem A.86 (page 45) and illustrated in Figure 15 (page 40). bounded (Definition A.39 page 34)

complemented (Definition A.63 page 38)

modular

orthocomplemented



distributive

orthomodular



modular orthocomplemented (Definition A.85 page 45)

boolean (Definition A.69 page 39)

Figure 15: relationships between selected lattice types (see Theorem A.86 page 45)

A.3.1

Definition

Definition A.72. 84 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice (Definition A.39 page 34). An element 𝑥⟂ ∈ 𝑋 is an orthocomplement of an element 𝑥 ∈ 𝑋 if ⟂⟂ 1. 𝑥 = 𝑥 ∀𝑥∈𝑋 (involutory) and ⟂ 2. 𝑥 ∧ 𝑥 = 0 (non-contradiction) and ∀𝑥∈𝑋 3. 𝑥 ≤ 𝑦 ⟹ 𝑦⟂ ≤ 𝑥⟂ ∀𝑥,𝑦∈𝑋 (antitone). The lattice 𝙇 is orthocomplemented (𝙇 is an orthocomplemented lattice) if every element 𝑥 in 𝑋 has an orthocomplement. The elements {𝑥, 𝑦} are orthocomplemented pairs in 𝙇 if 𝑦 = 𝑥⟂ . 84

📘 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⟩

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Definition A.73. 85 The O6 lattice is the ordered set ({0, 𝑝, 𝑞, 𝑝⟂ , 𝑞 ⟂ , 1}, ≤) with cover relation ≺= {(0, 𝑝) , (0, 𝑞) , (𝑝, 𝑞 ⟂ ) , (𝑞, 𝑝⟂ ) , (𝑝⟂ , 1) , (𝑞 ⟂ , 1)}. The 𝑂6 lattice is illustrated by the Hasse diagram to the right.

𝑞⟂

1

𝑝

𝑝⟂ 𝑞

0

Example A.74. 86 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⟂

𝑝

𝑝 𝑦

𝑥

𝑥

0

𝑂6 lattice

1.

2.

𝑝⟂ 𝑧

𝑦 0

𝑥

𝑝

3.

𝑥⟂ ⟂

𝑝 𝑦

𝑞

𝑧

𝑧

𝑥⟂ 𝑞⟂

𝑝⟂ 𝑦

𝑥 0

0

5.

𝑥⟂

4.

1

𝑦⟂

𝑤 𝑥

𝑦 0

0

𝑂8 lattice

𝑤

⟂ 𝑧⟂ 𝑦

⟂

𝑥

𝑦

𝑧⟂ 𝑦⟂

𝑥⟂

⟂ 𝑧⟂ 𝑦 𝑝

0

1

𝑦⟂ 𝑥⟂ 𝑝 𝑥 𝑦

⟂

1

1

1

𝑥⟂

6.

7.

Lattices that are orthocomplemented and orthomodular but not modular orthocomplemented and hence also non-Boolean: 1

1 𝑑⟂

𝑦⟂

𝑥⟂

𝑧⟂ 𝑝

𝑥

𝑧

𝑦

𝑝⟂

𝑎

𝑐

⟂

𝑧⟂

𝑦⟂

𝑥⟂

𝑏⟂

𝑏

𝑥

𝑦

𝑧

𝑐

0

𝑎⟂

𝑑

0

8.

9.

Lattices that are orthocomplemented, orthomodular, and modular orthocomplemented but non-Boolean: 1 𝑤

1 𝑦

𝑥

𝑧

𝑥

𝑦

0

𝑧

⟂

𝑧

𝑦⟂

𝑥⟂

0

10. 𝑀4 lattice 11. 𝑀6 lattice Lattices that are orthocomplemented, orthomodular, modular orthocomplemented and Boolean:

1 𝑟

1 1 0 1

0

⟂

𝑞

⟂

𝑝⟂

𝑝 𝑞

𝑝

𝑝⟂ 0

𝑟

0

85

📘 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. 86 📘 Beran (1985) pages 33–42, 📃 Maeda (1966) page 250, 📘 Kalmbach (1983) page 24 ⟨Figure 3.2⟩, 📘 Stern (1999) page 12, 📓 Holland (1970), page 50 Tuesday 14th February, 2017 9:12pm UTC Copyright © Daniel J. Greenhoe


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𝐿1 lattice

12.

A

𝐿2 lattice

13.

14.

𝐿22 lattice

ORDER STRUCTURES

𝐿32 lattice

15.

1

1 ⟂

𝑠

𝑟 𝑑

𝑏

𝑎

⟂

𝑑⟂ 𝑝

𝑞

𝑟

𝑞

𝑗⟂

𝑝⟂

⟂

𝑏⟂

𝑎

𝑟⟂

𝑠⟂

𝑡⟂ 𝑔⟂

ℎ⟂

⟂

𝑖

𝑑⟂

𝑓⟂ 𝑒⟂

𝑏

𝑐

𝑑

𝑒

𝑓

𝑟

𝑠

⟂

𝑎

𝑝

𝑠

𝑞⟂

𝑞

𝑔

𝑝⟂

𝑐⟂

𝑏⟂

ℎ

𝑖

𝑎⟂ 𝑗

𝑡

0

0 16.

𝐿42 lattice

17.

𝐿52 lattice 𝙔⟂

𝘕

Example A.75. The structure ( 𝟚ℝ , +, ∩, ∅, 𝙃 ; ⊆) is an orthocomplemented lattice where ℝ𝘕 is an Euclidean space with dimension 𝘕 and ℝ𝘕 𝘕 𝟚 is the set of all subspaces of ℝ and 𝙑 + 𝙒 is the Minkowski sum of subspaces 𝙑 and 𝙒 and 𝙑 ∩ 𝙒 is the intersection of subspaces 𝙑 and 𝙒 .

𝙕 𝙔 𝙕⟂

𝙓

Example A.76. The structure ( 𝟚𝙃 , ⊕, ∩, ∅, 𝙃 ; ⊆) is an orthocomplemented lattice where 𝙃 is a Hilbert space, 𝟚𝙃 is the set of all closed subspaces of 𝙃 , 𝙓 + 𝙔 is the Minkowski sum of subspaces 𝙓 and 𝙔 , 𝙓 ⊕ 𝙔 ≜ (𝙓 + 𝙔 )− is the closure of 𝙓 + 𝙔 , and 𝙓 ∩ 𝙔 is the intersection of subspaces 𝙓 and 𝙔 .

A.3.2

Properties

Theorem A.77. 87 Let 𝑥⟂ be the orthocomplement (Definition A.72 page 40) of an element 𝑥 in a bounded lattice 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤). 0⟂ = 1 (boundary condition) and ⎧ (1). ⟂ ⎪ (2). 1 = 0 (boundary condition) and 𝙇 is ⎫ ⎪ ⎪ ⟂ ⟂ ⟂ ortho⎬ ⟹ ⎨ (3). (𝑥 ∨ 𝑦) = 𝑥 ∧ 𝑦 ∀𝑥,𝑦∈𝑋 (disjunctive de morgan) and ⎪ (4). (𝑥 ∧ 𝑦)⟂ = 𝑥⟂ ∨ 𝑦⟂ ∀𝑥,𝑦∈𝑋 (conjunctive de morgan) and complemented ⎪ ⎭ ⎪ ⟂ ∀𝑥∈𝑋 (excluded middle). ⎩ (5). 𝑥 ∨ 𝑥 = 1 Let 𝑥⟂ ≜ ¬𝑥, where ¬ is an ortho negation function (Definition B.14 page 48). Then this theorem follows directly from Theorem B.21 (page 49). ✏

✎PROOF:

Corollary A.78. Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a lattice (Definition A.31 page 32). 𝙇 is orthocomplemented 𝙇 is complemented ⟹ { (Definition A.72 page 40) } { (Definition A.63 page 38) } 87

📘 Beran (1985) pages 30–31, 📃 Birkhoff and Neumann (1936) page 830 ⟨L74⟩, 📘 Cohen (1989) page 37 ⟨3B.13. Theorem⟩

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✎PROOF:

This follows directly from the definition of orthocomplemented lattices (Definition A.72 page 40) and complemented lattices (Definition A.63 page 38). ✏

Example A.79. 1

1

𝑎

𝑏

𝑝

𝑞 0

The 𝑂6 lattice (Definition A.73 page 41) illustrated to the left is both orthocomplemented (Definition A.72 page 40) and multiply complemented (Definition A.63 page 38). The lattice illustrated to the right is multiply complemented, but is non-orthocomplemented.

𝑎 𝑞

𝑝

𝑟

0

✎PROOF: 1. Proof that 𝑂6 lattice is multiply complemented: 𝑏 and 𝑞 are both complements of 𝑝. 2. Proof that the right side lattice is multiply complemented: 𝑎, 𝑝, and 𝑞 are all complements of 𝑟.

✏

A.3.3 Restrictions resulting in Boolean algebras Proposition A.80. 88 Let 𝙇 = ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice (Definition A.39 page 34). 1. 𝙇 is orthocomplemented (Definition A.72 page 40) and 𝙇 is Boolean ⟹ { 2. 𝙇 is distributive } { (Definition A.69 page 39) } (Definition A.53 page 36) ✎PROOF: 𝙇 is orthocomplemented { 𝙇 is distributive

and

}

⟹

𝙇 is complemented { 𝙇 is distributive

⟹ { 𝙇 is Boolean }

and

}

by Corollary A.78 by Definition A.69

✏ The center of an orthocomplemented lattice is defined later, but here is a characterization involving it now anyways. Proposition A.81. Let 𝙇 = ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a lattice (Definition A.31 page 32). and 𝙇 is 1. 𝙇 is orthocomplemented (Definition A.72 page 40) ⟺ } { Boolean } { 2. Every 𝑥 ∈ 𝙇 is in the center of 𝙇 (Definition C.15 page 54) ✎PROOF: 1. Proof that (1,2) ⟹ Boolean: 𝙇 is Boolean because it satisfies Huntington's Fourth Set page 39), as demonstrated by the following …

(Proposition A.71

(a) Proof that 𝑥 ∨ 𝑥 = 𝑥 (idempotent): 𝙇 is a lattice (by definition of 𝙇), and all lattices are idempotent (Definition A.31 page 32). (b) Proof that 𝑥 ∨ 𝑦 = 𝑦 ∨ 𝑥 (commutative): 𝙇 is a lattice (by definition of 𝙇), and all lattices are commutative (Definition A.31 page 32). 88

📘 Kalmbach (1983) page 22



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ORDER STRUCTURES

(c) Proof that (𝑥 ∨ 𝑦) ∨ 𝑧 = 𝑥 ∨ (𝑦 ∨ 𝑧) (associative): 𝙇 is a lattice (by definition of 𝙇), and all lattices are associative (Definition A.31 page 32). (d) Proof that (𝑥⟂ ∨ 𝑦⟂ )⟂ ∨ (𝑥⟂ ∨ 𝑦)⟂ = 𝑥 (Huntington's axiom): (𝑥⟂ ∨ 𝑦⟂ )⟂ ∨ (𝑥⟂ ∨ 𝑦)⟂ = (𝑥⟂ ⟂ ∧𝑦⟂ ⟂) ∨ (𝑥⟂ ⟂ ∧𝑦⟂ )

by de Morgan property (Theorem A.77 page 42)

= (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑦⟂ )

by involution property (Definition A.72 page 40)

=𝑥

by def. of center

(Definition C.15 page 54)

2. Proof that (1) ⟸ Boolean: (a) Proof that 𝑥 ∨ 𝑥⟂ = 1: by definition of Boolean algebras (Definition A.69 page 39). (b) Proof that 𝑥 ∧ 𝑥⟂ = 0: by definition of Boolean algebras (Definition A.69 page 39). (c) Proof that 𝑥⟂⟂ = 𝑥: by involutory property of Boolean algebra (Theorem A.70 page 39). (d) Proof that 𝑥 ≤ 𝑦 ⟹ 𝑦⟂ ≤ 𝑥⟂ : 𝑦⟂ ≤ 𝑥 ⟂ ⟺ 𝑦 ⟂

= 𝑦 ⟂ ∧ 𝑥⟂

by Lemma A.33 page 33 ⟂

⟺ 𝑦⟂⟂

= (𝑦⟂ ∧ 𝑥⟂ )

⟺ 𝑦⟂⟂

= 𝑦⟂⟂ ∨ 𝑥⟂⟂

by de Morgan property (Theorem A.70 page 39)

⟺ 𝑦

=𝑦∨𝑥

by involutory property (Theorem A.70 page 39)

⟺ 𝑦

=𝑦

by 𝑥 ≤ 𝑦 hypothesis

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 distributive property (Theorem A.70 page 39) by absorptive property (Theorem A.70 page 39) by distributive property (Theorem A.70 page 39)

= 𝑥 ∧ (𝑥 ∨ 𝑦 ) ∧ 1 =𝑥

by absorptive property (Theorem A.70 page 39)

⟹ 𝑥Ⓒ𝑦

by Definition C.9 page 53

by complement property (Theorem A.70 page 39)

∀𝑥, 𝑦 ∈ 𝙇

⟹ 𝑥 is in the center of 𝙇

by Definition C.15 page 54

✏ Example A.82. The 𝑂6 lattice (Definition A.73 page 41) illustrated to the left is orthocomplemented (Definition A.72 page 40) but non-join-distributive (Definition A.53 page 36),and hence non-Boolean. The lattice illustrated to the right is orthocomplemented and distributive and hence also Boolean (Proposition A.80 page 43).

A.3.4

Orthomodular lattices

Definition A.83. 89 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice (Definition A.39 page 34). 𝙇 is orthomodular if 1. 𝙇 is orthocomplemented and ⟂ 2. 𝑥 ≤ 𝑦 ⟹ 𝑥 ∨ (𝑥 ∧ 𝑦) = 𝑦 ∀𝑥,𝑦∈𝑋 (orthomodular identity) 89

📘 Kalmbach (1983) page 22, 📘 Lidl and Pilz (1998) page 90, 📃 Husimi (1937)

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Theorem A.84. 90 Let 𝙇 = ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an algebraic structure. lattice and 𝙇 is a ⎫ ⎧ 𝙇 is an orthomodular ⎧ ⎪ ⎪ 𝑥 ∧ 𝑦 ⟂ ⟂ = 𝑦 ∨ 𝑥⟂ ∧ 𝑦 ⟂ ⎪ ∀𝑥, 𝑦 ∈ 𝑋 ⟹ ( ) ( ) ⎨ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⎬ ⎨ Boolean algebra ⎪ ⎪ ⎪ ⎩ (Definition A.69 page 39) ⎩ Elkan's law ⎭

⎫ ⎪ ⎬ ⎪ ⎭

Definition A.85. Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice (Definition A.39 page 34). 𝙇 is a modular orthocomplemeted lattice if 1. 𝙇 is orthocomplemented (Definition A.72 page 40) and 2. 𝙇 is modular (Definition A.47 page 35) Theorem A.86. 91 Let 𝙇 be a lattice. {𝙇 is Boolean} ⟹ {𝙇 is modular orthocomplemented ⟹ {𝙇 is orthomodular ⟹ {𝙇 is orthocomplemented

Appendix B

(Definition A.85 page 45)} (Definition A.83 page 44)} (Definition A.72 page 40)}

Functions on lattices

B.1 Valuations Definition B.1. 92 Let 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) be a lattice (Definition A.31 page 32). A function v ∈ ℝ𝙓 is a valuation on 𝙇 if v(𝑥 ∨ 𝑦) + v(𝑥 ∧ 𝑦) = v(𝑥) + v(𝑦) ∀𝑥,𝑦∈𝑋 Proposition B.2. Let v ∈ ℝ𝙓 be a function on a lattice 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) (Definition A.31 page 32). ⟹ { 𝙇 is linear (Definition A.31 page 32) } { v is a valuation (Definition B.1 page 45) } ✎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

✏ Example B.3. 93 Consider the real valued lattice 𝙇 ≜ ( ℝ, max, min ; ≤). The absolute value function |⋅| is a valuation on 𝙇. ✎PROOF:

𝙇 is linear

(Definition A.31 page 32),

✏

so v is a valuation by Proposition B.2 (page 45).

Definition B.4. 94 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 2. 𝖽(𝑥, 𝑦) = 0 ⟺ 𝑥 = 𝑦 ∀𝑥,𝑦∈𝑋 (nondegenerate) and 3. 𝖽(𝑥, 𝑦) = 𝖽(𝑦, 𝑥) ∀𝑥,𝑦∈𝑋 (symmetric) and 4. 𝖽(𝑥, 𝑦) ≤ 𝖽(𝑥, 𝑧) + 𝖽(𝑧, 𝑦) ∀𝑥,𝑦,𝑧∈𝑋 (subadditive/triangle inequality).95 A metric space is the pair (𝑋, 𝖽). A metric is also called a distance function. 90

Renedo et al. (2003) page 72 📘 Kalmbach (1983) page 32 ⟨20.⟩, 📓 Iturrioz (1985), page 57 92 📘 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⟩ 93 📘 Khamsi and Kirk (2001) page 119 ⟨§5.7⟩ 94 📘 Dieudonné (1969), page 28, 📘 Copson (1968), page 21, 📘 Hausdorff (1937) page 109, 📘 Fréchet (1928), 📘 Fréchet (1906) page 30 📃

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FUNCTIONS ON LATTICES

Definition B.5. 96 Let (𝑋, 𝖽) be a metric space (Definition B.4 page 45). An open ball centered at 𝑥 with radius 𝑟 is the set 𝖡(𝑥, 𝑟) ≜ {𝑦 ∈ 𝑋|𝖽(𝑥, 𝑦) ⪇ 𝑟}. A closed ball centered at 𝑥 with radius 𝑟 is the set 𝖡 (𝑥, 𝑟) ≜ {𝑦 ∈ 𝑋|𝖽(𝑥, 𝑦) ≤ 𝑟}. A unit ball centered at 𝑥 is the set 𝖡(𝑥, 1). 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 A.31 page 32). 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 B.6. 1. 2.

97

Definition B.7. 98 Let v be a valuation (Definition B.1 page 45) on a lattice 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) (Definition A.31 page 32). Let 𝖽(𝑥, 𝑦) be the metric defined in Theorem B.6 (page 46). The pair (𝙇, 𝖽) is called a metric lattice. For finite modular lattices, the height function 𝗁(𝑥) (Definition A.42 page 34) can serve as the isotone valuation that induces a metric (next proposition). Proposition B.8. 99 Let 𝗁(𝑥) be the height (Definition A.42 page 34) of a point 𝑥 in a bounded lattice (Definition A.39 page 34) 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤). { 1. 𝙇 is modular and 2. 𝙇 is finite } 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 (positive) 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 B.9. 100 Let v be a valuation (Definition B.1 page 45) on a lattice 𝙇 ≜ ( 𝑋, ∨, ∧ ; ≤) (Definition A.31 page 32). Let 𝖽(𝑥, 𝑦) be the metric defined in Theorem B.6 (page 46). (𝙇, 𝖽) is a metric lattice 𝙇 is modular ⟹ { (Definition A.47 page 35) } { } (Definition B.7 page 46) Example B.10. The function 𝗁 on the Boolean (and thus also modular) lattice 𝙇32 illustrated to the right is a valuation (Definition B.1 page 45) that is positive (and thus also isotone, Proposition B.8 page 46). Therefore 𝖽(𝑥, 𝑦) ≜ 𝗁(𝑥 ∨ 𝑦) − 𝗁(𝑥 ∧ 𝑦) ∀𝑥,𝑦∈𝑋 3 is a metric (Definition B.7 page 46) on 𝙇2 . 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 B.5 page 46) and illustrated with solid dots to the right is 𝖡(𝑏, 1) ≜ {𝑥 ∈ 𝑋|𝖽(𝑏, 𝑥) ≤ 1} = {𝑏, 𝑝, 𝑟, 0}

𝗁(𝑝) = 2 𝗁(𝑎) = 1

𝗁(1) = 3 𝗁(𝑝) = 2 𝗁(𝑟) = 2 𝗁(𝑐) = 1 𝗁(𝑏) = 1 𝗁(0) = 0

Example B.11. 95

📘 Euclid (circa 300BC) ⟨Book I Proposition 20⟩ 📘 Aliprantis and Burkinshaw (1998), page 35 97 📘 Deza and Laurent (1997) page 105 ⟨(8.1.2)⟩, 📘 Birkhoff (1967) pages 230–231 98 📘 Deza and Laurent (1997) page 105, 📘 Birkhoff (1967) page 231 ⟨§X.2⟩ 99 📘 Birkhoff (1967) page 230 100 📘 Birkhoff (1967) page 232 ⟨Theorem X.2⟩, 📘 Deza and Laurent (1997) pages 105–106, 📘 Blyth (2005) page 58 ⟨Exercise 4.25⟩ 96

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Negation

Daniel J. Greenhoe

The height function 𝗁 (Definition A.42 page 34) 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.

page 47

𝗁(1) = 3 𝗁(𝑝) = 2 𝗁(𝑎) = 1

𝗁(𝑟) = 2 𝗁(𝑐) = 1 𝗁(0) = 0

B.2 Negation

subminimal negation

minimal negation

de Morgan negation

fuzzy negation

Kleene negation

intuitionalistic negation

ortho negation

orthomodular negation

Figure 16: lattice of negations

B.2.1 Definitions Definition B.12. 101 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice (Definition A.39 page 34). A function ¬ ∈ 𝑋 𝑋 is a subminimal negation on 𝙇 if 102 𝑥 ≤ 𝑦 ⟹ ¬𝑦 ≤ ¬𝑥 ∀𝑥,𝑦∈𝑋 (antitone). Definition B.13.

103

Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice (Definition A.39 page 34).

101

📘 Dunn (1996) pages 4–6, 📘 Dunn (1999) pages 24–26 ⟨2 THE KITE OF NEGATIONS⟩ In the context of natural language, D. Devidi has argued that, subminimal negation (Definition B.12 page 47) is “difficult to take seriously as” a negation. For further details see 📓 Devidi (2010), page 511, 📓 Devidi (2006), page 568, 📃 Greenhoe (2014b) ⟨§2.1.1⟩, 📘 Greenhoe (2016a) ⟨§11.1⟩ 103 📘 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⟩, 📃 BELLMAN AND GIERTZ (1973) PAGES 155–156 ⟨(N1) ¬0 = 1 AND ¬1 = 0, (N3) ¬¬𝑥 = 𝑥⟩ 102



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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). A minimal negation ¬ is an intuitionistic negation on 𝙇 if 3. 𝑥 ∧ ¬𝑥 = 0 ∀𝑥,𝑦∈𝑋 (non-contradiction). A minimal negation ¬ is a fuzzy negation on 𝙇 if 4. ¬1 = 0 (boundary condition). Definition B.14. 104 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be a bounded lattice (Definition A.39 page 34). A minimal negation ¬ is a de Morgan negation on 𝙇 if 5. 𝑥 = ¬¬𝑥 ∀𝑥∈𝑋 (involutory). A de Morgan negation ¬ is a Kleene negation on 𝙇 if 6. 𝑥 ∧ ¬𝑥 ≤ 𝑦 ∨ ¬𝑦 ∀𝑥,𝑦∈𝑋 (Kleene condition). A de Morgan negation ¬ is an ortho negation on 𝙇 if 7. 𝑥 ∧ ¬𝑥 = 0 ∀𝑥,𝑦∈𝑋 (non-contradiction). A de Morgan negation ¬ is an orthomodular negation on 𝙇 if 8. 𝑥 ∧ ¬𝑥 = 0 ∀𝑥,𝑦∈𝑋 (non-contradiction) and ⟂ 9. 𝑥 ≤ 𝑦 ⟹ 𝑥 ∨ (𝑥 ∧ 𝑦) = 𝑦 ∀𝑥,𝑦∈𝑋 (orthomodular). Remark B.15. 105 The Kleene condition is a weakened form of the non-contradiction and excluded middle properties in the sense 𝑥 ∧ ¬𝑥 = 0 ≤ 1⏟⏟⏟⏟⏟⏟⏟ = 𝑦 ∨ ¬𝑦 . ⏟⏟⏟⏟⏟⏟⏟ non-contradiction

excluded middle

Definition B.16. Let 𝙇 ≜ ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤) be a bounded lattice (Definition A.39 page 34) with a function ¬ ∈ 𝑋 𝑋 . If ¬ is a negation (Definition B.13 page 47), then 𝙇 is a lattice with negation.

B.2.2

Properties of negations

Theorem B.17. 106 Let ¬ ∈ 𝑋 𝑋 be a function on a bounded lattice 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤). ¬ is a ⟹ { ¬0 = 1 (boundary condition) } { fuzzy negation } Let ¬ ∈ 𝑋 𝑋 be a function on a bounded lattice 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤). (a) ¬1 = 0 (boundary condition) and ⎫ ⎧ ¬ is an ⎪ ⎪ ⟹ ⎨ (b) ¬0 = 1 (boundary condition) and ⎬ { intuitionistic negation } ⎪ ⎪ ⎩ (c) ¬ is a fuzzy negation ⎭

Theorem B.18.

107

Theorem B.19. 108 Let ¬ ∈ 𝑋 𝑋 be a function on a bounded lattice 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤). ¬ is a ⎧ ⎫ ¬𝑥 ∨ ¬𝑦 ≤ ¬(𝑥 ∧ 𝑦) ∀𝑥,𝑦∈𝑋 (conjunctive de Morgan inequality) and ⎪ ⎪ minimal ⎨ ⎬ ⟹ { ¬(𝑥 ∨ 𝑦) ≤ ¬𝑥 ∧ ¬𝑦 ∀𝑥,𝑦∈𝑋 (disjunctive de Morgan inequality) } ⎪ ⎩ negation ⎪ ⎭ Theorem B.20. 109 Let ¬ ∈ 𝑋 𝑋 be a function on a bounded lattice 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤). ¬ is a ¬(𝑥 ∨ 𝑦) = ¬𝑥 ∧ ¬𝑦 ∀𝑥,𝑦∈𝑋 (disjunctive de Morgan) and ⟹ { ¬(𝑥 ∧ 𝑦) = ¬𝑥 ∨ ¬𝑦 ∀𝑥,𝑦∈𝑋 (conjunctive de Morgan) de Morgan negation } 104

📘 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) 105 📘 Cattaneo and Ciucci (2009) page 78 106 📃 Greenhoe (2014b) ⟨§2.1.2⟩, 📘 Greenhoe (2016a) ⟨§11.2⟩ 107 📃 Greenhoe (2014b) ⟨§2.1.2⟩, 📘 Greenhoe (2016a) ⟨§11.2⟩ 108 📃 Greenhoe (2014b) ⟨§2.1.2⟩, 📘 Greenhoe (2016a) ⟨§11.2⟩ 109 📃 Greenhoe (2014b) ⟨§2.1.2⟩, 📘 Greenhoe (2016a) ⟨§11.2⟩ CC BY 4.0


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B.3

Projections

Theorem B.21. ¬ is an ⎧ ⎪ ⎨ ortho ⎪ ⎩ negation

Daniel J. Greenhoe

110

⎫ ⎪ ⎬ ⎪ ⎭

page 49

Let ¬ ∈ 𝑋 𝑋 be a function on a bounded lattice 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤). 1. ¬0 = 1 (boundary condition) and ⎧ ⎫ ⎪ ⎪ 2. ¬1 = 0 (boundary condition) and ⎪ ⎪ ⎪ 3. ¬(𝑥 ∨ 𝑦) = ¬𝑥 ∧ ¬𝑦 ∀𝑥,𝑦∈𝑋 (disjunctive de Morgan) and ⎪ ⟹ ⎨ ⎬ ⎪ 4. ¬(𝑥 ∧ 𝑦) = ¬𝑥 ∨ ¬𝑦 ∀𝑥,𝑦∈𝑋 (conjunctive de Morgan) and ⎪ ∀𝑥∈𝑋 (excluded middle) and ⎪ ⎪ 5. 𝑥 ∨ ¬𝑥 = 1 ⎪ ⎪ 6. 𝑥 ∧ ¬𝑥 ≤ 𝑦 ∨ ¬𝑦 ∀𝑥,𝑦∈𝑋 (Kleene condition). ⎩ ⎭

B.3 Projections Definition B.22. 111 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an orthocomplemented lattice (Definition A.72 page 40). A function 𝜙𝑥 ∈ 𝑋 𝑋 is a Sasaki projection on 𝑥 ∈ 𝑋 if 𝜙𝑥 (𝑦) ≜ (𝑦 ∨ 𝑥⟂ ) ∧ 𝑥. The Sasaki projections 𝜙𝑥 and 𝜙𝑦 are permutable if 𝜙𝑥 ∘ 𝜙𝑦 (𝑢) = 𝜙𝑦 ∘ 𝜙𝑥 (𝑢) ∀𝑢 ∈ 𝑋. Proposition B.23. Let 𝜙𝑥 (𝑦) be the Sasaki projection of 𝑦 onto 𝑥 (Definition B.22 page 49) in an orthocomplemented lattice 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤). (1). 𝑥 ≤ 𝑦 ⟹ 𝜙𝑥 (𝑦) = 𝑥 ∀𝑥,𝑦∈𝑋 (2). 𝑦 ≤ 𝑥 ⟹ 𝑦 ≤ 𝜙𝑥 (𝑦) ≤ 𝑥 ∀𝑥,𝑦∈𝑋 (3). 𝑦 ≤ 𝑥 and 𝙇 is Boolean ⟹ 𝜙𝑥 (𝑦) = 𝑦 ∀𝑥,𝑦∈𝑋 ✎PROOF: (1) ⟹ 𝜙𝑥 (𝑦) ≜ (𝑦 ∨ 𝑥⟂ ) ∧ 𝑥

by definition of Sasaki projection (Definition B.22 page 49)

=1∧𝑥

by 𝑥 ≤ 𝑦 hypothesis and Proposition C.1 page 51

=𝑥

by property of bounded lattices (Proposition A.41 page 34)

(2) ⟹ 𝑦 = 𝑦 ∧ 𝑥

by 𝑦 ≤ 𝑥 hypothesis ⟂

≤ (𝑦 ∨ 𝑥 ) ∧ 𝑥

by definition of ∨ (Definition A.27 page 31)

= 𝜙𝑥 (𝑦)


≤ (𝑦 ∨ 𝑥⟂ ) ∧ 𝑥


≤ 𝑥

by definition of ∧ (Definition A.28 page 32)

(3) ⟹ 𝜙𝑥 (𝑦) = (𝑦 ∨ 𝑥⟂ ) ∧ 𝑥 ⟂


= (𝑦 ∧ 𝑥) ∨ (𝑥 ∧ 𝑥)

by distributive property of Boolean lattices (Theorem A.70 page 39)

= (𝑦 ∧ 𝑥) ∨ 0

by non-contradiction of Boolean lattices (Theorem A.70 page 39)

= (𝑦 ∧ 𝑥)

by boundary property of bounded lattices (Proposition A.41 page 34)

=𝑦

by 𝑦 ≤ 𝑥 hypothesis and definition of ∧ (Definition A.28 page 32)

✏ Proposition B.24. Let 𝜙𝑥 (𝑦) be the Sasaki projection of 𝑦 onto 𝑥 (Definition B.22 page 49) in an orthocomplemented lattice ( 𝑋, ∨, ∧, 0, 1 ; ≤). (1). 𝜙0 (𝑦) = 0 ∀𝑦∈𝑋 (2). 𝜙𝑥 (0) = 0 ∀𝑥∈𝑋 (3). 𝜙1 (𝑦) = 1 ∀𝑦∈𝑋 (4). 𝜙𝑥 (1) = 𝑥 ∀𝑥∈𝑋 ⟂ (5). 𝜙𝑥 (𝑥 ) = 0 ∀𝑥∈𝑋 110

Greenhoe (2014b) ⟨§2.1.2⟩, 📘 Greenhoe (2016a) ⟨§11.2⟩ 📘 Nakamura (1957) pages 158–159 ⟨equation (S)⟩, 📘 Sasaki (1954) page 300 ⟨Def.5.1, cf Foulis 1962⟩, 📘 Kalmbach (1983) page 117 📃

111



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✎PROOF: 𝜙0 (𝑦) = 0

because 0 ≤ 𝑦 and by Proposition B.23 page 49 ⟂

𝜙𝑥 (0) ≜ (0 ∨ 𝑥 ) ∧ 𝑥 = 𝑥⟂ ∧ 𝑥

by definition of Sasaki projection (Definition B.22 page 49) by property of bounded lattices (Proposition A.41 page 34)

=0

by definition of orthocomplemented (Definition A.72 page 40) ⟂

𝜙1 (𝑦) ≜ (𝑦 ∨ 1 ) ∧ 1 = (𝑦 ∨ 0) ∧ 1

by definition of Sasaki projection (Definition B.22 page 49) by boundary condition (Theorem B.21 page 49)

=𝑦∧1


=1


𝜙𝑥 (1) = 𝑥

because 𝑥 ≤ 1 and by Proposition B.23 page 49

𝜙𝑥 (𝑥⟂ ) ≜ (𝑥⟂ ∨ 𝑥⟂ ) ∧ 𝑥 ⟂


=𝑥 ∧𝑥

by idempotency of lattices (Theorem A.32 page 32)

=0

by non-contradiction prop. of orthocomplemented lattice (Definition A.72 page 40)

✏ Example B.25. 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) 1 ⟂ 𝜙𝑝 (𝑝 ) ≜ (𝑝⟂ ∨ 𝑝⟂ ) ∧ 𝑝 = 𝑝⟂ ∧ 𝑝 = 0 (because 𝑝 ⟂ 𝑝⟂ ) 𝑞⟂ 𝜙𝑝 (𝑞 ⟂ ) ≜ (𝑞 ⟂ ∨ 𝑝⟂ ) ∧ 𝑝 = 1 ∧ 𝑝 = 𝑝 (because 𝑝 ≤ 𝑞⟂ ) 𝑝 𝜙𝑞⟂ (𝑝) ≜ (𝑝 ∨ 𝑞) ∧ 𝑞 ⟂ = 1 ∧ 𝑞 ⟂ = 𝑞 ⟂ (because 𝑞⟂ ≤ 1) ⟂ 𝜙𝑝 (1) ≜ (1 ∨ 𝑝 ) ∧ 𝑝 = 1 ∧ 𝑝 = 𝑝 (because 𝑝 ≤ 1) 0 ≜ (0 ∨ 𝑝⟂ ) ∧ 𝑝 = 𝑝⟂ ∧ 𝑝 = 0 (because 𝑝 ⟂ 0) 𝜙𝑝 (0) Example B.26. Let ℝ3 be the 3-dimensional Euclidean space (Example A.75 page 42) with subspaces 𝙕 and 𝙑 . Then the projection operator 𝑃𝙕 ⟂ onto 𝙕 ⟂ is a sasaki projection 𝜙𝙕 ⟂ . In particular, and as illustrated to the right, 𝐏𝙕 ⟂ 𝙑 ≜ 𝜙𝙕 ⟂ (𝙑 ) ≜ (𝙑 + 𝙕 ⟂⟂ ) ∩ 𝙕 ⟂ = (𝙑 + 𝙕 ) ∩ 𝙕 ⟂

B.4

𝑝⟂ 𝑞

𝙕 +𝙑

𝙕 𝙑

𝐏𝙕 ⟂ 𝙑 𝙕⟂

Logics

Definition B.27. 112 Let → be an implication function defined on a lattice with negation 𝙇 ≜ ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤) (Definition B.16 page 48). ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤, →) is a logic if ¬ is a minimal negation. ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤, →) is a fuzzy logic if ¬ is a fuzzy negation. ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤, →) is an intuitionalistic logic if ¬ is an intuitionalistic negation. ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤, →) is a de Morgan logic if ¬ is a de Morgan negation. ( 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. 112

📃

Straßburger (2005) page 136 ⟨Definition 2.1⟩, 📃 de Vries (2007) page 11 ⟨Definition 16⟩, 📃 Greenhoe (2014b)

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logic fuzzy logic intuitionalistic logic de Morgan logic ortho logic Boolean logic / classic logic

Figure 17: lattice of logics For examples and a definition of implication, see 📃 Greenhoe (2014b) ⟨§3.1⟩.

Appendix C

Relations on lattices

The relations in this section are typically defined on an orthocomplemented lattice (Definition A.72 page 40). Here, some relations are generalized to a lattice with negation (Definition B.16 page 48). A lattice (Definition A.31 page 32) with an ortho negation successfully defined on it is an orthocomplemented lattice (Definition A.72 page 40). In many cases, these relations only work well on an orthocomplemented lattice, and thus many results are restricted to orthocomplemented lattices.

C.1 Orthogonality Proposition C.1. Let ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an orthocomplemented lattice (Definition A.72 page 40). 𝑥⟂ ∨ 𝑦 = 1 and 𝑥≤𝑦 ⟹ ∀𝑥,𝑦∈𝑋 { 𝑥 ∧ 𝑦⟂ = 0 } ✎PROOF: 𝑥 ≤ 𝑦 ⟹ 𝑥 ∨ 𝑥⟂ ≤ 𝑦 ∨ 𝑥 ⟂ ⟹ 1≤𝑦∨𝑥

⟂

by excluded middle property (Definition A.72 page 40)

⟹ 𝑥⟂ ∨ 𝑦 = 1 ⟂

𝑥≤𝑦 ⟹ 𝑥∧𝑦 ≤𝑦∧𝑦

by monotone property of lattices (Proposition A.34 page 33) by upper bounded property of bounded lattices (Definition A.39 page 34)

⟂

⟹ 𝑥 ∧ 𝑦⟂ ≤ 0 ⟂

⟹ 𝑥∧𝑦 =0

by monotone property of lattices (Proposition A.34 page 33) by non-contradiction property (Definition A.72 page 40) by lower bounded property of bounded lattices (Definition A.39 page 34)

✏ Definition C.2. 113

113

Let ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤) be a lattice with negation (Definition B.16 page 48).

📘 Stern (1999) page 12, 📘 Loomis (1955) page 3



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RELATIONS ON LATTICES

The orthogonality relation ⟂∈ 𝟚𝑋𝑋 is defined as def

𝑥⟂𝑦 ⟺ 𝑥 ≤ ¬𝑦 If 𝑥 ⟂ 𝑦, we say that 𝑥 is orthogonal to 𝑦. Lemma C.3. Let ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤) be a lattice with negation (Definition B.16 page 48). ⟹ { 𝑥 ⟂ 𝑦 (orthogonal Definition C.2 page 51) } { 𝑦 ⟂ 𝑥 (symmetric) } ✎PROOF: 𝑥 ⟂ 𝑦 ⟹ 𝑥 ≤ ¬𝑦

by definition of ⟂ (Definition C.2 page 51)

⟹ (¬¬𝑦) ≤ ¬𝑥

by antitone property (Definition A.72 page 40)

⟹ 𝑦 ≤ ¬𝑥

by weak double negation property of negation (Definition B.13 page 47)

⟹ 𝑦⟂𝑥

by definition of ⟂ (Definition C.2 page 51)

✏ 114

Let ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an orthocomplemented lattice (Definition A.72 page 40). 1. 𝑥 ∧ 𝑦 = 0 and 𝑥⏟ ⟂𝑦 ⟹ { 2. 𝑥⟂ ∨ 𝑦⟂ = 1 }

Lemma C.4.

orthogonal (Definition C.2 page 51)

Remark C.5. 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 A.74 (page 41). 𝑎 ⟂ 𝑝 because 𝑎⟂ ≤ 𝑝⟂ . 𝑝 ⟂ 𝑟 because 𝑝 ≤ 𝑟⟂ . But yet 𝑎⟂ is not orthogonal to 𝑟 because 𝑎⟂ ≰ 𝑟⟂ .

✎PROOF: ⟂

✏

6! Example C.6. In the 𝑂6 lattice (Definition A.73 page 41), there are a total of (62) = (6−2)!2! = 15 distinct = 6×5 2 unordered (the ⟂ relation is symmetric by Lemma C.3 page 52 so the order doesn't matter) pairs of elements. Of these 15 pairs, 8 are orthogonal to each 𝑥 ⟂ 𝑦 𝑥 ⟂ 0 𝑦⟂ ⟂ 0 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

Example C.7. In lattice 5 of Example A.74 (page 41), there are a total of (10 2) distinct unordered pairs of elements. 𝑝 ⟂ 𝑝⟂ 𝑥 ⟂ 𝑥⟂ 𝑦 ⟂ 𝑧 𝑥⟂ Of these 45 pairs, 18 are orthogo𝑝 ⟂ 𝑥 ⟂ 𝑥 ⟂ 𝑦 𝑦 ⟂ 0 𝑦⟂ nal to each other, and 0 is orthog𝑝 ⟂ 𝑦 𝑥 ⟂ 𝑧 𝑧 ⟂ 𝑧 ⟂ 𝑧⟂ onal to itself, making a total of 19 𝑝 ⟂ 𝑧 𝑥 ⟂ 0 𝑧 ⟂ 0 0 orthogonal pairs: 𝑝 ⟂ 0 𝑦 ⟂ 𝑦⟂ 𝑝⟂ ⟂ 0

=

10! (10−2)!2!

⟂ ⟂ ⟂ ⟂

=

10×9 2

= 45

0 0 0 0

Example C.8. In the ℝ3 Euclidean space illustrated in Example A.75 (page 42), 𝙓 ⊆ 𝙔⟂ ⟹ 𝙓 ⟂ 𝙔 𝙔 ⊆ 𝙓⟂ ⟹ 𝙔 ⟂ 𝙓 𝙓 ⊆ 𝙕⟂ ⟹ 𝙓 ⟂ 𝙕 𝙔 ⊆ 𝙕⟂ ⟹ 𝙔 ⟂ 𝙕 𝙓 ∧𝙔 =𝙓 ∧𝙕 =𝙔 ∧𝙕 =𝟬 114

📘 Holland (1963), page 67, 📘 Greenhoe (2016a) ⟨Lemma 13.2⟩

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C.2

Commutativity

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C.2 Commutativity The commutes relation is defined next. Motivation for the name “commutes” is provided by Proposition C.14 (page 53) 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 C.9. 115 Let 𝙇 ≜ ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤) be a lattice with negation (Definition B.16 page 48). The commutes relation Ⓒ is defined as def

𝑥Ⓒ𝑦 ⟺ 𝑥 = (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)} Proposition C.10. 116 Let 𝙇 ≜ ( 𝑋, 𝑥Ⓒ0 and 0Ⓒ𝑥 ∀𝑥∈𝑋 𝑥Ⓒ1 and 1Ⓒ𝑥 ∀𝑥∈𝑋 𝑥Ⓒ𝑥 ∀𝑥∈𝑋

∨, ∧, 0, 𝑥Ⓒ𝑦 𝑥≤𝑦 𝑥⟂𝑦

1 ; ≤) be an orthocomplemented lattice. ⟺ 𝑥Ⓒ𝑦⟂ ∀𝑥,𝑦∈𝑋 ⟹ 𝑥Ⓒ𝑦 ∀𝑥,𝑦∈𝑋 ⟹ 𝑥Ⓒ𝑦 ∀𝑥,𝑦∈𝑋

Definition C.11. Let Ⓒ be the commutes relation (Definition C.9 page 53) on a lattice with negation 𝙇 ≜ ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤) (Definition B.16 page 48). 𝙇 is symmetric if 𝑥Ⓒ𝑦 ⟹ 𝑦Ⓒ𝑥 ∀𝑥,𝑦∈𝑋 In general, the commutes relation is not symmetric. But Proposition C.12 (next) describes some conditions under which it is symmetric. Proposition C.12. 117 Let ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an orthocomplemented lattice (Definition A.72 page 40). {𝑥Ⓒ𝑦 ⟹ 𝑦Ⓒ𝑥} ⟺ {𝑥 ≤ 𝑦 ⟹ 𝑦 = 𝑥 ∨ (𝑥⟂ ∧ 𝑦)} (orthomodular identity) (2) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⟂ (𝑥) ⟺ 𝑥 ≤ 𝑦 ⟹ 𝑥 = 𝑦 ∧ 𝑥 ∨ 𝑦 (𝑥 = 𝜙 (Sasaki projection) ) (3) 𝑦 { ( )} Ⓒ is symmetric at (𝑥, 𝑦) (1) ⟺ {𝑦 = (𝑥 ∧ 𝑦) ∨ [𝑦 ∧ (𝑥 ∧ 𝑦)⟂ ]} (4) ⟂ ⟺ {𝑥 = (𝑥 ∨ 𝑦) ∧ [𝑥 ∨ (𝑥 ∨ 𝑦) ]} (5) Theorem C.13. 118 Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an orthocomplemented lattice (Definition A.72 page 40). ⟂ {𝑥Ⓒ𝑐 ∀𝑥 ∈ 𝑋} ⟺ {𝙇 is isomorphic to [0 ∶ 𝑐] × [0 ∶ 𝑐 ]} with isomorphism 𝜃(𝑥) ≜ ([0 ∶ 𝑐], [0 ∶ 𝑐 ⟂ ]). Proposition C.14. 𝑥Ⓒ𝑦 ⟺

119

Let ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an orthomodular lattice. ∀𝑥,𝑦∈𝑋 𝜙𝑥 (𝑦) = 𝜙𝑦 (𝑥) = 𝑥 ∧ 𝑦

C.3 Center An element in an orthocomplemented lattice (Definition A.72 page 40) is in the center of the lattice if that element commutes (Definition C.9 page 53) 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 A.81 page 43). 115

📘 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 (1936) page 833 ⟨“𝑎 = (𝑎 ∩ 𝑥) ∪ (𝑎 ∩ 𝑥′ )”⟩ 116 📘 Holland (1963), page 67, 📘 Greenhoe (2016a) ⟨Proposition 13.2⟩ 117 📘 Holland (1963) page 68, 📃 Nakamura (1957) page 158, 📘 Greenhoe (2016a) ⟨Proposition 13.3⟩ 118 📘 Kalmbach (1983) page 20, 📃 MacLaren (1964) 119 📃 Foulis (1962) page 66, 📃 Sasaki (1954) ⟨cf Foulis 1962⟩ Tuesday 14th February, 2017 9:12pm UTC Copyright © Daniel J. Greenhoe


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1

⟂

𝑧

𝑥

⟂

𝑧 𝑥⟂

𝑦⟂

𝑝

𝑧 0

𝑥

𝑝 𝑧

𝑦

𝑥

⟂

𝑧

𝑦⟂

𝑝

⟂

𝑦

𝑥

0

(A) O6 lattice

1

1 ⟂

⟂

𝑥

𝑝 𝑧

𝑧⟂

⟂

(C)

𝑦⟂ 𝑥⟂

𝑥

𝑦

𝑧

0

0

(B)

RELATIONS ON LATTICES

(D)

𝙇32

Boolean lattice

Figure 18: Lattices with centers marked with solid dots (see Example C.17 page 54) Definition C.15. 120 Let Ⓒ be the commutes relation (Definition C.9 page 53) on a lattice with negation 𝙇 ≜ ( 𝑋, ∨, ∧, ¬, 0, 1 ; ≤) (Definition B.16 page 48). The center of 𝙇 is defined as {𝑥 ∈ 𝑋|𝑥Ⓒ𝑦 ∀𝑦 ∈ 𝑋} Proposition C.16. Let 𝙇 ≜ ( 𝑋, ∨, ∧, 0, 1 ; ≤) be an orthocomplemented lattice (Definition A.72 page 40). The elements 0 and 1 are in the center of 𝙇. ✎PROOF:

✏

This follows directly from Definition C.9 (page 53) and Proposition C.10 (page 53).

Example C.17. The centers of the lattices in Figure 18 (page 54) are illustrated with solid dots. Note that in the case of the Boolean lattice in (D), every dot is in the center (Proposition A.81 page 43).

C.4

D-Posets

Definition C.18. 121 Let 1 be the upper bound of an ordered set ( 𝑋, ≤). An operation⧵is a difference on ( 𝑋, ≤) if 1. 𝑥 ≤ 𝑦 ⟹ 𝑦⧵𝑥 ≤ 𝑦 ∀𝑥,𝑦∈𝑋 and 2. 𝑥 ≤ 𝑦 ⟹ 𝑦⧵(𝑦⧵𝑥) = 𝑥 ∀𝑥,𝑦∈𝑋 and 3. 𝑥 ≤ 𝑦 ≤ 𝑧 ⟹ 𝑧⧵𝑦 ≤ 𝑧⧵𝑥 ∀𝑥,𝑦,𝑧∈𝑋 and 4. 𝑥 ≤ 𝑦 ≤ 𝑧 ⟹ (𝑧⧵𝑥)⧵(𝑧⧵𝑦) = 𝑦⧵𝑥 ∀𝑥,𝑦,𝑧∈𝑋 . The structure (𝑋, ≤,⧵, 1) is called a D-poset. Proposition C.19.

122

Let 𝑋 be a set.

(𝑋, ≤,⧵, 1) is a ⎧ ⎪ ⎨ D-poset ⎪ ⎩ (Definition C.18 page 54)

⎧ ⎫ ⎪ ⎪ ⎬ ⟹ ⎨ ⎪ ⎪ ⎭ ⎩

1. 2. 3. 4.

𝑥≤𝑦≤𝑧 𝑥≤𝑦≤𝑧 𝑥≤𝑦≤𝑧 𝑥≤𝑦≤𝑧

⟹ ⟹ ⟹ ⟹

𝑦⧵𝑥 ≤ 𝑧⧵𝑥 𝑥 ≤ 𝑧⧵(𝑦⧵𝑥) (𝑧⧵𝑥)⧵(𝑦⧵𝑥) = 𝑧⧵𝑦 [𝑧⧵(𝑦⧵𝑥)] ⧵𝑥 = 𝑧⧵𝑦


and


and


and


.

⎫ ⎪ ⎬ ⎪ ⎭

Example C.20. 123 The structure (ℝ+ , −, ≤) is a D-poset where ℝ+ is the set of positive real numbers, − is the standard subtraction operation on ℝ, and ≤ is the standard ordering relation on ℝ+ . Example C.21. 124 The structure (𝟚𝑋 ,⧵, ⊆) is a D-poset where 𝟚𝑋 is the power set of a set 𝑋,⧵is the set difference operator, and ⊆ is the set inclusion relation. 120

📘 Holland (1970), page 80 📃 Kôpka and Chovanec (1994) page 22,24 ⟨DEFINITIONS 1,2⟩ 122 📃 Kôpka and Chovanec (1994) page 23 ⟨PROPOSITION 1.⟩ 123 📃 Kôpka and Chovanec (1994) page 22 ⟨Example 1⟩ 124 📃 Kôpka and Chovanec (1994) page 24 ⟨Example 4⟩ 121

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Appendix D

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MRA-wavelet subspace structure

D.1 Transversal Operators Definition D.1. 125 ℂ 1. 𝐓 is the translation operator on ℂ defined as 𝐓𝜏 𝖿(𝑥) ≜ 𝖿(𝑥 − 𝜏) and 𝐓 ≜ 𝐓1 ∀𝖿 ∈ℂℂ ℂ 2. 𝐃 is the dilation operator on ℂ defined as 𝐃𝛼 𝖿(𝑥) ≜ 𝖿(𝛼𝑥) and 𝐃 ≜ √2𝐃2 ∀𝖿 ∈ℂℂ 𝐓 −1 𝖿 (𝑥)

𝖿(𝑥)

𝐓𝖿 (𝑥)

𝐃𝖿 (𝑥) 𝖿(𝑥) 𝐃−1 𝖿(𝑥) 𝑥 2

𝑥 −2

Proposition D.2. ∑

𝑛∈ℤ

𝑛

𝐓 𝖿(𝑥) =

126

∑

−1

0

1

2

−2

−1

0

1

Let 𝐓 be the translation operator (Definition D.1 page 55).

𝐓 𝑛 𝖿(𝑥 + 1)

∀𝖿∈ℝℝ

𝑛∈ℤ

∑ (𝑛∈ℤ

𝐓𝑛 𝖿 (𝑥) is periodic with period 1

)

Proposition D.3. 127 Let 𝐓 and 𝐃 be as defined in Definition D.1 page 55. 𝐓 has an inverse 𝐓 −1 in ℂℂ expressed by the relation 𝐓 −1 𝖿(𝑥) = 𝖿 (𝑥 + 1) ∀𝖿 ∈ℂℂ (translation operator inverse). −1 ℂ 𝐃 has an inverse 𝐃 in ℂ expressed by the relation 𝐃−1 𝖿 (𝑥) =

√2 2

𝖿 ( 12 𝑥)

∀𝖿 ∈ℂℂ

(dilation operator inverse).

Proposition D.4. 128 Let 𝐓 and 𝐃 be as defined in Definition D.1 identity operator. 𝐃𝑗 𝐓 𝑛 𝖿 (𝑥) = 2𝑗/2 𝖿 (2𝑗 𝑥 − 𝑛) ∀𝑗,𝑛∈ℤ, 𝖿 ∈ℂℂ

page

Example D.5 (linear functions). 129 Let 𝐓 be the translation operator be the set of all linear functions in 𝙇𝟤ℝ . 1. {𝑥, 𝐓𝑥} is a basis for ℒ(ℂ, ℂ) and 2. 𝖿(𝑥) = 𝖿 (1)𝑥 − 𝖿(0)𝐓𝑥 ∀𝖿 ∈ ℒ(ℂ, ℂ)

55. Let 𝐃0 = 𝐓 0 ≜ 𝐈 be the

(Definition D.1 page 55).

Let ℒ(ℂ, ℂ)

125

📘 Walnut (2002) pages 79–80 ⟨Definition 3.39⟩, 📘 Christensen (2003) pages 41–42, 📘 Wojtaszczyk (1997) page 18 ⟨Definitions 2.3,2.4⟩, 📘 Kammler (2008) page A-21, 📘 Bachman et al. (2000) page 473, 📘 Packer (2004) page 260, 📘 Benedetto and Zayed (2004) page , 📘 Heil (2011) page 250 ⟨Notation 9.4⟩, 📘 Casazza and Lammers (1998) page 74, 📘 Goodman et al. (1993), page 639, 📘 Dai and Lu (1996), page 81, 📘 Dai and Larson (1998) page 2, 📘 Greenhoe (2013) page 2 126 📘 Greenhoe (2013) page 3 127 📘 Greenhoe (2013) page 3 128 📘 Greenhoe (2013) page 4 129 📘 Higgins (1996) page 2



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✎PROOF:

Daniel J. Greenhoe

D

MRA-WAVELET SUBSPACE STRUCTURE

By left hypothesis, 𝖿 is linear; so let 𝖿(𝑥) ≜ 𝑎𝑥 + 𝑏

𝖿 (1)𝑥 − 𝖿(0)𝐓𝑥 = 𝖿 (1)𝑥 − 𝖿(0)(𝑥 − 1)

by Definition D.1 page 55

= (𝑎𝑥 + 𝑏)|𝑥=1 𝑥 − (𝑎𝑥 + 𝑏)|𝑥=0 (𝑥 − 1)

by left hypothesis and definition of 𝖿

= (𝑎 + 𝑏)𝑥 − 𝑏(𝑥 − 1) = 𝑎𝑥 + 𝑏𝑥 − 𝑏𝑥 + 𝑏 = 𝑎𝑥 + 𝑏 = 𝖿(𝑥)

by left hypothesis and definition of 𝖿

✏ Example D.6 (Cardinal Series). Let 𝐓 be the translation operator (Definition D.1 page 55). The PaleyWiener class of functions 𝙋𝙒𝜎2 are those functions which are “bandlimited” with respect to their Fourier transform. The cardinal series forms an orthogonal basis for such a space. The Fourier coefficients for a projection of a function 𝖿 onto the Cardinal series basis elements is particularly simple—these coefficients are samples of 𝖿(𝑥) taken at regular intervals. In fact, one could represent the coefficients using inner product notation with the Dirac delta distribution 𝛿 as follows: Let 𝛼𝑛 ≜ ⟨𝖿(𝑥) | 𝐓 𝑛 𝛿(𝑥)⟩ ≜ 1.

2.

∫ ℝ

𝖿(𝑥)𝛿(𝑥 − 𝑛) 𝖽𝑥 = 𝖿(𝑛)

𝗌𝗂𝗇(𝜋𝑥) 𝑛∈ℕ is a basis for 𝙋𝙒𝜎2 } 𝜋𝑥 | ∞ 𝗌𝗂𝗇(𝜋𝑥) ∀𝖿 ∈ 𝙋𝙒𝜎2 , 𝜎 ≤ 12 𝖿(𝑥) = 𝛼 𝐓𝑛 ∑ 𝑛 𝜋𝑥 𝑛=1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝐓𝑛 {

∀𝑛∈ℤ.

Then

and

Cardinal series

Example D.7 (Fourier Series). 𝑖𝑥 1. {𝐃𝑛 𝑒 | 𝑛∈ℤ} is a basis for 𝙇(0 ∶ 2𝜋) 1 2. 𝖿(𝑥) = 𝛼𝑛 𝐃𝑛 𝑒𝑖𝑥 ∀𝑥∈(0∶2𝜋), 𝖿 ∈𝙇(0∶2𝜋) √2𝜋 ∑ 𝛼𝑛 ≜

3.

1 √2𝜋

𝑛∈ℤ 2𝜋

∫ 0

𝖿(𝑥)𝐃𝑛 𝑒−𝑖𝑥 𝖽𝑥

and where

∀𝖿∈𝙇(0∶2𝜋)

Example D.8 (Fourier Transform). 𝑖𝑥 𝟤 1. {𝐃𝜔 𝑒 |𝜔∈ℝ} is a basis for 𝙇 ℝ 𝑖𝜔 1 ̃ 2. 𝖿 (𝑥) = 𝖿(𝜔)𝐃 𝑥 𝑒 𝖽𝜔 √2𝜋 ∫ ℝ 1 ̃ 3. 𝖿(𝜔) ≜ 𝖿(𝑥)𝐃𝜔 𝑒−𝑖𝑥 𝖽𝑥 ∫ √2𝜋 ℝ

and

∀𝖿∈𝙇𝟤ℝ

where


Example D.9 (Gabor Transform). 130 𝑖𝑥 𝟤 −𝜋𝑥2 1. )(𝐃𝜔 𝑒 )| 𝜏,𝜔∈ℝ} is a basis for 𝙇ℝ {(𝐓𝜏 𝑒 2.

𝖿 (𝑥) =

3.

𝖦 (𝜏, 𝜔) ≜

∫ ℝ

𝖦 (𝜏, 𝜔) 𝐃𝑥 𝑒𝑖𝜔 𝖽𝜔

and

∀𝑥∈ℝ, 𝖿∈𝙇𝟤ℝ

2

𝖿 (𝑥)(𝐓𝜏 𝑒−𝜋𝑥 )(𝐃𝜔 𝑒−𝑖𝑥 ) 𝖽𝑥 ∫ ℝ

where

∀𝑥∈ℝ, 𝖿∈𝙇𝟤ℝ

Example D.10 (wavelets). Let 𝜓(𝑥) be a mother wavelet. 𝑘 𝑛 𝟤 1. and {𝐃 𝐓 𝜓(𝑥)| 𝑘,𝑛∈ℤ} is a basis for 𝙇ℝ 2.

𝖿(𝑥) =

𝛼𝑘,𝑛 𝐃𝑘 𝐓𝑛 𝜓(𝑥)


𝖿(𝑥)𝐃𝑘 𝐓𝑛 𝜓 ∗ (𝑥) 𝖽𝑥


∑∑

where

𝑘∈ℤ 𝑛∈ℤ 3. 130

𝛼𝑛 ≜

∫ ℝ

📘 Qian and Chen (1996) ⟨Chapter 3⟩, 📘 Forster and Massopust (2009) page 32 ⟨Definition 1.69⟩

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The Structure of Wavelets

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D.2 The Structure of Wavelets In Fourier analysis, continuous dilations (Definition D.1 page 55) of the complex exponential form a basis for the space of square integrable functions 𝙇𝟤ℝ such that 𝙇𝟤ℝ = 𝗌𝗉𝖺𝗇{𝐃𝜔 𝑒𝑖𝑥 |𝜔∈ℝ}. In Fourier series analysis , discrete dilations of the complex exponential form a basis for 𝙇𝟤ℝ (0 ∶ 2𝜋) such that 𝙇𝟤ℝ (0 ∶ 2𝜋) = 𝗌𝗉𝖺𝗇{𝐃𝑗 𝑒𝑖𝑥 | 𝑗∈ℤ}. In Wavelet analysis, for some mother wavelet 𝙇𝟤ℝ = 𝗌𝗉𝖺𝗇{𝐃𝜔 𝐓𝜏 𝜓(𝑥)|𝜔, 𝜏 ∈ ℝ}.

(Definition D.18 page 61)

𝜓(𝑥),

However, the ranges of parameters 𝜔 and 𝜏 can be much reduced to the countable set ℤ resulting in a dyadic wavelet basis such that for some mother wavelet 𝜓(𝑥), 𝙇𝟤ℝ = 𝗌𝗉𝖺𝗇{𝐃𝑗 𝐓 𝑛 𝜓(𝑥)|𝑗, 𝑛 ∈ ℤ}. Wavelets that are both dyadic and compactly supported have the attractive feature that they can be easily implemented in hardware or software by use of the Fast Wavelet Transform (Figure 20 page 64). In 1989, Stéphane G. Mallat introduced the Multiresolution Analysis (MRA, Definition D.12 page 58) method for wavelet construction. The MRA has since become the dominate wavelet construction method. Moreover, P.G. Lemarié has proved that all wavelets with compact support are generated by an MRA.131 The MRA is an analysis of the linear space 𝙇𝟤ℝ . An analysis of a linear space 𝙓 is any sequence ⦅𝙑𝑗 ⦆𝑗∈ℤ of linear subspaces of 𝙓 . The partial or complete reconstruction of 𝙓 from ⦅𝙑𝑗 ⦆𝑗∈ℤ is a

synthesis.132 Some analyses are completely characterized by a transform. For example, a Fourier analysis is a sequence of subspaces with sinusoidal bases. Examples of subspaces in a Fourier anal-

ysis include 𝙑1 = 𝗌𝗉𝖺𝗇{𝑒𝑖𝑥 }, 𝙑2.3 = 𝗌𝗉𝖺𝗇{𝑒𝑖2.3𝑥 }, 𝙑√2 = 𝗌𝗉𝖺𝗇{𝑒𝑖√2𝑥 }, etc. A transform is loosely defined as a function that maps a family of functions into an analysis. A very useful transform (a “Fourier transform”) for Fourier Analysis is 1 𝖿(𝑥)𝑒−𝑖𝜔𝑥 𝖽𝑥 [𝐅𝖿̃ ](𝜔) ≜ ∫ √2𝜋 ℝ An analysis can be partially characterized by its order structure with respect to an order relation such as the set inclusion rela- scaling subspace tion ⊆. Most transforms have a very simple M-𝑛 order structure, as illustrated to the right.133 The M-𝑛 lattices for 𝑛 ≥ 3 are modu𝙑0 𝙑1 lar (Lemma A.56 page 37) but not distributive (Theorem A.57 page 37). Analyses typically have one subspace that is a scaling subspace; and this subspace is often simply a family of constants (as is the case with Fourier Analysis).

analysis of 𝙓 𝙓 𝙑2

⋯

𝙑𝑛−1

𝟬

An analysis can be represented using three different structures: 131

📃 Lemarié (1990), 📘 Mallat (1999) page 240 The word analysis comes from the Greek word ἀvάλυσις, meaning “dissolution” (📘 Perschbacher (1990), page 23 ⟨entry 359⟩), which in turn means “the resolution or separation into component parts” (📘 Black et al. (2009), http: 132

//dictionary.reference.com/browse/dissolution) 133

📘 Greenhoe (2013) page 29 ⟨§2.2⟩



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D


➀ sequence of subspaces ➁ sequence of basis vectors ➂ sequence of basis coefficients These structures are isomorphic to each other, and can therefore be used interchangeably. Example D.11. 134 Some examples of the order structures of some analyses are illustrated in Figure 1 (page 4) and Figure 4 (page 8).

D.3 Multiresolution analysis A multiresolution analysis provides “coarse” approximations of a function in a linear space 𝙇𝟤ℝ at multiple “scales” or “resolutions”. Key to this process is a sequence of scaling functions. Most traditional transforms feature a single scaling function 𝜙(𝑥) set equal to one (𝜙(𝑥) = 1). This allows for convenient representation of the most basic functions, such as constants.135 A multiresolution system, on the other hand, uses a generalized form of the scaling concept:136 1. Instead of the scaling function simply being set equal to unity (𝜙(𝑥) = 1), a multiresolution analysis (Definition D.12 page 58) is often constructed in such a way that the scaling function 𝜙(𝑥) forms a partition of unity such that ∑𝑛∈ℤ 𝐓 𝑛 𝜙(𝑥) = 1. 2. Instead of there being just one scaling function, there is an entire sequence of scaling functions ⦅𝐃𝑗 𝜙(𝑥)⦆𝑗∈ℤ , each corresponding to a different “resolution”. Definition D.12.

137

Let ⦅𝙑𝑗 ⦆𝑗∈ℤ be a sequence of subspaces on 𝙇𝟤ℝ . Let 𝐴− be the closure of a set

𝐴. The sequence ⦅𝙑𝑗 ⦆𝑗∈ℤ is a multiresolution analysis on 𝙇𝟤ℝ if

∀𝑗∈ℤ (closed) and 𝙑𝑗 = 𝙑𝑗 − 2. 𝙑𝑗 ⊂ 𝙑𝑗+1 ∀𝑗∈ℤ (linearly ordered) and − ⎞ ⎛ 𝙑𝑗 ⎟ = 𝙇𝟤ℝ (dense in 𝙇𝟤ℝ ) and 3. ⎜ ⋃ ⎜𝑗∈ℤ ⎟ ⎠ ⎝ 4. 𝖿 ∈ 𝙑𝑗 ⟺ 𝐃𝖿 ∈ 𝙑𝑗+1 ∀𝑗∈ℤ, 𝖿 ∈𝙇𝟤ℝ (self-similar) and 𝑛 5. ∃𝜙 such that {𝐓 𝜙| 𝑛∈ℤ} is a Riesz basis for 𝙑0 . A multiresolution analysis is also called an MRA. An element 𝙑𝑗 of ⦅𝙑𝑗 ⦆𝑗∈ℤ is a scaling subspace of 1.

the space 𝙇𝟤ℝ . The pair (𝙇𝟤ℝ , ⦅𝙑𝑗 ⦆) is a multiresolution analysis space, or MRA space. The function 𝜙 is the scaling function of the MRA space. The traditional definition of the MRA also includes the following: 𝑛 6. 𝖿 ∈ 𝙑𝑗 ⟺ 𝐓 𝖿 ∈ 𝙑𝑗 ∀𝑛,𝑗∈ℤ, 𝖿 ∈𝙇𝟤ℝ (translation invariant) 7. 𝙑 = {𝟘} (greatest lower bound is 𝟬) ⋂ 𝑗 𝑗∈ℤ

134

📘 Greenhoe (2013) pages 30–31 Jawerth and Sweldens (1994) page 8 136 The concept of a scaling space was perhaps first introduced by Taizo Iijima in 1959 in Japan, and later as the Gaussian Pyramid by Burt and Adelson in the 1980s in the West. 📘 Mallat (1989) page 70, 📘 Iijima (1959), 📘 Burt and Adelson (1983), 📘 Adelson and Burt (1981), 📘 Lindeberg (1993), 📘 Alvarez et al. (1993), 📘 Guichard et al. (2012), 📘 Weickert (1999) ⟨historical survey⟩ 137 📘 Hernández and Weiss (1996) page 44, 📘 Mallat (1999) page 221 ⟨Definition 7.1⟩ , 📘 Mallat (1989) page 70, 📘 Meyer (1992) page 21 ⟨Definition 2.2.1⟩, 📘 Christensen (2003) page 284 ⟨Definition 13.1.1⟩, 📘 Bachman et al. (2000) pages 451–452 ⟨Definition 7.7.6⟩, 📘 Walnut (2002) pages 300–301 ⟨Definition 10.16⟩, 📘 Daubechies (1992) pages 129– 140 ⟨Riesz basis: page 139⟩ 135

📃

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Multiresolution analysis

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Daniel J. Greenhoe

However, these follow from the MRA as defined in Definition D.12 (Proposition D.13 page 59, Proposition D.14 page 59). Proposition D.13. 138 Let MRA be defined as in Definition D.12 page 58. 𝖿 ∈ 𝙑𝑗 ⟺ 𝐓 𝑛 𝖿 ∈ 𝙑𝑗 ∀𝑛,𝑗∈ℤ, 𝖿∈𝙇𝟤ℝ } ⟹ { ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ {⦅𝙑𝑗 ⦆𝑗∈ℤ is an MRA} translation invariant

Proposition D.14.

139

Let MRA be defined as in Definition D.12 page 58. 𝙑 = {𝟘} (greatest lower bound is 𝟬) ⟹ ⋂ 𝑗 {⦅𝙑𝑗 ⦆𝑗∈ℤ is an MRA} { 𝑗∈ℤ

}

The MRA (Definition D.12 page 58) is more than just an interesting mathematical toy. Under some very “reasonable” conditions (next proposition), as 𝑗 → ∞, the scaling subspace 𝙑𝑗 is dense in 𝙇𝟤ℝ …meaning that with the MRA we can represent any “reasonable” function to within an arbitrary accuracy. Proposition D.15. ⎧ ⎪ ⎨ ⎪ ⎩

(1). (2). (3).

140

⦅𝐓 𝑛 𝜙⦆ is a Riesz sequence ̃ 𝜙(𝜔) is continuous at 0 ̃ 𝜙(0) ≠ 0

and and

−

⎞ ⎫ ⎧ ⎪ ⎛⎜ ⎪ 𝟤 𝙑 ⟹ ⎨ ⎜ ⋃ 𝑗 ⎟⎟ = 𝙇ℝ ⎬ ⎪ ⎪ ⎩ ⎝𝑗∈ℤ ⎠ ⎭

A multiresolution analysis (Definition D.12 page 58) together with the set inclusion relation ⊆ form the linearly ordered set (Definition A.4 page 27) (⦅𝙑𝑗 ⦆ , ⊆), illustrated to the right by a Hasse diagram (Definition A.6 page 28). Subspaces 𝙑𝑗 increase in “size” with increasing 𝑗. That is, they contain more and more vectors (functions) for larger and larger 𝑗—with the upper limit of this sequence being 𝙇𝟤ℝ . Alternatively, we can say that approximation within a subspace 𝙑𝑗 yields greater “resolution” for increasing 𝑗.141

(dense in 𝙇𝟤ℝ )

𝙇𝟤ℝ

⎫ ⎪ ⎬ ⎪ ⎭ entire linear space

⋮

𝙑2 𝙑1 𝙑0 𝙑−1

larger subspaces smaller subspaces

⋮ 𝟬

smallest subspace

Remark D.16. 142 Note that the greatest lower bound (g.l.b.) of the linearly ordered set (⦅𝙑𝑗 ⦆ , ⊆) is 𝟬 (Proposition D.14 page 59): All linear subspaces contain the zero vector. So the intersection of any two subspaces must at least contain 𝟘. If the intersection of any two linear subspaces 𝙓 and 𝙔 is exactly {𝟘}, then for any vector in the sum of those subspaces (𝒖 ∈ 𝙓 ⨣ 𝙔 ) there are unique vectors 𝖿 ∈ 𝙓 and 𝗀 ∈ 𝙔 such that 𝒖 = 𝖿 + 𝗀. This is not necessarily true if the intersection contains more than just {𝟘} . Example D.17. 138

📘 Hernández and Weiss (1996) page 45 ⟨Theorem 1.6⟩, 📘 Greenhoe (2013) pages 32–33 ⟨Proposition 2.1⟩ 📘 Wojtaszczyk (1997) pages 19–28 ⟨Proposition 2.14⟩, 📘 Hernández and Weiss (1996) page 45 ⟨Theorem 1.6⟩, 📘 Pinsky (2002) pages 313–314 ⟨Lemma 6.4.28⟩, 📘 Greenhoe (2013) pages 33–35 ⟨Proposition 2.2⟩ 140 📘 Wojtaszczyk (1997) pages 28–31 ⟨Proposition 2.15⟩, 📘 Greenhoe (2013) pages 35–37 ⟨Proposition 2.3⟩ 141 📘 Michel and Herget (1993) page 83 ⟨Theorem 3.2.12⟩, 📘 Kubrusly (2001) page 67 ⟨Theorem 2.14⟩, 📘 Greenhoe (2014a) ⟨Theorem 7.1⟩ 142 📘 Greenhoe (2013) page 38 ⟨§2.3.2 Order structure⟩ 139



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transform −3

−1

2 𝜋

−2

approximation 3

1 −2 𝜋

2

2 𝜋

𝑥

𝑛 −3

−2

−1

−2 𝜋

1

2

3

𝙑0 √2 𝜋

√2 𝜋

𝑛

𝑥

−√2 𝜋

−√2 𝜋

𝙑1 𝑥

𝑛

𝙑2 Figure 19: Example approximations of 𝗌𝗂𝗇(𝜋𝑥) in 3 Haar scaling subspaces (see Example D.17 page 59)

In the Haar MRA, the scaling function 𝜙(𝑥) is the pulse function 1

1 for 𝑥 ∈ [0 ∶ 1) 𝜙(𝑥) = { 0 otherwise. −1

0

1

2

In the subspace 𝙑𝑗 (𝑗 ∈ ℤ) the scaling functions are (2)𝑗/2 for 𝑥 ∈ [0 ∶ (2−𝑗 )) 𝐃𝑗 𝜙(𝑥) = { 0 otherwise.

2𝑗/2

2−𝑗

The scaling subspace 𝙑0 is the span 𝙑0 ≜ 𝗌𝗉𝖺𝗇{𝐓 𝑛 𝜙| 𝑛∈ℤ}. The scaling subspace 𝙑𝑗 is the span 𝙑𝑗 ≜ 𝗌𝗉𝖺𝗇{𝐃𝑗 𝐓𝑛 𝜙|𝑛 ∈ ℤ}. Note that ‖𝐃𝑗 𝐓 𝑛 𝜙‖ for each resolution 𝑗 and shift 𝑛 is unity: 2 𝑗 𝑛 2 ‖𝐃 𝐓 𝜙‖ = ‖𝜙‖ 1

=

∫ 0

|1|2 𝖽𝑥

by definition of ‖⋅‖ on 𝙇𝟤ℝ

=1

1

Let 𝖿(𝑥) = 𝗌𝗂𝗇(𝜋𝑥). Suppose we want to project 𝖿(𝑥) onto the subspaces 𝙑0 , 𝙑1 , 𝙑2 , ….

The values of the transform coefficients for the subspace 𝙑𝑗 are given by

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D.4

Wavelet analysis

Daniel J. Greenhoe

1 𝑗 𝑛 ⟨𝖿(𝑥) | 𝐃 𝐓 𝜙⟩ 2 𝑗 𝑛 ‖𝐃 𝐓 𝜙‖ 1 𝑗/2 𝑗 = ⟨1𝖿 (𝑥) | 2 𝜙(2 𝑥 − 𝑛)⟩ * 2 ‖𝜙‖ = 2𝑗/2 ⟨𝖿(𝑥) | 𝜙(2𝑗 𝑥 − 𝑛)⟩

page 61

[𝐑𝑗 𝖿(𝑥)](𝑛) =

=2

𝑗/2

=2

𝑗/2

=2

𝑗/2

by Proposition D.4 page 55

2−𝑗 (𝑛+1)

∫ 2−𝑗 𝑛

𝖿(𝑥) 𝖽𝑥

2−𝑗 (𝑛+1)

∫ 2−𝑗 𝑛

𝗌𝗂𝗇(𝜋𝑥) 𝖽𝑥 −𝑗

=

2 (𝑛+1) 1 − 𝖼𝗈𝗌(𝜋𝑥) ( 𝜋) |2−𝑗 𝑛

2𝑗/2 𝖼𝗈𝗌 2−𝑗 𝑛𝜋 ) − 𝖼𝗈𝗌(2−𝑗 (𝑛 + 1)𝜋 )] 𝜋 [ (

And the projection 𝐀 𝑛 𝖿 (𝑥) of the function 𝖿(𝑥) onto the subspace 𝙑𝑗 is 𝐀𝑗 𝖿(𝑥) =

𝖿(𝑥) | 𝐃𝑗 𝐓 𝑛 𝜙⟩ 𝐃𝑗 𝐓 𝑛 𝜙 ∑⟨

𝑛∈ℤ 𝑗/2

=

2 𝖼𝗈𝗌 2−𝑗 𝑛𝜋 ) − 𝖼𝗈𝗌(2−𝑗 (𝑛 + 1)𝜋 )]2𝑗/2 𝜙(2𝑗 𝑥 − 𝑛) ∑[ ( 𝜋 𝑛∈ℤ

=

2𝑗 𝖼𝗈𝗌(2−𝑗 𝑛𝜋 ) − 𝖼𝗈𝗌(2−𝑗 (𝑛 + 1)𝜋 )]𝜙(2𝑗 𝑥 − 𝑛) [ ∑ 𝜋 𝑛∈ℤ

The transforms into the subspaces 𝙑0 , 𝙑1 , and 𝙑2 , as well as the approximations in those subspaces are as illustrated in Figure 19 (page 60).

D.4 Wavelet analysis The term “wavelet” comes from the French word “ondelette”, meaning “small wave”. And in essence, wavelets are “small waves” (as opposed to the “long waves” of Fourier analysis) that form a basis for the Hilbert space 𝙇𝟤ℝ .143 Definition D.18. 144 Let 𝐓 and 𝐃 be as defined in Definition D.1 page 55. A function 𝜓(𝑥) in 𝙇𝟤ℝ is a wavelet function for 𝙇𝟤ℝ if 𝑗 𝑛 𝟤 {𝐃 𝐓 𝜓|𝑗,𝑛∈ℤ} is a Riesz basis for 𝙇ℝ . In this case, 𝜓 is also called the mother wavelet of the basis {𝐃𝑗 𝐓 𝑛 𝜓|𝑗,𝑛∈ℤ}. The sequence of subspaces ⦅𝙒𝑗 ⦆𝑗∈ℤ is the wavelet analysis induced by 𝜓, where each subspace 𝙒𝑗 is defined as 𝙒𝑗 ≜ 𝗌𝗉𝖺𝗇{𝐃𝑗 𝐓 𝑛 𝜓 | 𝑛∈ℤ} .

A wavelet analysis ⦅𝙒𝑗 ⦆ is often constructed from a multiresolution anaysis (Definition D.12 page 58) ⦅𝙑𝑗 ⦆ under the relationship 143 144

📘 Strang and Nguyen (1996) page ix, 📘 Atkinson and Han (2009) page 191 📘 Wojtaszczyk (1997) page 17 ⟨Definition 2.1⟩, 📘 Greenhoe (2013) page 50 ⟨Definition 2.4⟩



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𝙑𝑗+1 = 𝙑𝑗 ⨣ 𝙒𝑗 , where ⨣ is subspace addition (Minkowski addition). By this relationship alone, ⦅𝙒𝑗 ⦆ is in no way uniquely defined in terms of a multiresolution analysis ⦅𝙑𝑗 ⦆. In general there are many possible complements of a subspace 𝙑𝑗 . To uniquely define such a wavelet subspace, one or more additional constraints are required. One of the most common additional constraints is orthogonality, such that 𝙑𝑗 and 𝙒𝑗 are orthogonal to each other. Definition D.19. Let (𝙇𝟤ℝ , ⦅𝙑𝑗 ⦆ , 𝜙, ⦅ℎ𝑛 ⦆) be a multiresolution system (Definition D.12 page 58) and ⦅𝙒𝑗 ⦆𝑗∈ℤ a wavelet analysis (Definition D.18 page 61) with respect to ⦅𝙑𝑗 ⦆𝑗∈ℤ . Let ⦅𝑔𝑛 ⦆𝑛∈ℤ be a sequence of coefficients such that 𝜓 = ∑𝑛∈ℤ 𝑔𝑛 𝐃𝐓𝑛 𝜙. A wavelet system is the tuple 𝟤 (𝙇ℝ , ⦅𝙑𝑗 ⦆ , ⦅𝙒𝑗 ⦆ , 𝜙, 𝜓, ⦅ℎ𝑛 ⦆ , ⦅𝑔𝑛 ⦆) and the sequence ⦅𝑔𝑛 ⦆𝑛∈ℤ is the wavelet coefficient sequence. Theorem D.20. 145 Let (𝙇𝟤ℝ , ⦅𝙑𝑗 ⦆ , ⦅𝙒𝑗 ⦆ , 𝜙, 𝜓, ⦅ℎ𝑛 ⦆ , ⦅𝑔𝑛 ⦆) be a wavelet system (Definition D.19 page 62). Let 𝙑1 ⨣ 𝙑2 represent Minkowski addition of two subspaces 𝙑1 and 𝙑2 of a Hilbert space 𝙃 . (𝙇𝟤ℝ is equivalent to one very large scaling subspace) 𝙇𝟤ℝ = lim 𝙑𝑗 𝑗→∞

= 𝙑𝑗 ⨣ 𝙒𝑗 ⨣ 𝙒𝑗+1 ⨣ 𝙒𝑗+2 ⨣ ⋯ = ⋯ ⨣ 𝙒−2 ⨣ 𝙒−1 ⨣ 𝙒0 ⨣ 𝙒1 ⨣ 𝙒2 ⨣ ⋯

𝙇𝟤ℝ is equivalent to one scaling space ( and a sequence of wavelet subspaces ) (𝙇𝟤ℝ is equivalent to a sequence of wavelet subspaces)

✎PROOF: 1. Proof for (1): 𝙇𝟤ℝ = lim 𝙑𝑗

by Definition D.12 page 58

𝑗→∞

2. Proof for (2): 𝙑 𝑗 ⨣ 𝙒𝑗 ⨣ 𝙒𝑗+1 ⨣ 𝙒𝑗+2 ⨣ ⋯ = 𝙑 𝑗+1 ⨣ 𝙒𝑗+1 ⨣ 𝙒𝑗+2 ⨣ 𝙒𝑗+3 ⨣ ⋯ ⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝙑𝑗+1

𝙑𝑗+2

=𝙑 𝑗+2 ⨣ 𝙒𝑗+2 ⨣ 𝙒𝑗+3 ⨣ 𝙒𝑗+4 ⨣ ⋯ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝙑𝑗+3

=𝙑 𝑗+3 ⨣ 𝙒𝑗+3 ⨣ 𝙒𝑗+4 ⨣ 𝙒𝑗+5 ⨣ ⋯ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝙑𝑗+4

=𝙑 𝑗+5 ⨣ 𝙒𝑗+5 ⨣ 𝙒𝑗+6 ⨣ 𝙒𝑗+6 ⨣ ⋯ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝙑𝑗+5

= lim 𝙑𝑗+5 ⨣ 𝙒𝑗+5 ⨣ 𝙒𝑗+6 ⨣ 𝙒𝑗+6 ⨣ ⋯ 𝑗→∞

= 𝙇𝟤ℝ 145

📘 Greenhoe (2013) page 53 ⟨Theorem 2.8⟩

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3. Proof for (3): 𝙇𝟤ℝ =

𝙑 ⏟0

⨣ 𝙒0 ⨣ 𝙒1 ⨣ 𝙒 2 ⨣ 𝙒3 ⨣ ⋯

by (2)

𝙑−1 ⨣ 𝙒−1

=

𝙑 ⏟ −1 𝙒−1 ⨣ 𝙒0 ⨣ 𝙒1 ⨣ 𝙒2 ⨣ 𝙒3 ⨣ ⋯ 𝙑−2 ⨣ 𝙒−2

=

𝙑 ⏟ −2 𝙒−2 ⨣ 𝙒−1 ⨣ 𝙒0 ⨣ 𝙒1 ⨣ 𝙒2 ⨣ 𝙒3 ⨣ ⋯ 𝙑−3 ⨣ 𝙒−3

=

𝙑 −3 𝙒−3 ⨣ 𝙒−2 ⨣ 𝙒−1 ⨣ 𝙒0 ⨣ 𝙒1 ⨣ 𝙒2 ⨣ 𝙒3 ⨣ ⋯ ⏟ 𝙑−4 ⨣ 𝙒−4

⋮ = ⋯ ⨣ 𝙒−3 ⨣ 𝙒−2 ⨣ 𝙒−1 ⨣ 𝙒0 ⨣ 𝙒1 ⨣ 𝙒2 ⨣ 𝙒3 ⨣ ⋯

✏ Remark D.21. In the special case that two subspaces 𝙒1 and 𝙒2 are orthogonal to each other, then the subspace addition operation 𝙒1 ⨣ 𝙒2 is frequently expressed as 𝙒1 ⊕ 𝙒2 . In the case of an orthonormal wavelet system, the expressions in Theorem D.20 (page 62) could be expressed as 𝙇𝟤ℝ = lim 𝙑𝑗 𝑗→∞

= 𝙑𝑗 ⊕ 𝙒𝑗 ⊕ 𝙒𝑗+1 ⊕ 𝙒𝑗+2 ⊕ ⋯ = ⋯ ⊕ 𝙒−2 ⊕ 𝙒−1 ⊕ 𝙒0 ⊕ 𝙒1 ⊕ 𝙒2 ⊕ ⋯ . .

D.5 Fast Wavelet Transform (FWT) Filter banks can be used to implement a “Fast Wavelet Transform” (FWT ). This is illustrated in Figure 20 page 64.146

References 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. Edward H. Adelson and Peter J. Burt. Image data compression with the laplacian pyramid. In Proceedings of the Pattern Recognition and Information Processing Conference, pages 218–223, Dallas Texas, 1981. IEEE Computer Society Press. URL citeseerx.ist.psu.edu/viewdoc/summary? doi=10.1.1.50.791. 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. Luis Alvarez, Frédéric Guichard, Pierre-Louis Lions, and Jean Michel Morel. Axioms and fundamental equations of image processing. Archive for Rational Mechanics and Analysis, 123(3):199–257, 1993. URL http://link.springer.com/article/10.1007/BF00375127. 146

📘 Mallat (1999) page 257 ⟨Figure 7.12⟩, 📘 Greenhoe (2013) pages 371–372 ⟨Figure L.1⟩



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𝑣𝑗 (𝑛) = ⟨𝑓 (𝑥) | 𝜙𝑗,𝑛 (𝑥)⟩ ?

?

̄ 𝗁(𝑛)

̄ 𝗀(𝑛)

?

?

↓2

↓2 -

𝑣𝑘−1 (𝑛) = ⟨𝖿 (𝑥) | 𝜙𝑘−1,𝑛 (𝑥)⟩ ?

?

̄ 𝗁(𝑛)

̄ 𝗀(𝑛)

?

?

↓2

↓2 -

𝑣𝑘−2 (𝑛) = ⟨𝖿 (𝑥) | 𝜙𝑘−2,𝑛 (𝑥)⟩ ?

𝑤𝑘−1 (𝑛) = ⟨𝖿(𝑥) | 𝜓𝑘−1,𝑛 (𝑥)⟩

𝑤𝑘−2 (𝑛) = ⟨𝖿(𝑥) | 𝜓𝑘−2,𝑛 (𝑥)⟩

?

⋮

⋮

𝑣1 (𝑛) = ⟨𝖿(𝑥) | 𝜙1,𝑛 (𝑥)⟩

𝑤1 (𝑛) = ⟨𝖿(𝑥) | 𝜓1,𝑛 (𝑥)⟩ ?

?

̄ 𝗁(𝑛)

̄ 𝗀(𝑛)

?

?

↓2

↓2

𝑣0 (𝑛) = ⟨𝖿 (𝑥) | 𝜙(𝑥 − 𝑛)⟩

?

-

𝑤0 (𝑛) = ⟨𝖿(𝑥) | 𝜓(𝑥 − 𝑛)⟩

Figure 20: 𝑘-Stage Fast Wavelet Transform (FWT) Ivan Amato. Expanding the genetic alphabet: New symbols added to genetic molecules form unknown messages. Science News, 137(6):88–90+94, February 10 1990. URL http://www.jstor. org/stable/3974601. Kendall E. Atkinson and Weimin Han. Theoretical Numerical Analysis: A Functional Analysis Framework, volume 39 of Texts in Applied Mathematics. Springer, 3 edition, 2009. ISBN 9781441904584. URL http://books.google.com/books?vid=ISBN1441904581. George Bachman, Lawrence Narici, and Edward Beckenstein. Fourier and Wavelet Analysis. Universitext Series. Springer, 2000. ISBN 9780387988993. URL http://books.google.com/books? vid=ISBN0387988998. 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). 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. John Benedetto and Ahmed I. Zayed, editors. A Prelude to Sampling, Wavelets, and Tomography, pages 1–32. Applied and Numerical Harmonic Analysis. Springer, 2004. ISBN 9780817643041. URL http://books.google.com/books?vid=ISBN0817643044. 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. CC BY 4.0


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Patrick Billingsley. Probability And Measure. Wiley series in probability and mathematical statistics. Wiley, 3 edition, 1995. ISBN 978-0471007104. URL http://books.google.com/books?vid= ISBN0471007102. Garrett Birkhoff. On the combination of subalgebras. Mathematical Proceedings of the Cambridge Philosophical Society, 29:441–464, October 1933. doi: 10.1017/S0305004100011464. URL http: //adsabs.harvard.edu/abs/1933MPCPS..29..441B. Garrett Birkhoff. The logic of quantum mechanics. Annals of Mathematics, 37(4):823–843, October 1936. URL http://www.jstor.org/stable/1968621. 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, 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 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. Duncan Black, Ian Brooks, and et al. Robert Groves, editors. Collins English Dictionary—Complete and Unabridged. HarperCollins Publishers, 10 edition, 2009. ISBN 978-0-00-732119-3. URL http://books.google.com/books?vid=ISBN9780007321193. Thomas Scott Blyth. Lattices and ordered algebraic structures. Springer, London, 2005. ISBN 1852339055. URL http://books.google.com/books?vid=ISBN1852339055. 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. Peter J. Burt and Edward H. Adelson. The laplacian pyramid as a compact image code. IEEE Transactions On Communications, COM-3L(4):532–540, April 1983. URL http://citeseer.ist.psu. edu/burt83laplacian.html. Peter G. Casazza and Mark C. Lammers. Bracket Products for Weyl-Heisenberg Frames, pages 71–98. Applied and Numerical Harmonic Analysis. Birkhäuser, 1998. ISBN 9780817639594. 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. Ole Christensen. An Introduction to Frames and Riesz Bases. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston/Basel/Berlin, 2003. ISBN 0-8176-4295-1. URL http://books. google.com/books?vid=ISBN0817642951. Tuesday 14th February, 2017 9:12pm UTC Copyright © Daniel J. Greenhoe


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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. Anthanasios Papoulis. Probability, Random Variables, and Stochastic Processes. McGraw-Hill, New York, 3 edition, 1991. ISBN 0-07-048477-5. URL http://books.google.com.tw/books?vid= ISBN0070484775. C.S. Peirce. On the algebra of logic. American Journal of Mathematics, 3(1):15–57, March 1880. URL http://www.jstor.org/stable/2369442. Wesley J. Perschbacher, editor. The New Analytical Greek Lexicon. Hendrickson Publishers, Peabody, Mass., 1990. ISBN 978-0-943575-33-9. URL http://www.amazon.com/dp/0943575338. Mark A. Pinsky. Introduction to Fourier Analysis and Wavelets. Brooks/Cole, Pacific Grove, 2002. ISBN 0-534-37660-6. URL http://www.amazon.com/dp/0534376606. Jeffrey C. Pommerville. Fundamentals of Microbiology. Jones & Bartlett Publishers, January 1 2013. ISBN 9781449647964. URL http://books.google.com/books?vid=isbn1449647960. Shie Qian and Dapang Chen. Joint time-frequency analysis: methods and applications. PTR Prentice Hall, 1996. ISBN 9780132543842. URL http://books.google.com/books?vid= ISBN0132543842. 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/. 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. 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. et. al Runtao He. Analysis of multimerization of the sars coronavirus nucleocapsid protein. Biochemical and Biophysical Research Communications, 316(2):476–483, April 2 2004. URL http: //www.sciencedirect.com/science/article/pii/S0006291X04003250. et. al. S. G. Gregory. The dna sequence and biological annotation of human chromosome 1. Nature: International Weekly Journal of Science, 441:315–321, May 18 2006. doi: 10.1038/nature04727. URL http://www.nature.com/nature/journal/v441/n7091/abs/nature04727.html. 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. CC BY 4.0


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Ernst Schröder. Vorlesungen über die Algebra der Logik: Exakte Logik, volume 1. B. G. Teubner, Leipzig, 1890. URL http://www.archive.org/details/vorlesungenberd02mlgoog. 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. Neil J. A. Sloane. On-line encyclopedia of integer sequences. World Wide Web, 2014. URL http: //oeis.org/. 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. 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. Gilbert Strang and Truong Nguyen. Wavelets and Filter Banks. Wellesley-Cambridge Press, Wellesley, MA, 1996. ISBN 0-9614088-7-1. URL http://books.google.com/books?vid= ISBN0961408871. 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. 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/. 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. Ulrike von Krosigk and Steven A. Benner. Expanding the genetic alphabet: Pyrazine nucleosides that support a donor[bond]donor[bond]acceptor hydrogen-bonding pattern. Helvetica Chimica Acta, 87(6):1299–1324, June 24 2004. doi: 10.1002/hlca.200490120. URL http://onlinelibrary. wiley.com/doi/10.1002/hlca.200490120/abstract. John von Neumann. Continuous Geometry. Princeton mathematical series. Princeton University Press, Princeton, 1960. URL http://books.google.com/books?id=3bjqOgAACAAJ. David F. Walnut. An Introduction to Wavelet Analysis. Applied and numerical harmonic analysis. Springer, 2002. ISBN 0817639624. URL http://books.google.com/books?vid=ISBN0817639624. J. D. Watson and F. H. C. Crick. Molecular structures of nucleic acids: A struture for deoxyribose nucleic acid. Nature, pages 737–738, April 25 1953a. doi: 10.1038/171737a0. URL http://www. nature.com/nature/journal/v171/n4356/pdf/171737a0.pdf. Tuesday 14th February, 2017 9:12pm UTC Copyright © Daniel J. Greenhoe


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J. D. Watson and F. H. C. Crick. Genetical implications of the structure of deoxyribonucleic acid. Nature, pages 964–967, May 30 1953b. doi: 10.1038/171964b0. URL http://www.nature.com/ nature/journal/v171/n4361/pdf/171964b0.pdf. Joachim Weickert. Linear scale-space has first been proposed in japan. Journal of Mathematical Imaging and Vision, 10:237–252, May 1999. URL http://dl.acm.org/citation.cfm?id=607668. Alfred North Whitehead. A Treatise on Universal Algebra with Applications, volume 1. University Press, Cambridge, 1898. URL http://resolver.library.cornell.edu/math/1927624. P. Wojtaszczyk. A Mathematical Introduction to Wavelets, volume 37 of London Mathematical Society student texts. Cambridge University Press, February 13 1997. ISBN 9780521578943. URL http://books.google.com/books?vid=ISBN0521578949.

Reference Index Aliprantis and Burkinshaw (1998), 46 Adams (1990), 39 Adelson and Burt (1981), 4, 58 Alvarez et al. (1993), 4, 58 Amato (1990), 27 Atkinson and Han (2009), 5, 61 Bachman et al. (2000), 55, 58 Balbes and Dwinger (1975), 36 Bellman and Giertz (1973), 47 Beran (1985), 27, 34, 36, 37, 40–42 Billingsley (1995), 17 Birkhoff (1933), 28, 32, 33, 36 Birkhoff and Hall (1934), 37 Birkhoff (1936), 53 Birkhoff (1938), 32, 35 Birkhoff (1948), 32, 33, 36–38 Birkhoff (1967), 29, 34, 35, 45, 46 Birkhoff and Neumann (1936), 40, 42 Blyth (2005), 45, 46 Comnish-Bowden (1985), 27 Burris and Sankappanavar (1981), 32, 33, 36, 37 Burt and Adelson (1983), 4, 58 Casazza and Lammers (1998), 55 Cattaneo and Ciucci (2009), 48 Christensen (2003), 55, 58 Cifarelli and Regazzini (1996), 17 CC BY 4.0

Cohen (1989), 42 Black et al. (2009), 3, 57 Copson (1968), 23, 45 Cristianini and Hahn (2007), 25 Dai and Lu (1996), 55 Dai and Larson (1998), 55 Daubechies (1992), 58 Davey and Priestley (2002), 33 Davis (1955), 36 Dedekind (1900), 27, 32, 33, 36 de Finetti (1930a), 17 de Finetti (1930b), 17 de Finetti (1930c), 17 Devidi (2006), 47 Devidi (2010), 47 de Vries (2007), 47, 50 Deza and Laurent (1997), 45, 46 Deza and Deza (2006), 45 Deza and Deza (2009), 45 DiBenedetto (2002), 23 Dieudonné (1969), 45 Dilworth (1945), 38, 39 Dilworth (1950b), 29 Dilworth (1950a), 29 Dilworth (1984), 36 Doner and Tarski (1969), 33 Dunn (1996), 47 Dunn (1999), 47, 48 Elnitski et al. (2003), 27 Erné et al. (2002), 37 Euclid (circa 300BC), 46 Farley (1997), 29, 31 Feller (1970), 19 Forster and Massopust (2009), 56


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Foulis (1962), 36, 53 Fréchet (1906), 45 Fréchet (1928), 45 Fréchet (1930a), 17 Fréchet (1930b), 17 GenBank (2014), 25 Giles (1987), 23 Givant and Halmos (2009), 33, 39 Goodman et al. (1993), 55 Gottwald (1999), 47 Grätzer (1971), 36 Grätzer (2003), 28, 29, 33, 38 Grätzer (2007), 39 Greechie (1971), 18 Greenhoe (2013), 3, 4, 6, 7, 55, 57–59, 61–63 Greenhoe (2014a), 4, 59 Greenhoe (2014b), 21, 34, 47–51 Greenhoe (2014c), 5 Greenhoe (2016b), 33–38 Greenhoe (2016a), 32, 47–49, 52, 53 S. G. Gregory (2006), 25 Gudder (1988), 40 Guichard et al. (2012), 4, 58 Halmos (1960), 30 Hausdorff (1937), 30, 45 Runtao He (2004), 25 Heil (2011), 55 Heitzig and Reinhold (2002), 37 Hernández and Weiss (1996), 58, 59 Higgins (1996), 55 Holland (1963), 52, 53 Holland (1970), 33, 41, 53, 54 Huntington (1904), 39 Tuesday 14th February, 2017 9:12pm UTC Copyright © Daniel J. Greenhoe

Subject Index Huntington (1933), 39 Husimi (1937), 44, 48 Hutter and Benner (2003), 27 Iijima (1959), 4, 58 Istrǎţescu (1987), 45 Iturrioz (1985), 45 Jawerth and Sweldens (1994), 4, 58 Jenei (2003), 48 Jevons (1864), 39 Jiang and Seela (2010), 27 Kalmbach (1983), 40, 41, 43– 45, 48, 49, 53 Kalmbach (1986), 17 Kammler (2008), 55 Khamsi and Kirk (2001), 23, 45 Kolmogorov (1933b), 17, 18 Kolmogorov (1933a), 17 Kôpka and Chovanec (1994), 54 Korselt (1894), 27, 33, 36, 37 von Krosigk and Benner (2004), 27 Kubrusly (2001), 4, 59 Kubrusly (2011), 23 Larson and Andima (1975), 35 Lemarié (1990), 5, 57 Lidl and Pilz (1998), 44, 48 Lindeberg (1993), 4, 58 Loomis (1955), 40, 51

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MacLane and Birkhoff (1999), 27, 32, 37, 39 MacLaren (1964), 53 Maeda (1958), 53 Maeda (1966), 35, 41 Maeda and Maeda (1970), 32, 35, 36 Mallat (1989), 4, 58 Mallat (1999), 3, 5, 57, 58, 63 Malyshev et al. (2009), 27 McKenzie (1970), 34 Mendel (1853), 25 Meyer (1992), 58 Michel and Herget (1993), 4, 59 Müller (1909), 39 Müller-Olm (1997), 33 Munkres (2000), 23 Nakamura (1957), 49, 53 Narens (2014), 18 Nguyen and Walker (2006), 47 Novák et al. (1999), 47 Ore (1935), 27, 32, 35, 36 Ore (1940), 36 Packer (2004), 55 Padmanabhan and Rudeanu (2008), 34–36 Pap (1995), 17 Papoulis (1991), 18, 19 Peirce (1880), 27 Perschbacher (1990), 3, 57

Pinsky (2002), 59 Pommerville (2013), 25 Qian and Chen (1996), 56 Renedo et al. (2003), 45 Riečan (1957), 36 Roman (2008), 39 Rota (1964), 32 Rota (1997), 32 Saliǐ (1988), 38, 39 Sasaki (1954), 49, 53 Schröder (1890), 36, 39 Shen and Vereshchagin (2002), 29, 30 Stanley (1997), 29–31 Steen and Seebach (1978), 23 Stern (1999), 35, 38, 40, 41, 51 Strang and Nguyen (1996), 5, 61 Straßburger (2005), 50 Thakare et al. (2002), 37 Troelstra and van Dalen (1988), 47 von Neumann (1960), 36 Walnut (2002), 55, 58 Watson and Crick (1953b), 25 Watson and Crick (1953a), 25 Weickert (1999), 4, 58 Whitehead (1898), 39 Wojtaszczyk (1997), 55, 59, 61 Benedetto and Zayed (2004), 55

absorptive, 32, 34, 39, 44 additive, 17–19 additivity, 18 alphabetic order relation, 30 analysis, 3, 57 analyzed, 20 anti-atom, 35 anti-atomic, 35, 35 anti-chain, 31 anti-symmetric, 27 antichain, 29, 29, 31 antitone, 12, 40, 47, 48, 52 arithmetic axiom, 36 associative, 30, 32, 34, 39, 40, 44 atom, 7, 35 atomic, 7, 35, 35, 39 Axiom of Continuity, 17

base set, 27 basis, 55–57 Benzene ring, 41 Birkhoff distributivity criterion, 10, 37 Boole's inequality, 19 Boolean, 12, 17–19, 21, 39, 41, 43–46, 49, 50, 53 boolean, 40 Boolean algebra, 39, 39, 40, 44, 45 Boolean lattice, 1, 9, 10, 12, 14, 15, 17–20, 31, 39, 39, 49, 52 boolean lattice, 9, 10 Boolean logic, 20, 24, 50, 51 Boolean probability space, 22 bound greatest lower bound, 32 infimum, 32 least upper bound, 31

Subject Index 𝐿1 lattice, 42 𝐿2 lattice, 42 𝐿22 lattice, 42 𝐿32 lattice, 42 𝐿42 lattice, 42 𝐿52 lattice, 42 𝑀4 lattice, 41 𝑀6 lattice, 41 𝑂6 lattice, 41, 43, 44, 50, 52 𝑂8 lattice, 41 ℝ3 Euclidean space, 52 𝙇32 Boolean lattice, 54 GLB, 32 LUB, 31 σ-additive, 17 σ-additivity, 17, 18 3-dimensional Euclidean space, 50

ball a primorial lattice generated by 𝙇52 , 13 absolute value, 45 Tuesday 14th February, 2017 9:12pm UTC Copyright © Daniel J. Greenhoe

closed, 46 open, 46 bandlimited, 56 vel MRA-Wa et subspace architecture …

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supremum, 31 boundary, 49 boundary condition, 42, 48, 49 boundary condition (Theorem B.21 page 49), 50 bounded, 14, 34, 34, 39, 40 bounded lattice, 9, 10, 12, 34, 35, 38–40, 42–49 bounded lattice difference, 10, 10, 12 bounded lattice difference operator, 6 bounded lattices, 51 cardinal arithmetic, 30 Cardinal Series, 56 Cardinal series, 56 cartesian product, 30 center, 43, 44, 53, 54, 54 chain, 27, 29, 31, 32, 34 characterized, 3, 57 classic 10 Boolean properties, 39 classic logic, 51 closed, 58 closed ball, 14, 46 closed unit ball, 46, 46 closure, 42, 58 commutative, 30, 32, 34, 39, 40, 43 commutes, 53, 53, 54 compact support, 5, 57 compactly supported, 57 comparable, 27, 27, 29 complement, 38, 38, 39, 44 lattice, 38 complemented, 8, 10, 38, 39, 40, 42, 43 complemented lattice, 43 complements, 43 complex exponential, 57 complex plane, 23 conjunctive de Morgan, 48, 49 conjunctive de morgan, 42 conjunctive de Morgan inequality, 48 conjunctive distributive, 36 continuous, 57, 59 coordinate wise, 30 cosets, 7 cover relations, 28 covered by, 35 covering relation, 28 covers, 28, 35 D-poset, 12, 54, 54 de Morgan, 39, 44 de Morgan logic, 50, 51 de Morgan negation, 48, 48, 50 CC BY 4.0

Subject Index definitions anti-atom, 35 anti-atomic, 35 antichain, 29 atom, 35 atomic, 35 base set, 27 Benzene ring, 41 Boolean algebra, 39 Boolean lattice, 39 Boolean logic, 50 bounded, 34 center, 54 chain, 27 closed ball, 46 closed unit ball, 46 commutes, 53 D-poset, 54 de Morgan logic, 50 diamond, 37 dilation operator inverse, 55 distance function, 45 dual, 30, 32 fully ordered set, 27 fuzzy logic, 50 Hasse diagram, 28 hexagon, 41 intuitionalistic logic, 50 Kleene logic, 50 lattice, 32 lattice with negation, 48 linearly ordered set, 27 logic, 50 lower bounded, 34 M3 lattice, 37 metric, 45 metric space, 45 modular orthocomplemeted lattice, 45 mother wavelet, 61 MRA, 58 MRA space, 58 multiresolution analysis, 58 multiresolution analysis space, 58 N5 lattice, 36 O6 lattice, 41 open ball, 46 ordered set, 27 ortho logic, 50 orthocomplemented lattice, 40 orthogonal, 52 orthogonality, 52 p-lattice, 7 partially ordered set, 27 pentagon, 36 permutable, 49 poset, 27


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primorial lattice, 7 primorial lattice generated by 𝙇𝘕2 , 12 probability space, 17 reduction of 𝙇𝘕2 , 9 Sasaki projection, 49 scaling function, 58 scaling subspace, 58 subposet, 28 totally ordered set, 27 translation operator inverse, 55 unit ball, 46 upper bounded, 34 wavelet analysis, 61 wavelet coefficient sequence, 62 wavelet function, 61 wavelet system, 62 dense, 58, 59 diamond, 37 dictionary order relation, 30 difference, 54 difference operator, 1, 6 Digital signal processing, 24 digital signal processing, 22 dilation operator, 55 dilation operator inverse, 55 Dilworth's theorem, 29, 29, 31 Dirac delta distribution, 56 direct product, 29 direct sum, 29 discrete, 57 discrete metric, 23 discrete topology, 23 discrete-time signal analysis, 22 discrete-time signal processing, 22, 24 disjoint union, 29 disjunctive de Morgan, 48, 49 disjunctive de morgan, 42 disjunctive de Morgan inequality, 48 disjunctive distributive, 36 distance function, 45 distributive, 14, 18, 19, 30, 33, 36, 36–40, 43, 44, 49, 57 distributive inequalities, 33 distributive lattice, 35 distributive pseudocomplemented lattice, 18 distributivity, 17, 36 divides, 7 DSP, 22, 24 dual, 27, 30, 32 dual distributive, 36, 36 dual distributivity, 36 dual modular, 35 dual modularity, 35 Tuesday 14th February, 2017 9:12pm UTC Copyright © Daniel J. Greenhoe

Subject Index dyadic, 57 Elkan's law, 45 Euclidean space, 42 examples Cardinal Series, 56 Fourier Series, 56 Fourier Transform, 56 Gabor Transform, 56 Haar scaling function, 59 linear functions, 55 wavelets, 56 excluded middle, 19, 38, 42, 48, 49, 51 Fast Wavelet Transform, 24, 57, 63 fast wavelet transform, 63 field of probability, 17 Filter bank, 63 finite, 7, 8, 29, 34, 46 finite modular lattice, 18 finite width, 39 Fourier Analysis, 3, 57 Fourier coefficients, 56 Fourier Series, 56 Fourier Transform, 56 Fourier transform, 3, 57 fully ordered set, 27 function, 45, 47, 48 functions absolute value, 45 complement, 38, 38 complex exponential, 57 de Morgan negation, 48, 48, 50 discrete metric, 23 Fourier coefficients, 56 function, 45 fuzzy negation, 48, 48, 50 generalized probability, 18, 18 height, 14, 34, 46, 47 implication, 22, 50, 51 intuitionalistic negation, 50 intuitionistic negation, 48 Kleene negation, 20, 48, 50 lattice metric, 14 length, 34 maxmini, 14 measure, 17 measure-theoretic, 18 measure-theoretic probability, 17 membership functions, 21 metric, 45, 46 Tuesday 14th February, 2017 9:12pm UTC Copyright © Daniel J. Greenhoe

Daniel J. Greenhoe metric primorial projection, 14 minimal negation, 48, 48, 50 Minkowski addition, 62 mother wavelet, 56, 57 negation, 48, 48, 52 ortho negation, 19, 42, 48, 50, 51 orthomodular negation, 48 probability, 17, 17, 22 probability function, 17 quantum probability, 18, 18 Sasaki primorial projection, 14 Sasaki projection, 14 scaling function, 5 subminimal negation, 47, 47 subspace addition, 63 traditional probability, 18, 18 translation operator, 55 unique complement, 38 valuation, 14, 45, 45, 46 zero primorial projection, 14 Fuzzy logic, 21 fuzzy logic, 50, 51 fuzzy negation, 48, 48, 50 fuzzy subset logic, 21 FWT, 63 g.l.b., 59 Gabor Transform, 56 Gaussian Pyramid, 4, 58 generalized probability, 18, 18 genome, 25 Genomic Signal Processing, 25 genomic signal processing, 1, 6 greatest lower bound, 32, 32, 58, 59 GSP, 25 Haar, 60 Haar scaling function, 59 Hasse diagram, 4, 28, 28, 59 Hasse diagrams, 28 height, 14, 34, 34, 46, 47 hexagon, 41 Hilbert space, 42 Huntington properties, 39, 39 Huntington's axiom, 40, 44 Huntington's Fourth Set, 43 Huntington's fourth set, 39 vel MRA-Wa et subspace architecture …

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idempotency, 50 idempotent, 32, 39, 40, 43 identity, 39 identity operator, 55 implication, 22, 50, 51 incomparable, 27, 29 independent, 33 indiscrete topology, 23 inequalities distributive, 33 minimax, 33 modular, 33 infimum, 32 intersection, 42 intuitionalistic logic, 50, 51 intuitionalistic negation, 50 intuitionistic negation, 48, 48 involution, 44 involutory, 12, 39, 40, 44, 48 isotone, 46 join, 32 join super-distributive, 33 join-identity, 34 Kleene condition, 48, 49 Kleene logic, 50 Kleene negation, 20, 48, 50 lattice, 1, 7, 32, 32–34, 36, 37, 39, 42–46, 51 complemented, 38 distributive, 36 M3, 37 N5, 36 relatively complemented, 38 lattice metric, 14 lattice projection, 24 lattice reduction operator, 9, 9, 10 lattice with negation, 17, 20, 22, 48, 50–54 least element, 7 least upper bound, 31, 32, 34 length, 29, 31, 34 lexicographical, 30 linear, 32, 45, 55, 56 linear functions, 55 linear order relation, 27 linearly ordered, 58 linearly ordered set, 4, 27, 59 logic, 1, 6, 22, 50, 51 lower bound, 32, 32, 34 lower bounded, 34, 34, 51 M𝑛 lattice, 7 M3 lattice, 37, 37 maxmini, 14 measure, 17 measure-theoretic, 18 version 0.80

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measure-theoretic probability, 17 median inequality, 33 median property, 36 meet, 32 meet sub-distributive, 33 meet-identity, 34 membership functions, 21 metric, 45, 45, 46 metric lattice, 14, 46 metric primorial projection, 14 metric space, 45, 46 minimal negation, 48, 48, 50 Minimax inequality, 33 minimax inequality, 33 Minkowski addition, 62 Minkowski sum, 42 modular, 3, 18, 33, 35, 35–40, 45, 46, 57 Modular inequality, 33 modular inequality, 33 modular lattice, 36 modular orthocomplemented, 40, 41, 45 modular orthocomplemeted lattice, 45 modularity, 35 modularity inequality, 33 monotone, 17–19, 51 monotonic, 21 Monotony laws, 33 mother wavelet, 56, 57, 61 MRA, 1, 6, 58, 58, 59 MRA space, 58, 58 multiply complemented, 38, 38, 43 Multiresolution Analysis, 3, 5, 57 multiresolution analysis, 1, 4, 58, 58, 59 multiresolution analysis space, 58 multiresolution anaysis, 61 N5 sublattice, 10 N5 lattice, 8, 36, 36 negation, 48, 48, 52 non-associative, 52 non-Boolean, 17, 41, 44 non-boolean, 10 non-complemented, 38 non-contradiction, 12, 38, 40, 48–51 non-distributive, 3, 10, 37, 38 non-join-distributive, 44 non-modular, 37, 38, 47 non-monotonic, 21 non-negative, 45 non-orthocomplemented, 43 non-orthomodular, 41 CC BY 4.0

Subject Index nonBoolean, 8 noncomplemented, 8 noncontradiction, 14 nondegenerate, 17, 18, 45 nondistributive, 8 nonmodular, 8 nonnegative, 17, 18 nonorthocomplemented, 8 normalized, 17, 18 not modular orthocomplemented, 41 not uniquely complemented, 8 number of lattices, 37 O6 lattice, 18, 41, 41, 54 O6 sublattice, 10 ondelette, 5, 61 open ball, 46 operations bounded lattice difference, 10, 10, 12 bounded lattice difference operator, 6 cartesian product, 30 closure, 42 coordinate wise, 30 difference, 54 difference operator, 1, 6 Digital signal processing, 24 dilation operator, 55 direct product, 29 direct sum, 29 discrete-time signal processing, 24 disjoint union, 29 divides, 7 DSP, 24 genomic signal processing, 1, 6 identity operator, 55 intersection, 42 join, 32 lattice projection, 24 lattice reduction operator, 9, 9, 10 meet, 32 metric primorial projection, 14 Minkowski addition, 62 Minkowski sum, 42 ordinal product, 30 ordinal sum, 30 primorial projection, 15 projection operator, 20, 21, 27 reduction of 𝙇32 , 9 reduction of 𝙇42 , 9, 10 reduction of 𝙇52 , 10, 11 reduction operator, 6


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Sasaki primorial projection, 14, 14 Sasaki projection, 49, 50, 53 sasaki projection, 50 Sasaki projection of 𝑦 onto 𝑥, 49 set difference, 10, 27 set difference operator, 54 set inclusion, 54 supremum, 32 symbolic sequence analysis, 1, 6 symbolic sequence processing, 1, 6, 24 symbolic signal processing, 24 transform, 3, 57 translation operator, 55 zero primorial projection, 14 order preserving, 30 order relation, 9, 27, 27, 28 order relations, 28 ordered set, 27, 27, 29, 32, 54 linearly, 27 totally, 27 ordered set of partitions of an integer, 32 ordering relation, 12 ordinal product, 30 ordinal sum, 30 ortho logic, 20, 50, 51 ortho logics, 20 ortho negation, 19, 42, 48, 50, 51 orthocomplement, 40, 40, 42 orthocomplemented, 9, 10, 12, 19, 21, 39, 40, 40–45, 47, 50 Orthocomplemented lattice, 40 orthocomplemented lattice, 1, 6, 12, 22, 40, 42, 43, 49–54 orthocomplemented lattices, 41 orthocomplemented O6 lattice, 20 orthocomplemented pair, 9, 12 orthocomplemented pairs, 12, 40 orthogonal, 52, 52, 63 orthogonality, 52, 52, 62 orthomodular, 18, 40, 41, 44, 45, 48, 53 orthomodular identity, 44, 53 orthomodular lattice, 18, 45 orthomodular negation, 48 Tuesday 14th February, 2017 9:12pm UTC Copyright © Daniel J. Greenhoe

Subject Index orthonormal wavelet system, 63 p-lattice, 7 Paley-Wiener, 56 partial order relation, 27 partially ordered set, 27 partition, 7, 29, 31 partition of unity, 4, 5, 58 pentagon, 36 periodic, 55 permutable, 49 pointwise order relation, 30 poset, 27 positive, 46 power set, 32, 54 prime numbers, 7 primoral lattice, 6 primorial, 7, 7, 8 primorial lattice, 1, 6, 7, 7, 12, 14, 15, 20–22 primorial lattice generated by 𝙇52 , 12 primorial lattice generated by 𝙇𝘕2 , 12, 20 primorial lattices, 7 primorial numbers, 7 primorial projection, 15 probability, 1, 6, 17, 17, 22 probability function, 17 probability space, 17, 19, 20 projection operator, 20, 21, 27 properties σ-additive, 17 σ-additivity, 17, 18 absorptive, 32, 34, 39, 44 additive, 17–19 additivity, 18 anti-atomic, 35 anti-symmetric, 27 antitone, 12, 40, 47, 48, 52 associative, 30, 32, 34, 39, 40, 44 atomic, 7, 35, 39 Boole's inequality, 19 Boolean, 12, 17–19, 21, 39, 41, 43–46, 49, 50, 53 boolean, 40 Boolean lattice, 19 boundary, 49 boundary condition, 42, 48, 49 boundary condition (Theorem B.21 page 49), 50 bounded, 14, 34, 39, 40 closed, 58 commutative, 30, 32, 34, 39, 40, 43 compact support, 5, 57 compactly supported, Tuesday 14th February, 2017 9:12pm UTC Copyright © Daniel J. Greenhoe

Daniel J. Greenhoe 57 comparable, 27, 27, 29 complement, 44 complemented, 8, 10, 38, 39, 40, 42, 43 conjunctive de Morgan, 48, 49 conjunctive de morgan, 42 conjunctive de Morgan inequality, 48 conjunctive distributive, 36 continuous, 57, 59 covered by, 35 covers, 28, 35 de Morgan, 39, 44 dense, 58, 59 discrete, 57 disjunctive de Morgan, 48, 49 disjunctive de morgan, 42 disjunctive de Morgan inequality, 48 disjunctive distributive, 36 distributive, 14, 18, 19, 30, 33, 36, 36–40, 43, 44, 49, 57 dual distributive, 36, 36 dual modular, 35 dyadic, 57 Elkan's law, 45 excluded middle, 19, 38, 42, 48, 49, 51 finite, 7, 8, 29, 34, 46 finite width, 39 greatest lower bound, 32 Huntington properties, 39 Huntington's axiom, 40, 44 idempotency, 50 idempotent, 32, 39, 40, 43 identity, 39 incomparable, 27, 29 independent, 33 intuitionistic negation, 48 involution, 44 involutory, 12, 39, 40, 44, 48 isotone, 46 join super-distributive, 33 join-identity, 34 Kleene condition, 48, 49 least upper bound, 32 length, 29, 31 vel MRA-Wa et subspace architecture …

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linear, 32, 45, 55, 56 linearly ordered, 58 lower bounded, 34, 51 median inequality, 33 median property, 36 meet sub-distributive, 33 meet-identity, 34 modular, 3, 18, 33, 35, 35–40, 45, 46, 57 modular orthocomplemented, 40, 41, 45 monotone, 17–19, 51 monotonic, 21 multiply complemented, 38, 38, 43 non-associative, 52 non-Boolean, 17, 41, 44 non-boolean, 10 non-complemented, 38 non-contradiction, 12, 38, 40, 48–51 non-distributive, 3, 10, 37, 38 non-join-distributive, 44 non-modular, 37, 38, 47 non-monotonic, 21 non-negative, 45 non-orthocomplemented, 43 non-orthomodular, 41 nonBoolean, 8 noncomplemented, 8 noncontradiction, 14 nondegenerate, 17, 18, 45 nondistributive, 8 nonmodular, 8 nonnegative, 17, 18 nonorthocomplemented, 8 normalized, 17, 18 not modular orthocomplemented, 41 not uniquely complemented, 8 order preserving, 30 orthocomplemented, 9, 10, 12, 19, 21, 39, 40, 40–45, 47, 50 orthogonal, 52, 63 orthomodular, 18, 40, 41, 44, 45, 48, 53 orthomodular identity, 44, 53 Paley-Wiener, 56 partition of unity, 5 periodic, 55 positive, 46 primorial, 7, 7, 8 version 0.80

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primorial lattice, 12 reflexive, 27 self-similar, 58 subadditive, 18, 45 symmetric, 45, 52, 53, 53 transitive, 27 translation invariant, 58, 59 triangle inequality, 45 unbounded, 34 unique, 59 uniquely complemented, 38, 38, 39 upper bounded, 18, 34, 51 valuation, 46, 47 weak double negation, 48, 52 width, 29, 29, 31 pulse function, 60 quantum probability, 18, 18 real valued lattice, 45 reduction of 𝙇22 , 9 reduction of 𝙇23 , 9 reduction of 𝙇24 , 9, 10 reduction of 𝙇25 , 10, 11 reduction of 𝙇2𝘕 , 9 reduction operator, 6 reflexive, 27 relation, 35, 36 relations alphabetic order relation, 30 commutes, 53, 53, 54 covering relation, 28 dictionary order relation, 30 distributivity, 17, 36 dual, 27 dual distributivity, 36 dual modularity, 35 lexicographical, 30 linear order relation, 27 modularity, 35 order relation, 9, 27, 28 orthogonality, 52 partial order relation, 27 pointwise order relation, 30 set inclusion, 1 resolution, 4, 58, 59 Riesz basis, 58, 61 Riesz sequence, 59 Sasaki primorial projection, 14, 14 Sasaki projection, 14, 49, 49, 50, 53 sasaki projection, 50 CC BY 4.0

Subject Index Sasaki projection of 𝑦 onto 𝑥, 49 scaling, 3, 57 scaling function, 4, 5, 58, 58 scaling functions, 4, 58 scaling subspace, 58, 59 self-similar, 58 sequence, 14, 20 sequence analysis, 22 sequence processing, 22 set, 54 set difference, 10, 27 set difference operator, 54 set inclusion, 1, 54 space metric, 45, 46 space of square integrable functions, 57 spline, 5 structures 𝐿1 lattice, 42 𝐿2 lattice, 42 𝐿22 lattice, 42 𝐿32 lattice, 42 𝐿42 lattice, 42 𝐿52 lattice, 42 𝑀4 lattice, 41 𝑀6 lattice, 41 𝑂6 lattice, 41, 43, 44, 50, 52 𝑂8 lattice, 41 ℝ3 Euclidean space, 52 𝙇32 Boolean lattice, 54 3-dimensional Euclidean space, 50 a primorial lattice generated by 𝙇52 , 13 anti-chain, 31 antichain, 29, 29, 31 base set, 27 basis, 55–57 Boolean algebra, 39, 39, 40, 44, 45 Boolean lattice, 1, 9, 10, 12, 14, 15, 17–20, 31, 39, 39, 49, 52 boolean lattice, 9, 10 Boolean logic, 20, 24, 50, 51 Boolean probability space, 22 bounded lattice, 9, 10, 12, 34, 35, 38–40, 42–49 bounded lattice difference, 12 bounded lattices, 51 Cardinal series, 56 center, 43, 44, 53, 54, 54 chain, 27, 29, 31, 32, 34 classic logic, 51 closed ball, 14, 46


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closed unit ball, 46, 46 closure, 58 complemented lattice, 43 complements, 43 complex plane, 23 cosets, 7 cover relations, 28 D-poset, 12, 54, 54 de Morgan logic, 50, 51 diamond, 37 Dirac delta distribution, 56 discrete topology, 23 distributive lattice, 35 distributive pseudocomplemented lattice, 18 dual, 30 Euclidean space, 42 Fast Wavelet Transform, 24, 57, 63 field of probability, 17 Filter bank, 63 finite modular lattice, 18 fully ordered set, 27 function, 47, 48 Fuzzy logic, 21 fuzzy logic, 50, 51 fuzzy subset logic, 21 FWT, 63 g.l.b., 59 Gaussian Pyramid, 4, 58 genome, 25 Genomic Signal Processing, 25 greatest lower bound, 58, 59 GSP, 25 Hasse diagram, 4, 28, 28, 59 Hilbert space, 42 indiscrete topology, 23 intuitionalistic logic, 50, 51 Kleene logic, 50 lattice, 1, 7, 32–34, 36, 37, 39, 42–46, 51 lattice with negation, 17, 20, 22, 48, 50–54 least element, 7 linearly ordered set, 4, 27 logic, 1, 6, 22, 50, 51 lower bound, 34 M𝑛 lattice, 7 M3 lattice, 37 metric lattice, 14, 46 metric space, 46 modular lattice, 36 modular orthocomplemeted lattice, 45 Tuesday 14th February, 2017 9:12pm UTC Copyright © Daniel J. Greenhoe

Subject Index MRA, 1, 6, 58, 59 MRA space, 58 Multiresolution Analysis, 3, 5, 57 multiresolution analysis, 1, 4, 58, 59 multiresolution anaysis, 61 N5 sublattice, 10 N5 lattice, 8, 36, 36 O6 lattice, 18, 41, 41, 54 O6 sublattice, 10 open ball, 46 order relation, 27 order relations, 28 ordered set, 27, 27, 29, 32, 54 ordered set of partitions of an integer, 32 ordering relation, 12 ortho logic, 20, 50, 51 ortho logics, 20 Orthocomplemented lattice, 40 orthocomplemented lattice, 1, 6, 12, 22, 40, 42, 43, 49–54 orthocomplemented lattices, 41 orthocomplemented O6 lattice, 20 orthocomplemented pair, 12 orthocomplemented pairs, 12 orthomodular lattice, 18, 45 orthonormal wavelet system, 63 p-lattice, 7 partially ordered set, 27 partition, 7, 29, 31 pentagon, 36 poset, 27 power set, 32, 54 prime numbers, 7 primoral lattice, 6 primorial lattice, 1, 6, 7, 7, 12, 14, 15, 20–22 primorial lattice generated by 𝙇52 , 12 primorial lattice generated by 𝙇𝘕2 , 12, 20 primorial lattices, 7 primorial numbers, 7 probability, 1, 6, 22


Daniel J. Greenhoe probability space, 17, 19, 20 real valued lattice, 45 reduction of 𝙇22 , 9 reduction of 𝙇𝘕2 , 9 relation, 35, 36 Riesz basis, 58, 61 Riesz sequence, 59 scaling subspace, 59 sequence, 14, 20 set, 54 space of square integrable functions, 57 spline, 5 subposet, 28 Symbolic Sequence Processing, 25 totally ordered set, 27 translation operator, 55, 56 unit ball, 46 upper bound, 34, 54 wavelet analysis, 1, 61 wavelet system, 62 subadditive, 18, 45 subminimal negation, 47, 47 subposet, 28 subspace addition, 63 supremum, 32 symbol sequence analysis, 23 symbolic sequence analysis, 1, 6, 23 Symbolic Sequence Processing, 25 symbolic sequence processing, 1, 6, 23, 24 symbolic signal processing, 23, 24 symmetric, 45, 52, 53, 53 synthesis, 3, 57 theorems Birkhoff distributivity criterion, 10, 37 classic 10 Boolean properties, 39 Dilworth, 29 Dilworth's theorem, 29, 31 distributive inequalities, 33 Huntington properties, 39 Huntington's Fourth Set, 43 Huntington's fourth set,


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39 Minimax inequality, 33 minimax inequality, 33 Modular inequality, 33 Monotony laws, 33 totally ordered set, 27 traditional probability, 18, 18 transform, 3, 3, 57, 57 transitive, 27 translation invariant, 58, 59 translation operator, 55, 55, 56 translation operator inverse, 55 triangle inequality, 45 unbounded, 34 unique, 59 unique complement, 38 uniquely complemented, 38, 38, 39 unit ball, 46 upper bound, 31, 31, 34, 54 upper bounded, 18, 34, 34, 51 valuation, 14, 45, 45–47 values GLB, 32 LUB, 31 atom, 7 complement, 39 greatest lower bound, 32 height, 34 infimum, 32 least upper bound, 31, 34 lower bound, 32, 32 orthocomplement, 40, 40, 42 orthocomplemented pair, 9 orthocomplemented pairs, 40 upper bound, 31, 31 wavelet analysis, 1, 61, 61 wavelet coefficient sequence, 62 wavelet function, 61 wavelet system, 62, 62 wavelets, 56 weak double negation, 48, 52 width, 29, 29, 31 zero primorial projection, 14, 14

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License

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MRA-Wavelet subspace architecture for logic

MRA-Wavelet subspace architecture for logic

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