De Morgan Algebras - University of Illinois at Chicago

De Morgan Algebras - Completeness and Recursion

Louis H. Kauffman

University of Illinois at Chicago Circle interest because it is quite elementary, and it is a generalization of a corres-

An elementary proof is given of a completeness theorem for De Morgan Algebras.

ponding argument for Boolean algebras. De

The proof involves a construction that

Morgan algebras grow out of Boolean alge-

associates to a De Morgan algebra

bras.

new De Morgan algebra tion of

~

~.

B, a

In section 2 we define De Morgan Al-

The construc-

gebras and give a construction that asso-

bears a close analogy to the

construction of the complex numbers from

ciates to a De Morgan algebra

the real numbers.

De Morgan algebra

Similarly, De Morgan

B.

a new

The construction

algebras may be constructed from Boolean

of

algebras.

struction of the complex numbers from the

Relationships with recursion

~

B

bears a close analogy to the con-

real numbers.

and periodic sequences are discussed.

With the aid of this con-

struction, the completeness result (Theorem 2.5) is proved. how

~

In section 3 we show

is related to recursion in

B.

Section 4 outlines the construction of an algebra of periodic sequences, and delineates directions for further investigation. I.

Introduction 2.

A D_e Morsan algebra is an algebra that

The Completeness Theorem

satisfies most of the properties of a Boo-

It is possible to choose a very con-

lean algebra except for the law of the ex-

cise set of axioms for the algebras we

cluded middle, expressed as

shall study.

or

xx' = 0

tion.

x + x' = 1

in the usual Boolean nota-

These algebras have been studied by

various authors

(see [1],[2],[5]).

I shall give a longer llst,

and thereby avoid extremely detailed demonstrations.

The

purpose of this paper is to give an ele-

Definition 2.1.

mentary proof of a completeness theorem

is a set

for De Morgan algebras, and to indicate

tion

some interesting examples of these alge-

operations

bras. The completeness theorem that we prove

satisfy the following axioms:

B

algebra

(inversion), and two binary a,b ~ a+b, a,b ~ ab

that

The binary operations are each

may be deduced at once from deeper results about the structure of De Morgan al-

commutative and associative. (ii) (a')' = a, aa = a, a+a = a

gebras (see Remark 2.10).

all

Nevertheless, I

believe that the proof given here is of

82

B

together with a unary opera-

a ~ a'

(i)

A De Morgan

a E B.

for

(ill)

(a+b)' = a'b',

for all

(ab)' = a'+b'

(i)

a,b E B.

(iv) There exist elements O,1 E B such that aO = O, a+O = a, al = a, a+l = 1 (v)

for all

(ii) If

a E B.

xy

a(b+c) = (ab)+(ac), a+(bc) =

(a+b)(a+c)

for all

Let

a,b,c E B.

that is generated by the axioms

all

and

aa' = 0

algebra and let

~ = BxB.

= [(a,b)ia,b E Bj. B as follows:

B

Thus

Define operations

mapping such that

(a,b)+(c,d) :

for all

and ~ ' = ~

is a non-trivial ~

.

(0 ~ I)

Hence, if

V = {o,~,{,~]

(a,a) E B.

other words, the diagonal map

model

A:B ~ ~,

for

free De Morgan algebra o__ B

~ = B

is

V.

as Lemma 2.6.

Let

sion involving The

f(x) E E(S) x E S.

Then

where

be an expresf(x) = A,B,C,D

are

expressions that do not contain the element

S, denoted

B(S), is obtained as follows:

if and only if

In order to prove Theorem 2.5 we shall

a,b E B.

be any set.

Then

for every homomorphism

Ax + Bx' + Cxx t + D S

S.

need some preliminary lemmas.

a subalgebra of

a{ + b ~

be a

algebra if and only if it is true about the

In

algebra homomorphism that exhibits B ^ B. Every element of

be the

a consequence of the axioms for a De Morgan

A(a) = (a,a), is a De Morgan

Let

B(S)

In other words, an equality

A

with

s E S.

:B(S)

, ~+~.~..

a(b,c) = (ab,ac), then we may

Definition 2~3.

Let

a,B E E(S), ~ = B

¢(s) = ¢(B)

a,b,c E B, we adopt the con-

is of the form

for

four element De Morgan algebra described

for

Note that

defined by

¢(s)

free De Morgan algebra on a set

is a small De Morgan algebra.

a E B

Let

after Definition 2.2.

is the smallest non-

and ~ = O

B(S), ¢:B(S) - C, is determined

Theorem 2. 5 .

Boolean al-

V = [o,~,t.~]

identify

¢(x') = ¢(x)'

pleteness theorem.

trivial Boolean algebra, then

vention

and

We are now prepared to state the com-

is a De Morgan algebra

V = [O,I}

If, for

is a set-

~(xy) = ¢(x)¢(y),

uniquely by its values

it is easy to see

becomes a De Morgan algebra.

,~=~

C

x,y E B.

algebra

that is not Boolean.

~

and

¢:B - C

As usual, a homomorphism from a free

Note that Z I = Z

If

B

¢(x+y) = ¢(x) + ¢(y)

With these definitions,

gebra, then

A homomorphism

of De Morgan algebras

(a,b)' : (b',a'), O = (0,0), i = (l,1), @ = (o,l) for a,b,c,d E B.

B

(ii).

Definition 2.4.

(a+c,b+d)

B

It is easy to verify that

tions inherited from the formal operations

be a De Morgan

(a,b)(c,d) = (ac,bd),

that

.

Let

is a De Morgan algebra with opera-

of rule Let

in

for

a E B.

Definition 2.2.

of Definition 2.1.

B(S) = E(S)/= B(S)

a+a' = 1

x', x+y, and

E(S).

denote the equivalence relation on

(i) - (v)

that satisfies the axioms above plus

and

x,y E E(S), then

belong to

=

E(S)

A Boolean algebra is a De Morgan algebra

(vi)

Primitive expressions are in

E(S). That is, O,1 E E(S) s c E(S).

X.

The primi-

The proof of this lemma proceeds Just

tive expressions in B(S) are O,1 and elements s E S. Let E(S) denote the

as in ordinary Boolean algebra, and will therefore be omitted.

set of expressions defined by the followins rules:

83

Lemma 2.7. in E(S):

The following

(Ax + Bx' + Cxx')'

equivalence

holds

Proof of T h e o r e m 2. 5 . Let m,6 E E(S). By Lemma 2.6 we may assume that ~ = Ax

= A'x + B'x' + xx' +

+ Bx' + Cxx' + D

and that where

A'B'C'

.

+ ~x' + ~xx' + ~

Proof:

(Ax + Bx' + Cxx') a

A,B,~,D tain

= (Ax)' (Bx ' (Cxx')' + x'')(C'

=

+ x)(C'

+ (xx')')

ables

+ X' + x)

= (A'B' + A'X + B'X' + X'x)(C'

and

= A'B'C' + (A'C' + A'B' + A')X

to

+ (B'C' + A'B' + B')X'

I.

~

of vari-

that occur in the

If

N=O,

then

D

and

D

Since N=0.

contains

a = D

are equal

0 ~ 1

in a free

the theorem is trivial If

N > O, then either

a variable

x

and we

indicated

Lemma 2.8) that the following

f(x)

as a function

hold in

f(O),

f(~), f(2),

(,)

above. (using

equivalences

E(S):

+ D_

This can all be made more precise

in terms of homomorphisms,

N

Thus we may assume by induction

the proof of the lemma.

x, and evaluating

S)

may use the equivalence

For the next lemma, we shall proceed

f(~).

or

or

= A'B'C' + A'X + B'X' + XX'

of

of

where

for the case

+ B'(C' + A' + 1)x' + xx'

regarding

~ = ~ 0

= A'B'C' + A'(C' + B' + 1)x

informally,

(elements

De M o r g a n algebra,

+ (A' + B' + C' + 1 + 1)xx'

This completes

The proof will proceed by in-

two expressions.

+ x' + x)

and

that do not con-

duction on the total number

= (A' + x')(B' (A' + X ' ) ( B '

A,B,C,D

are expressions

x.

~ = Ax +

A +

=D

but only at

A+

B+

C +D

= A+~+

~+D

some loss in clarity. We now use Lemma 2.8.

Let

be an element Lemma 2.6.

f(x) = Ax + Bx' + Cxx' + D

of

E(S)

as described

(*)

to show that

~ = ~

E(S).

in

= Ax + Bx' + Cxx' + D

Then

= (A'x + B'x' + xx' + A'B'C')'

f(O) = B + D

: = D

(A'x)'(B'x')'(xx')'(A'B'C')'

= ((A'X)' + D)((B'x')'

f(~) + f(~) = A + B + C + D.

+ D,

(Z.7)

by

f(~) = A + D +i

in

((A'B'C')'

+ D

+ D)((XX')'

+ D)

+ D)

A

Here equality means as functions

on

Proof:

and

Certainly

f(0) = B + D

+O)(A+B+C+D) Now make the substitutions

f(1) = A + O. Now f(Z) = (A + B + C)Z + D ( s i n c e ~ t = Z ) , f(~) : (A + B + C ) ~ + f(Z) + ~

D.

= ((A + D) + X')((B + D) + X)((XX')'

V.

(*), reverse steps,

and

= 6.

Hence

(f(~) + Z ) ( f ( ~ )

This completes

that

the induction step

Note that T h e o r e m 2.5 has the following corollary:

Therefore +&)

= (Z+ D ) ( ~ + = ~+2D = 0 +

+~D (~+~)D

D)

Corollary + D

+ D

=D+D=D. Finally,

and conclude

and the proof of the theorem.

= (A + B + C + I)~ + D = i + D,

and f(~) +~,= ~.~ + D.

indicated by

2.9.

Let

B(S)

be a free De

Morgan algebra.

Then

to a sub-algebra

of a product

B(S)

the four element algebra

is isomorphic of copies

V = [o~,{,~

of J.

f(i) + f(~) = (A + B + C)(#+~D+ D Proof:

=A+B+C+D. This completes

Let

~ = (¢:B(S) - ~I ¢

homomorphism~.

the proof of the lemma.

$4

Let

V¢

is a

be a copy of

V

indexed by an element of

T(x) : aT(x) + bT2(x),, T2(x) : aT2(x)

~, and let

A

=

U V¢.

Define

F : B(S) - W

F(e) =

U ¢(a)

by 2.5

F(e) = F(0)

Hence of

F

where

injects

+ bT(x) '. Let X = (T(x), T2(x)). Then ~ ( X ) = aX + bX' = a(T(x), T2(x)) + b(T2(x)',

by

¢ : B(S) - V¢. Then if and only if

B(S)

T(x)')

c~-6. = (aT(x) + bT2(x)', aT2(x) + bT(x)')

as a subalgebra

= (T(x), T2(x))

~.

Remark 2.!0.

---- X

In fact, Corollary 2.9 is

•

Thus the algebraic structure of

true for arbitrary De Morgan algebras.

B

re-

[5].

flects the properties of period two sequences recurslvely generated from B. The

~.

Morgan algebra of sequences.

This deeper result may be found in [2] or

next section describes a more general De Recursion and Fixed Points The A

construction leading from a

Boolean algebra

B

~.

to its corresponding A

De Morgan algebra

B

Sequence Models Corresponding to a De Morgan algebra

is closely related

to the structure of recursion in the given

let

Boolean algebra.

of elements of

Let form

T : B - B

period

be a mapping of the

T(x) = ax + bx'

where

a,b E B.

T

has period two; in fact

T2(x) = T(T(x))

8(B)

= (a + b)x + abx'

B

denote the set of all sequences B

with an assigned even

(possibly of period

0).

8(B)

consists of sequences

that

n

That is

b = [bnJ such

ranges over the integers and

bn+ p = b n

for all

n, where

p = p(b)

is

an even non-negative integer associated

T3(x) = ax + bx' = T(x)

with the sequence. o,.

Tn+2(x )

=

8(B) has the structure of an algebra as follows:

Tn(x).

Hence there may be no element that

T(X) = X.

However, B

such fixed points. T(x) = x', then {

such

(1)

does contain and

For example, if

x = x'

and &

(ab) n = (anbn),

Ca + b) n = a n + b n

PCab) = P(a + b) =

Icm(p(a), p(b))

if

has no solutions

in the Boolean algebra fied by

X E B

B, but is satis-

and

gebra, and let

Let

B

T : B - B

be a Boolean al-

Here the symbol

be the mapping

Let

~ : B - B

be the

corresponding mapping on

B.

exist elements

such that

X

of

B

lcm

otherwise,

denotes least

common multiple.

A

described above.

p(b) + O,

in 0

proposition 3.I.

p(a) ~ o

(ii) (a')n = (an_k)' where

Then there

k = p(a)/2,

p(a') = p(a).

~(X) = X. In particular, we may take X = (T(x), T2(x)) or X = (T2(x), T(x))

Note that in the sequence algebra, inver-

for any

plus a half-period shift.

sion is obtained by ordinary inversion

x E B.

of period two sequences in Proof:

The following identities in

B

morphic to

are easily verified:

B.

The sub-algebra 8(B)

is iso-

As it stands, the sequence

algebra is not a De Morgan algebra.

8S

forms and interference phenomena. This temporal, musical aspect is precisely what is prohibited by the stark all or

Axioms (1), (ll), (lii), and (v) are satisfied but there is no available choice for 0

or

1.

If

8 (B)p

denotes the sub-

algebra of sequences of period we may take

0p

and

lp

p

then

nothing of two-valued logic. these restrictions,

to be the cons-

fuzziness and ambiguity, but the precise

tant sequences of zeroes and ones respectively, with assigned period p. This gives 8(B)p

emergence of patterned forms, spatial and temporal.

the structure of a De Morgan alge-

bra for each

p.

We could force

8(B)

When we drop

the result is not

inBibliography

to the mold by taking a quotient construction, but I believe it is more interesting to leave it as it stands.

Our rules

[1]

Bialynicki-Birula, A. and Raslowa, H., On the representation of quasiBoolean algebras. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. (1957), 259-261.

[2]

Balbes, R. and Dwinger, P., Distributive Lattices, Univ. of Missouri Press (1974).

[3]

Berman, J. and Dwinger, P., De Morgan algebras - free products and free algebras, (unpublished).

[4]

Brown, G. Spencer, Laws of Form, The Julian Press, Inc., New York (1972).

[5]

Kalman, J. A., Lattices with involution, Trans. Amer. Math. Soc. 87

for combining sequences of different periods provide a simple model of interference phenomena that bears investigation on its own grounds. We may regard

8 (B)

as a set of peri-

odic oscillations. In this regard it is interesting to compare our ideas with the suggestions

of G. Spencer Brown in his

book Laws of Form ([4]).

Spencer Brown

suggests that an element

X

satisfying

X' = X might be seen as an oscillation (if its O, then its l, then its O, then its

l, ...).

(1958), 485-#91.

Taking this literally, we

might form a diagram:

-n vq F

[6]

Kauffman, L. H., Network synthesis and Varela's calculus, (to appear in Int. J. Gen. Syst.).

[7]

Varela, F. J., A calculus for self-reference, Int. J. Gen. Syst. 2 (1975), 5-2#.

[8]

Varela, F. J., The arithmetic of closure, Pro6ress i__n Cybernetics and Systems Research, Vol. 3, edit. by R. Treppl, G. Klir and L. Ricclardi, Hemisphere Publications (Wiley) N.Y. (1977).

-LY-L_

In each case, the spatial sense in which

i

and

~t = ~

involves ordinary in-

version plus a left-right shift. and ~

synchronized

With Z"

(and note that here

the synchronization has become the spatial relationship of the two sequences) as This is the motiabove, w e have ~ = O . vation for our construction of also for the construction

B ~

8 (B), and of sec-

tion 2. There are many connections between our discussion and Brown's work. These will be explored in another paper. I would like to remark here that it is striking that once one lifts the law of the excluded middle from Boolean algebra, there is opened up the possibility of infinite models involving simple analogs of wave-

86