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