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From: everest@netcom.com (Wlodzimierz Holsztynski)
Subject: tautologies in Kleene algebras (fuzzy logic and set theory)
Message-ID: <everestDn3939.3rB@netcom.com>
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Date: Tue, 20 Feb 1996 19:06:44 GMT
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It is well known that for arbitrary boolean algebra B
strings of the form  f=g  which are
tautologies in B  are the same as tautologies in the 2-element boolean
algebra {0,1}  (it coincides with Lukasiewicz algebra L_2).
This means that  2^n  substitutions are sufficient to verify
a tautology in any (even infinite) boolean algebra.  All this
is a classical result.  I'd like to ask if the similar results were
*published* for fuzzy logic and set theory, i.e. for Kleene algebras.
I have discovered and proved them recently but they have such a "classical"
feel that I expect them to be known.  If this is the case
I would appreciate specific references (rather than intuitive opinions).

Here are the main results.

Definition:

A = (K, \cup, \cap, ~,0,1) is a Kleene algebra
if  K is a set, \cup and \cap are commutative and associative
binary operations such that

	a \cap (\a \cup b) = b  and
	a \cup (\a \cap b) = a  for every  a b \in A,

and the distributivity of \cap w.r. \cup holds (then the other
distributivity law holds too);  and  ~  is a unary operation which
such that ~~a = a for every  a \in K,  and ~ is a dual isomorphism
between (K, \cap) and (K, \cup), i.e. De Morgan law:

	~(a \cap b) = ~a \cup ~b  (then the other law holds too),

and finally, the Kleene axiom should hold:

	(a \cap ~a) \cup (b \cup ~b) =  b \cup ~b  for every  a b \in K

(i.e.  a \cap ~a \le b \cup ~b  or what is still equivalent


	(a \cup ~a) \cap (b \cap ~b) =  b \cap ~b  for every  a b \in K).

	
We see that Kleene algebras have the same signature (operation symbols)
as boolean algebras, they form a larger class than boolean algebras.

NOTATION:  [0;1] := { x : 0 \le x \le 1}  is the unit interval of reals.

EXAMPLE:  ([0;1], max, min, \, 0, 1)  where  \a := 1-a  for every
a \in [0;1], is a Kleene algebra.  It contains Lukasiewicz algebras
as subalgebras,  L_n := (T_n, max, min, \, 0, 1),  where

	T_{n+1} := { 0/n, ..., n/n }

is the (n+1)-element set of fractions k/n  (k=0,...,n).

THEOREM 1.  Every Kleene algebra A admits an injective homomorphism
into  L_3^V  for a certain set  V  (the cardinality of V depends on A).

THEOREM 2.  Every Kleene algebra which is not boolean admits
a surjective homomorphism onto  L_3.

THEOREM 3.  Let  A  be a Kleene algebra.  If  A is boolean than
the set of tautologies  f=g  in  A is the same as for  L_2,
and if  A  is not boolean that its set of tautologies is the same
as for  L_3  (the former set of tautologies, for L_2, contains
the later one, that for  L_3).

In the above theorem  f  and  g  stand for arbitrary strings which
are what is commonly called boolean expressions which produce values
from a given Kleene algebra A after replacing
variables with the elements of the given algebra A.
And  f=g  is called a tautology in  A  if  all possible substitutions
of variables by elements of A in the string  f=g  produce an equality
in A each time.

Thank you,

	Wlodzimierz Holsztynski
	  everest@netcom.com

