Some Algebraic Aspects of Fuzzy Sets and Fuzzy Logic!

Some Algebraic Aspects of Fuzzy Sets and Fuzzy Logic Mai Gehrke, Carol Walker, Elbert Walker New Mexico State University Las Cruces, New Mexico 88003, USA [email protected], [email protected], [email protected]

1

Introduction

This paper is an informal discussion of some of the algebraic systems that arise in the beginnings of fuzzy set theory and logic. Our main concern is with the unit interval endowed with its natural order structure, a t-norm, and a negation. A fundamental problem is to determine when two such algebraic systems are isomorphic. Principal references for our work are the papers [8, 9, 10, 11, 12], and the proofs of the results stated here may be found in these papers..

2

De Morgan systems on the unit interval

A fuzzy subset A of a set S is a mapping A : S ! [0; 1]. Operations on the set F(S) of all such fuzzy subsets of S come from operations on [0; 1]. The usual ones, introduced by Zadeh [18] are ^, _, and 0 given by (A ^ B)(s) = minfA(s); B(s)g (A _ B)(s) = maxfA(s); B(s)g A0 (s) = 1 A(s) What we have here is an algebra, the set F(S) with the operations ^, _, and 0 on it. The …rst two operations are binary— that is, functions of two variables, and the last is unary, a function of one variable. For technical reasons, we want two more This paper is a condensed version of an invited address given at the International Conference on Intelligent Technologies (InTech’2000), hosted by the Faculty of Science and Technology, Assumption University, Bangkok, Thailand. We thank the organizers and the hosts for an excellent meeting and for the opportunity to make a presentation.

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operations on F(S), which boil down to specifying the two special elements 0 and 1 of of F(S) given by 0(s) = 0 and 1(s) = 1 for all s in S. These are nullary operations, or constants. Now our algebra is the set F(S) together with the operations ^, _, 0 , 0, and 1 on it, and is denoted (F(S), ^, _, 0 , 0, 1). In general, an algebra is a set together with a family of operations on it. Our algebra, which we denote simply by F(S), is the algebra of all fuzzy subsets of S. The …rst order of business in fuzzy set theory is to determine its basic properties. The following hold for fuzzy subsets A, B, and C of a set S. 1. A ^ A = A, A _ A = A (^ and _ are idempotent.) 2. A ^ B = B ^ A, A _ B = B _ A (^ and _ are commutative.) 3. A^(B ^C) = (A^B)^C, A_(B _C) = (A_B)_C (^ and _ are associative.) 4. A ^ (A _ B) = A, A _ (A ^ B) = A (The absorption laws hold.) 5. A ^ (B _ C) = (A ^ B) _ (A ^ C), A _ (B ^ C) = (A _ B) ^ (A _ C) (^ and _ distribute over each other.) 6. 1 ^ A = A, 0 _ A = A (1 and 0 are identities for ^ and _, respectively.) 7. (A ^ B)0 = A0 _ B 0 , (A _ B)0 = A0 ^ B 0 (The De Morgan laws hold.) 8. 00 = 1, 10 = 0 9. A00 = A (0 is an involution.) 10. (A ^ A0 ) ^ (B _ B 0 ) = (A ^ A0 ) (The Kleene equation holds.) The …rst 4 properties say that F(S) is a lattice, the …rst 5 that it is a distributive lattice, property 6 says that the lattice is bounded, the …rst 9 say that it is a De Morgan algebra, and all 10 say that it is a Kleene algebra. With this terminology, we could say that a De Morgan algebra is a bounded distributive lattice with an involution that satis…es De Morgan’s laws, and that a Kleene algebra is a De Morgan algebra that satis…es the Kleene equation. If instead of property 10 we have (A ^ A0 ) = 0 and (B _ B 0 ) = 1, then we have a Boolean algebra But F(S) is not a Boolean algebra.. De…ning A B if and only if A ^ B = A gives us a partial order in which any two elements X and Y have a least upper bound and greatest lower bound, which in this case are X _ Y and X ^ Y respectively. Conversely, for a partial order on any set in which any two elements X and Y have a least upper bound X _ Y and greatest lower bound X ^ Y , the operations ^ and _ satisfy the …rst 4 properties. Whenever we have a lattice, it is often convenient to express properties in terms of its partial order rather than strictly in terms of the operations ^ and _. For example, instead of writing X ^ Y = X, we could write X Y . 2

The operations above on F(S) come from operations on the interval [0; 1]. With the operations min and max and the constants 0 and 1, [0; 1] becomes a bounded lattice. This algebra ([0; 1]; ^; _; 0; 1) we denote simply by I. It is this algebra, together with other operations put on it, that forms the basis of much of the theory of fuzzy sets. First we examine brie‡y the automorphisms of the algebra I. De…nition 1 An automorphism of I is a one-to-one mapping f of I onto I such that 1. f (a ^ b) = f (a) ^ f (b) 2. f (a _ b) = f (a) _ f (b) 3. f (0) = 0, f (1) = 1 This is expressed by saying that f is a one-to-one and onto mapping that preserves the operations. Applying operations and then taking function values gets the same result as taking function values and then applying the operations. Any continuous strictly increasing map connecting (0; 0) and (1; 1) in the plane is an automorphism of I. These automorphisms play a signi…cant role the theory. Let Aut(I) be the set of all automorphisms of I. Its elements are functions, and may be composed. That is, if f and g are in Aut(I), f g is the element of Aut(I) given by (f g)(x) = f (g(x)). With this operation, Aut(I) is a group— that is, composition of functions is a binary operation on Aut(I) that is associative, has an identity, and every element has an inverse. This means that f (gh) = (f g)h. There is an element id in Aut(I) such that id f = f id = f for all f . (The function id is the function given by id(x) = x for all x. It is called the identity of the group.) For each f 2 Aut(I), there is an element f 1 2 Aut(I) such that f f 1 = f 1 f = id. (The element f 1 is simply the inverse of f as a function on [0; 1].) A subset S of a group G is a subgroup of G if the restriction of the operation to S makes S into a group. There is a particularly important subgroup of Aut(I). First note that the set R+ of positive real numbers is a group under the operation of multiplication. Then note that each positive real number r gives an automorphism by r(x) = xr . Identifying r with this automorphism, the set R+ of positive real numbers is a subgroup of Aut(I).

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3

t-norms

We will put additional operations on the algebra I, and …rst we consider t-norms. They are generalizations of intersection and are one of the fundamental objects of interest in fuzzy set theory and logic. De…nition 2 A t-norm is a binary operation [0; 1],

on [0; 1] such that for all x; y; z 2

1. 1 x = x 2. x y = y x 3. (x y) z = x (y z) 4. (x ^ y) z = (x z) ^ (x z) and (x _ y) z = (x z) _ (x z) Thus a binary associative, and requirement that Here are a few

operation on [0; 1] is a t-norm if 1 is an identity, it is commutative, distributes over ^ and _. Actually, item 4 is equivalent to the is increasing in each variable. examples of t-norms.

minimum: x y =x^y multiplication: x y=x y ×ukasiewicz t-norm:

x N y = (x + y

1) _ 0

Hamacher t-norms: x 4H y =

xy a) (x + y

a + (1

xy)

for a

0

Frank t-norms: x 4F y = loga 1 +

(ax

1) (ay a 1

1)

for a > 0, a 6= 1

Yager t-norms: x 4Y y = 1

((1

x)a + (1

4

1

y)a ) a

_ 0 for a

1

Aczél-Alsina t-norms: x 4R y = e

((

ln x)a +(

1

ln y)a ) a

for a > 0

generalized ×ukasiewicz t-norms: x Na y = ((xa + y a

1

1) _ 0) a for a 6= 0

Let be a t-norm and consider the system (I; ). This algebra is simply I with an additional structure on it, namely the operation . Let be another t-norm on I. We need to make precise the notion of the systems (I; ) and (I; ) being structurally the same. One of our primary concerns is with such questions. De…nition 3 Let and be t-norms. The algebras (I; ) and (I; ) are isomorphic if there is an element h 2 Aut(I) such that h(x y) = h(x) h(y). We write (I; ) t (I; ). The mapping h is an isomorphism. The t-norms and are isomorphic if (I; ) t (I; ). An isomorphism of any algebra with itself is an automorphism of that algebra. Isomorphism of t-norms is an equivalence relation and partitions t-norms into equivalence classes. The t-norm min is rather special. A t-norm is idempotent if a a = a for all a 2 [0; 1]. If is idempotent, then for a b, a = a a a b a 1 = a, so = min. Thus min is the only idempotent t-norm. It is in an equivalence class all by itself. There are two other fundamental kinds of t-norms. De…nition 4 A t-norm is convex if whenever x y c x1 y1 , then there is an r between x and x1 and an s between y and y1 such that c = r s. A t-norm is Archimedean if for each a; b 2 (0; 1), there is a positive integer n such that n times }| { z an = a a a < b (n depends on a and b).

For t-norms, the condition of convexity is equivalent to continuity in the usual topology on the unit interval. We refer to the condition as convex. This formulation has the advantage of being strictly order theoretic, allowing us to remain within the algebraic context of I as a lattice. De…nition 5 A convex, Archimedean t-norm is nilpotent if for a 6= 1, an = 0 for some positive integer n, the n depending on a. So convex Archimedean t-norms fall naturally into two classes: nilpotent ones and those not nilpotent. Those not nilpotent are called strict. The fundamental fact about strict ones is the following.

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Theorem 6 A t-norm is strict if and only if it is isomorphic to multiplication. Thus a t-norm is strict if and only if there is an element f 2 Aut(I) such that f (x y) = f (x)f (y). Another element g 2 Aut(I) satis…es this condition if and only if g = rf for some r > 0. Corollary 7 Aut(I; ) = R+ . The isomorphism in the theorem is also called a generator of the t-norm. It is called a generator because such an automorphism f of I gives a t-norm via x y = f 1 f (x)f (y). The corollary follows immediately. The identity function id on [0; 1] is an automorphism of (I; ), and if g is another automorphism of (I; ), then g = r(id) = r. On the other hand, elements of R+ are automorphisms of (I; ). Corollary 8 For any strict t-norm , Aut(I; ) t Aut(I; ). Corollary 9 For any two strict t-norms

and , Aut(I; ) t Aut(I; ).

Among the examples of t-norms listed earlier, the strict t-norms are multiplication, the Hamacher t-norms, the Frank t-norms, the Aczél-Alsina t-norms, and the generalized ×ukasiewicz t-norms with a < 0. Call two automorphisms f and g equivalent if they give the same strict t-norm, and write f s g. Then s is an equivalence relation on Aut(I) and so induces a partition of the group Aut(I). The members of this partition are the right cosets fR+ f : f 2 Aut(I) of R+ . So the set of strict t-norms of I is in natural one-to-one correspondence with the right cosets in Aut(I) of the subgroup R+ . Rephrasing, we have Corollary 10 For an automorphism f of I, let Then + f ! R f

f

be given by x f y = f

1

(f (x) f (y)).

is a one-to-one correspondence between the strict t-norms on [0; 1] and the right cosets of the subgroup R+ in Aut(I). Now we turn to nilpotent t-norms. There are two basic facts about them: any two are isomorphic, and each has a trivial automorphism group. Theorem 11 An Archimedean t-norm is nilpotent if and only if there is an element f 2 Aut(I) such that f (x y) = (f (x) + f (y) 1)_0. The automorphism f is unique.

Thus, if t-norm

is a nilpotent t-norm, then (I; ) x N y = (x + y

(I; N) where N is the ×ukasiewicz 1) _ 0

The isomorphism f in the theorem is called an L-generator of the nilpotent t-norm. So there is a nilpotent t-norm for every automorphism of I. 6

Corollary 12 If

is a nilpotent t-norm, then Aut(I; ) = fidg.

Among the examples of t-norms listed earlier, the nilpotent t-norms are the ×ukasiewicz t-norm, the generalized ×ukasiewicz t-norms for a > 0, and the Yager t-norms.

4

Negations

An antiautomorphism of the bounded lattice I is a one-to-one mapping of [0; 1] onto itself that reverses order. That is, (x ^ y) = (x) _ (y) and (x _ y) = (x) ^ _ (y). This is equivalent to the condition that (x) (y) whenever x y. The set of all automorphisms and antiautomorphisms of I is a group denoted M ap(I). De…nition 13 A negation (or involution) on I is an antiautomorphism that for x 2 [0; 1], ( (x)) = x.

such

Thus a negation is an order-reversing, one-to-one mapping of I onto I such that = id. We reserve the notation for the negation such that (x) = 1 x. It is trivial that conjugates of negations are negations, that is, that for f 2 Aut(I) and a negation, then f 1 f is a negation. It is a little less trivial and a bit surprising that every negation is a conjugate of by an automorphism. 2

Theorem 14 Let

be a negation and de…ne f by (x) + x 2

f (x) = Then f 2 Aut(I), and

=f

1

f.

The automorphism f is called a generator of . Consider the two algebras (I; ) and (I; ) where and are negations. They are isomorphic if there is a map h 2 Aut(I) with h( (x)) = h(x), that is if h = h, or equivalently if = h 1 h. Theorem 15 For any negation , (I; ) is isomorphic to (I; ). The elements f in Aut(I) such that f = f form a subgroup Z( ) of Aut(I) called the centralizer of . The group Z( ) plays a role for negations analogous to that of R+ for strict t-norms. The following are easy consequences. Corollary 16 The set of isomorphisms from (I; ) to (I; ) is the right coset Z( )f of Z( ). In particular, the generator f of is an isomorphism from (I; ) to (I; ). Noting that f

1

Z( )f = Z( ), we have 7

Corollary 17 Aut(I; ) = f Since z ! f Aut(I; ), we get

1

1

Z( )f = Z( ). In particular, Aut(I; ) = Z( ).

zf is an isomorphism from Z( ) = Aut(I; ) to f

Corollary 18 For any two negations

1

Z( )f =

and , Aut(I; ) t Aut(I; ).

The upshot of all this is that furnishing I with any negation yields an algebra isomorphic to that gotten by furnishing I with the negation : x ! 1 x.

5

De Morgan systems

Let be a t-norm and a negation. Then de…ned by x y = ( (x) (y)) de…nes a binary operation on [0; 1] called a t-conorm. It has the following characterizing properties. 0 x = x. x y = y x. (x y) z = x (y z). is increasing in each variable. If a t-norm and t-conorm are related in this way by the negation , then (I; ; ; ) is a De Morgan system, and the t-norm and the t-conorm are said to be dual to one another via the negation . ((I; ; ; ) is not a De Morgan algebra; the operations and are not idempotent.) Now suppose that q : (I; ; ; ) ! (I; 4; ; 5) is an isomorphism. This means that q 2 Aut(I) and the following hold. q(x y) = q(x) 4 q(y) q( (x)) = (q(x)) q(x y) = q(x) 5 q(y) But since x y = ( (x) (y)) and x 5 y = ( (x) 4 (y)), if the …rst two equations hold, then so does the third. Therefore to be an isomorphism, q need only be required to satisfy the …rst two conditions. That is, isomorphisms from (I; ; ; ) to (I; 4; ; 5) are the same as isomorphisms from (I; ; ) to (I; 4; ). We will also call these systems De Morgan systems. For notational reasons, we are going to adorn our operators with their generators. Thus a strict t-norm will be written f , meaning that f is a generator of . Ordinary 8

multiplication is r for any r 2 R+ , but that will be denoted as usual by . Finally, f denotes the negation with generator f . We write 1 simply as . So a De Morgan system looks like (I; f ; g ). To determine the isomorphisms q from (I; f ; g ) to (I; u ; v ), we just note that such a q must be an isomorphism from (I; f ) to (I; u ) and from (I; g ) to (I; v ). Theorem 19 The set of isomorphisms from (I;

f;

g)

to (I;

u;

v)

is the set

u 1 R+ f \ v 1 Z( )g This intersection may be empty, of course. That is the case when the equation u rf = v 1 zg has no solution for r > 0 and z 2 Z( ). A particular example of this is the case where f = g = u = 1, v 2 = Z( ), and v 21 = 12 . Then r = v 1 z with r > 0 and z 2 Z( ). But then 1

r

1 2

1

=v z

1 2

=

1 2

r

=

1 2

Thus r = 1, and so v = z. But v 2 = Z( ). So there are De Morgan systems (I; f ; g ) and (I; u ; v ) which are not isomorphic. When two De Morgan systems are isomorphic, the isomorphism is unique and the situation is this. Theorem 20 (I; f ; g ) t (I; u ; v ) if and only if (I; u ; v ) = (I; f h ; gh ) for some h 2 Aut(I), in which case h 1 is the only such isomorphism. In particular, (I; f ; g ) t (I; ; gf 1 ). One implication of this theorem, taking f = g, is that the theory of the De Morgan system (I; f ; f ) is the same as that of (I; ; ). More generally this holds for (I; f ; g ) and (I; ; gf 1 ). This suggests that in applications of De Morgan systems, one may as well take the strict t-norm to be ordinary multiplication. Corollary 21 Aut((I;

f;

g ))

= fidg.

Corollary 22 (I; ; ) t (I; ; ) if and only if

=r

1

r for some r 2 R+ .

Among the negations = r 1 r for r 2 R+ there is exactly one with a given …xed point u 2 (0; 1), so for example, there is a one-to-one correspondence between isomorphism classes of De Morgan systems with strict t-norms and De Morgan systems of the form (I; ; ) where is multiplication and has …xed point 21 . Taking = in this last Corollary, we see that (I; ; ) t (I; ; ) if and only if = r 1 r for some r 2 R+ . So De Morgan systems isomorphic to (I; ; ) are exactly those of the form (I; ; r ) with r 2 R+ . Negations of the form r 1 r are Yager negations [17]. Thus we can state

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Corollary 23 De Morgan systems (I; ; ) which are isomorphic to (I; ; ) are precisely those with a Yager negation. The system ([0; 1]; ^; _;0 ), where ^, _, and 0 are max, min, and x0 = 1 x forms a De Morgan algebra in the usual lattice theoretic sense. If we replace 0 , which we have been denoting by , by any other involution , then the systems ([0; 1]; ^; _;0 ) and ([0; 1]; ^; _; ) are isomorphic. Isomorphisms between these algebras are exactly the isomorphisms between (I; ) and (I; ). This suggests that in applications of De Morgan systems where ^ and _ are taken for the t-norm and t-conorm, respectively, the negation may as well be (x) = 1 x. Now we consider nilpotent t-norms. Their duals are called nilpotent t-conorms. We will take for generators the isomorphism of the nilpotent t-norm with the ×ukasiewicz t-norm. Suppose that f is a generator of the nilpotent t-norm , is a negation, and x y = ( (x) (y)) 1 = f (((f (x)) + (f (y)) 1) _ 0) = (f ) 1 (((f (x)) + (f (y)) 1) _ 0) Thus the antiautomorphism f and the ×ukasiewicz t-norm determine the nilpotent t-conorm . The system (I; ; ; ) is a nilpotent De Morgan system. As with strict t-norms, we will adorn our operators with their generators. Thus a nilpotent t-norm will be written Nf , meaning that f is a generator of . Finally, f denotes the negation with generator f . We write Nid and id simply as N and . So a nilpotent De Morgan system looks like (I; Nf ; g ). To determine the isomorphisms q from (I; Nf ; g ) to (I; Nu ; v ), we just note that such a q must be an isomorphism from (I; Nf ) to (I; Nu ) and from (I; g ) to (I; v ). We get the following theorem. Theorem 24 The set of isomorphisms from (I; Nf ;

g)

to (I; Nu ;

v)

is the set

u 1 f \ v 1 Z( )g This intersection may be empty. That is the case when the equation u 1 f = v 1 zg has no solution for z 2 Z( ). A particular example of this is the case where f = g = u = id and v 2 = Z( ). Then id = v 1 z with z 2 Z( ). So v = z. But v2 = Z( ). So there are nilpotent De Morgan systems (I; Nf ; g ) and (I; Nu ; v ) which are not isomorphic. When two De Morgan systems are isomorphic, the isomorphism is unique and the situation is this. Theorem 25 (I; Nf ; g ) t (I; Nu ; some h 2 Aut(I), in which case h (I; Nf ; g ) t (I; N; gf 1 ).

if and only if (I; Nu ; v ) = (I; Nf h ; gh ) for is the only such isomorphism. In particular,

v) 1

10

One implication of this theorem, taking f = g, is that the theory of the De Morgan system (I; Nf ; f ) is the same as that of (I; N; ). More generally this holds for (I; Nf ; g ) and (I; N; gf 1 ). This suggests that in applications of De Morgan systems, one may as well take the nilpotent t-norm to be the ×ukasiewicz t-norm. Corollary 26 Aut((I; Nf ;

g ))

= fidg.

Corollary 27 (I; N; ) t (I; N; ) if and only if

= .

Thus there is a one-to-one correspondence between isomorphism classes of De Morgan systems with nilpotent t-norms and De Morgan systems of the form (I; N; ) where N is the ×ukasiewicz t-norm. In contrast to the strict t-norm case, the negation connecting a given t-norm and t-conorm is unique.

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