Classes of fuzzy sets with the same material conditional

By the representation's theorem it is known that a fuzzy relation R is a preorder if and only if R = inf~ e so If, where -~ is the set of logical states of I~ In this paper it is shown how, in almost all cases, both for the t-norm Min and archimedean t-norms T, it is possible to consider that the fuzzy sets I~ are normalized. Furthermore a partial order between logical states, defined from the elemental preorders If, is studied. ©1997 Elsevier Science Inc. K E Y W O R D S : fuzzy re/at/ons, f u z ~ mater/a/condit ionals or elemental preorders, fuzzy logical states. 1. I N T R O D U C T I O N In [4], a nonuniversal but systematic me thod for obtaining the fuzzy extension of a vague predicate on a set X was given. The te rm "vague Address correspondence to A. 1~ de Soto, F. Econ6micas y Empresariales, Campus de Vegazana, 24071 Le6rg Spain. E-mail: ddears@un±leon, es. Received May 1, 1996; accepted October 1, 1996. International Journal of Approximate Reasoning 1997; 16:225-233 © 1997 Elsevier Science Inc. 0888-613X/97/$17.00 655 Avenue of the Americas, New York, NY 10010 PlI S0888-613X(96)00122-3 226 A . R . de So to e t al. predicate" is used here to mean the name of a graded property which is without classical set extension. The method is available when the use of the predicate from both a set of graded IF-THEN rules and a prototype u ~ X is known, and the set of rules is captured by a fuzzy T-preorder, i.e., a fuzzy relation R : E × E ~ [0, 1] satisfying 1. R(x/x) = 1 and 2. T(R(z/y), R(y/x)) < R(z /x) for any x, y, z in E, and T a continuous t-norm [2]. As is well known [6, 7], a fuzzy relation R is a T-preorder if and only if a family ~r of fuzzy sets exists such that R(y /x ) = inf I f ( y / x ) , (1) ~t~,9 r where each I f ( y / x ) = sup{z ~ [0, 1] : T(/z(x), z) < Iz(y)} is an elemental preorder, i.e., it is generated by residuation. The biggest family ~ r o f fuzzy sets verifying the last equality, or representation theorem, is the set .W r of T-logical states of the relation R, which is defined as .Z, eR r = {/x ~ [0, 11E : T(Ix(x), R(y /x ) ) < Ix(y), Vx, y ~ E}. The method to obtain a fuzzy set as the extension of a vague predicate by means of a T-preorder, representing the set of rules, and a prototype u, consists, basically, in identifying the preordering class of the prototype with the extension q~e of the predicate P: ~oe(x) = R t~u ( x ) = R ( x / u ) . The classes /z~ g of a T-preorder R are always T-logical states of R [6] and are normalized I because of the preorder's reflexivity. Reciprocally, if a fuzzy set /z is normalized, for any u ~ [/x], it is verified that I T txh"(x) = I f ( x / u ) = sup{z ~ [0, 1] :T (z , 1) = z ~ /~(x)} = / ~ ( x ) , and so any normalized fuzzy set can be seen as a preordering class of a preorder. The elemental preorders are a generalization of the classical material conditional which is defined, for a crisp subset A, as ~A =A x A U ( X A ) ×X , x A fuzzy set is said to be normalized if the crisp subset [/.,] = {x e X :/~(x) = 1} is not empty. It is also said that /z "has points." Classes of Fuzzy Sets 227 because if ~0 A is the membership function of A, it is fulfilled that