Generalized Fuzzy Connectives onMV-Algebras

Abstract

w x The theory of MV-algebras, developed by C. C. Chang 8, 9 first w x appeared in 1958. Several years later Zadeh 14 published his fundamental paper on fuzzy set theory. The two theories had independent motivations. MV-algebras were developed to provide an algebraic proof of the completeness theorem for Łukasiewicz infinite valued propositional logic. Fuzzy set theory was proposed to provide a mathematical basis for sets where, in reality, set membership was vague. The logic of Łukasiewicz includes in some sense two valued classical logic; fuzzy set theory includes the usual two valued characteristic functions. Thus both theories are extensions of classical logic. Both of the above theories use as a basis of interpretation the real unit w x w x interval 0, 1 . The basic MV-algebraic operations on 0, 1 are a [ b s Ž . Ž . U min 1, a q b , a ? b s max 0, a q b y 1 , and a s 1 y a. The basic fuzzy w x Ž . theoretic operations on 0, 1 suggested by Zadeh are a k b s max a, b , Ž . U a n b s min a, b , and a s 1 y a. Now the operations k and n can be defined in terms of [ and U , thus the Łukasiewicz operations are more general and include the basic fuzzy operations. By pointwise extension we obtain, given an arbitrary set X / B, two w x X U algebraic structures on 0, 1 , say A determined by [, ?, and A 1 2 determined by k, n, . A is an MV-algebra and is often referred to as 1 the bold algebra of fuzzy subsets of X. A is the basic algebra of fuzzy 2 subsets of X. By the remark above that k, n can be defined in terms of [, U , we see that A is some type of substructure of A . 2 1