On P-fazzy correspondences and generalized associativity

Abstract

A P-fuzzy correspondence is a mapping Ā : A1 × A2 × ⋯ × An → P, where A1,A2, . . . , An are nonempty sets, and P a partially ordered set. In this paper we give necessary and sufficient conditions under which a family of ordinary correspondences on sets A1,A2, . . . , An is a family of levels of a P-fuzzy correspondence. A G-groupoid is a quadruple (S1, S2, S3, A), where S1, S2, S3 are sets, and A mapping, A : S1 × S2 → S3. We consider a type of P-fuzzy correspondences as a generalization of a notion of G-groupoids, called PG-fuzzy correspondences. That is a PG-fuzzy correspondence is a mapping Ā : A1 × A2 × ⋯ × An → P, where all sets A1, A2, . . . , An and P are partially ordered. Here, we deal with binary PG-fuzzy correspondences. We give a general solution of the functional equation of generalized associativity A(x,B(y,z)) = C(D(x,y)z), where A, B, C and D are binary PG-fuzzy correspondences. In particular, if A, B, C and D are semilattices which are solutions of the considered equation, then it turns out that they are equal, i.e., A = B = C = D.