Necessary and sufficient conditions for the existence of solution of generalized fuzzy relation equations A ⇔X = B

Abstract

In 2013 Li and Jin studied a particular type of fuzzy relational equations on finite sets, where the introduced min-bi-implication composition is based on Łukasiewicz equivalence. In this paper such fuzzy relation equations are studied on a more general level, namely complete residuated lattice valued fuzzy relation equations of type ⋀y∈Y(A(x,y)↔X(y)=B(x) are analyzed, and the existence of solutions S is studied. First a necessary condition for the existence of solution is established, then conditions for lower and upper limits of solutions are given, and finally sufficient conditions for the existence of the smallest and largest solutions, respectively, are characterized. If such general or global solutions do not exist, there might still be partial or point wise solutions; this is a novel way to study fuzzy relation equations. Such point wise solutions are studied on Łukasiewicz, Product and Gödel t-norm based residuated lattices on the real unit interval.