Solutions of matrix equations with weak fuzzy equivalence relations

Abstract

In this paper, we are solving matrix equations where the operation on matrices might not be a standard composition, but it can be any convenient binary operation. We use fuzzy (lattice valued) techniques in which a particular fuzzy weak equivalence relation and its compatibility with the matrix operation plays the central role. This fuzzy weak equivalence relation is weakly reflexive, symmetric, transitive, and compatible with the matrix operation. We obtain approximate solutions of two main types of matrix equations. We also tackle the question of the uniqueness of solutions (which are unique up to the fuzzy weak equivalence relation).