Fuzzy congruence relations

Abstract

Abstract If A = [X, F] is an universal algebra and if R(X) denotes the set of all fuzzy subsets of X × X, (i.e. fuzzy relations on X), then the algebraic operations fg ϵ F on X are extended to algebraic operations ∼fγ on [R(X), F] is an algebra similar to A. It is called relational algebra. A fuzzy equivalence relation which is at the same time a fuzzy subalgebra of [R(X), F], is called a fuzzy congruence relation on X. In this paper we study fuzzy congruence relations. Amongst other things, we prove that the set of all fuzzy congruence relations on an algebra is a complete lattice and an algebraic closure system.