Fuzzy Congruence Relations and Fuzzy Order Filters

In this paper, we introduce the notion of fuzzy congruence relation, fuzzy order filter of the adjoint semigroup M(X) of BCK-algebra X. We discuss some characterizations of fuzzy congruence relations and fuzzy order filters. In the adjoint semigroup of a positive implicative BCK-algebra with condition(S), every fuzzy congruence relation may induce a fuzzy order filter of M(X), and every fuzzy order filter may induce a fuzzy congruence relation. Moreover, we show that there is a bijection from the set of all fuzzy congruence relations to the set of all fuzzy order filters of semigroup M(X) which is the adjoint semigroup of a positive implicative BCK-algebra X with condition(S).