Abstract: The method of kernel-trace is successful in the study of congruence of the semigroup. The fuzzy set is the generalization of the classical set, while the fuzzy relation is a special function. In this paper we study the fuzzy kernel-trace of the fuzzy congruence relation on some class semigroup, that is, the notions of fuzzy kernel and fuzzy trace of a fuzzy congruence relation on a completely regular semigroup; we establish these notions by introducing fuzzy congruence pairs. Finally, we put emphasis on a one-to-one corre-spondence between fuzzy congruence pairs and fuzzy congruence on a completely regular semigroup.

Key words: completely regular semigroup, fuzzy kernel, fuzzy trace, fuzzy normal congruence, fuzzy congruence pairs