Abstract. We say that a semigroup $S$ is a permutable semigroup if, for every congruences $\alpha $ and $\beta $ of $S$, $\alpha\circ \beta = \beta\circ \alpha $. In [4], A. Nagy showed that every permutable semigroup satisfying an arbitrary non-trivial permutation identity is medial or an ideal extension of a rectangular band by a non-trivial commutative nil semigroup. The author raised the following problem: Is every permutable semigroup satisfying a non-trivial permutation identity medial? In the present paper we give a positive answer for this problem.

AMS Subject Classification (1991): 20M10

Received January 13, 2006, and in revised form June 7, 2006. (Registered under 5935/2009.)