Abstract. We say that a semigroup $S$ is a permutable semigroup if, for all congruences $\alpha $ and $\beta $ of $S$, $\alpha\circ \beta = \beta\circ \alpha $. In this paper we show that every permutable semigroup satisfying a permutation identity $x_1x_2\ldots x_n=x_{\sigma(1)}x_{\sigma(2)}\ldots x_{\sigma(n)}, \sigma(1)\not=1, \sigma(n)\not=n$ is commutative. We also prove 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 following problem is unsolved: Is every permutable semigroup satisfying a non-trivial permutation identity medial?

AMS Subject Classification (1991): 20M35

Received April 13, 2004, and in final form October 6, 2004. (Registered under 5856/2009.)