Module Amenability of Semigroup Algebras under Certain Module Actions

In this paper we define a congruence ? on inverse semigroup S such that amenability of S is equivalent to amenability of S/?. We study module amenability of semigroup algebra l1(S/?). when S is an inverse semigroup with idempotents E and prove that it is equivalent to module amenability of l1(S). The main difference of this action with the more studied trivial action is that in this case the corresponding homomorphic image is a Clifford semigroup rather than a discrete group.