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 l 1 (S/ ∼) when S is an inverse semigroup with idempotents E and prove that it is equivalent to module amenability of l 1 (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 grou