We argue that weak containment is an appropriate notion of amenability for inverse semigroups. Given an inverse semigroup S and a homomorphism φ of S onto a group G, we show, under an assumption on ker(φ), that S has weak containment if and only if G is amenable and ker(φ) has weak containment. Using Fell bundle amenability, we find a related result for inverse semigroups with zero. We show tha...

Let S be an inverse semigroup with the set of idempotents E and S/ ≈ be an appropriate group homomorphic image of S. In this paper we find a one-to-one correspondence between two cohomology groups of the group algebra `1(S) and the semigroup algebra `1(S/ ≈) with coefficients in the same space. As a consequence, we prove that S is amenable if and only if S/ ≈ is amenable. This could be consider...