  Let $S$ be an inverse semigroup with the set of idempotents $E$. In this
paper we define the module super-amenability of a Banach algebra which is a
Banach module over another Banach algebra with compatible actions, and show
that when $E$ is upward directed and acts on $S$ trivially from left and by
multiplication from right, the semigroup algebra $ \ell ^{1}(S)$ is
$\ell^{1}(E)$-module super-amenable if and only if an appropriate group
homomorphic image of $S$ is finite.

Functional Analysis