  Let $S$ be an inverse semigroup with an upward directed set of idempotents
$E$. In this paper we define the module topological center of second dual of a
Banach algebra which is a Banach module over another Banach algebra with
compatible actions, and find it for $ \ell ^{1}(S)^{**}$ (as an
$\ell^{1}(E)$-module). We also prove that $ \ell ^{1}(S)^{**}$ is
$\ell^{1}(E)$-module amenable if and only if an appropriate group homomorphic
image of $S$ is finite.
