  The notions of module pseudo-amenable and module pseudo-contractible Banach
algebras are introduced. For a Banach algebra with bounded approximate
identity, module pseudo-amenability and module approximate amenability are the
same properties. It is given a complete characterization of module
pseudo-amenability for a Banach algebra. For every inverse semigroup $S$ with
subsemigroup $E$ of idempotents, necessary and sufficient conditions are
obtained for the $\ell^1(S)$ and its second dual to be $\ell^1(E)$-module
pseudo-amenable.
