In this paper, the concepts of essential module amenability and the essential module \((\phi ,\varphi )\) -amenability for a Banach algebra \({\mathcal {A}}\) (with an extra module structure on a Banach algebra \({\mathfrak {A}}\) ) are introduced, where \(\varphi\) is a character on \({\mathfrak {A}}\) and \(\phi\) is a module character on \({\mathcal {A}}\) . For every inverse semigroup S with subsemigroup E of idempotents, necessary and sufficient conditions are obtained for the \(l^1(S)\) and its second dual to be essentially module amenable (as \(l^1(E)\) -module). An example shows the essential module amenability and the module amenability for Banach algebras are different. In analogy with the classical case, the essential module amenability of triangular Banach algebras and Lau-product Banach algebras are studied.
