In this paper, we introduce a new notion of module strong pseudoamenability for Banach algebras. We study the relation between this new concept to other various notions at this issue, module pseudo-contractibility, module pseudo-amenability and module approximate amenability. For an inverse semigroup S with the set of all idempotents E, we show that ` 1 (S) is module strong pseudoamenable as an ` 1 (E)-module if and only if S is amenable. For specific types of semigroups such as Brandt semigroups and bicyclic semigroups, we investigate the module strong pseudo-amenability of ` 1 (S). We show that for every non-empty set I, MI (C) under this new notion is forced to have a finite index as an A-module, where A = [ai,j ] ∈ MΛ(C) | ∀i 6= j, ai,j = 0 .
