Department of Mathematics, Faculty of Science, Islamic Azad University, Centeral Tehran Branch, P. O. Box 13185/768, Tehran, Iran.
Let A be a Banach algebra and E be a Banach A-bimodule then S = A E,
the l1-direct sum of A and E becomes a module extension Banach algebra when equipped
with the algebras product (a; x):(a′; x′) = (aa′; a:x′ + x:a′). In this paper, we investigate
△-amenability for these Banach algebras and we show that for discrete inverse semigroup S
with the set of idempotents ES, the module extension Banach algebra S = l1(ES) l1(S) is
△-amenable as a l1(ES)-module if and only if l1(ES) is amenable as Banach algebra.