Module Amenability of the Projective Module Tensor Product

Let S be an inverse semigroup with the set of idempotents E. In the current paper, we show that the projective module tensor product l^1 (S)? ?_(l^1 (E) ) l^1 (S) is l^1 (E)-module amenable when S is amenable. This could be considered as the module version (for inverse semigroups) of a result of Johnson (1972) which asserts that for any (discrete) amenable locally compact group G (when l^1 (E)=C, the set of complex numbers), the projective tensor product l^1 (G) ?? ?l?^1 (G)?l^1 (G×G) is amenable.