Module Biprojective and Module Biflat Banach Algebras

In this paper we define module biprojctivity and module biflatness for a Banach algebra which is a Banach module over another Banach algebra with compatible actions, and find their relation to classical biprojectivity and biflatness. As a typical example, We show that for an inverse semigroup $S$ with an upward directed set of idempotents $E$, the semigroup algebra $\ell ^{1}(S)$, as an $\ell ^{1}(E)$-module, is module biprojective if and only if an appropriate group homomorphic image of $S$ is finite. Also we show that $\ell ^{1}(S)$ is module biflat if and only if $S$ is amenable.