Some Characterizations for Approximate Biflatness of Semigroup Algebras

In this paper, we study an approximate biflatness of $l^1\left(S\right)$ , where $S$ is a Clifford semigroup. Indeed, we show that a Clifford semigroup algebra $l^1\left(S\right)$ is approximately biflat if and only if every maximal subgroup of $S$ is amenable, $E\left(S\right)$ is locally finite, and $l^1\left(S\right)$ has an approximate identity in $c_{00}\left(S\right)$ . Moreover, we prove that $l^1\left(S\right)$ is approximately biflat if and only if each maximal subgroup of $S$ is amenable for an inverse semigroup $S$ such that $E\left(S\right)$ , the set of its idempotent elements, is totally ordered and locally finite.