Identities of semigroup algebras of completely 0-simple semigroups

Let $H=M^0(G;I,\Delta;P)$ be a Rees semigroup of matrix type with sandwich matrix $P$ over a group $H^0$ with zero. If $F$ is a subgroup of $G$ of finite index and $X$ is a system of representatives of the left cosets of $F$ in $G$, then with the matrix $P$ there is associated in a natural way a matrix $P(F,X)$ over the group $F^0$ with zero. Our main result: the semigroup algebra $K[H]$ of $H$ over a field $K$ of characteristic 0 satisfies an identity if and only if $G$ has an Abelian subgroup $F$ of finite index and, for any $X$, the matrix $P(F,X)$ has finite determinant rank.