The congruence $\eta^{\ast}$ on semigroups

In this paper we define a congruence $\eta^{\ast}$ on semigroups. For the finite semigroups $S$, $\eta^{\ast}$ is the smallest congruence relation such that $S/\eta^{\ast}$ is a nilpotent semigroup (in the sense of Malcev). In order to study the congruence relation $\eta^{\ast}$ on finite semigroups, we define a $\textbf{CS}$-diagonal finite regular Rees matrix semigroup. We prove that, if $S$ is a $\textbf{CS}$-diagonal finite regular Rees matrix semigroup then $S/\eta^{\ast}$ is inverse. Also, if $S$ is a completely regular finite semigroup, then $S/\eta^{\ast}$ is a Clifford semigroup. We show that, for every non-null principal factor $A/B$ of $S$, there is a special principal factor $C/D$ such that every element of $A\setminus B$ is $\eta^{\ast}$-equivalent with some element of $C\setminus D$. We call the principal factor $C/D$, the $\eta^{\ast}$-root of $A/B$. All $\eta^{\ast}$-roots are $\textbf{CS}$-diagonal. If certain elements of $S$ act in the special way on the $\textbf{R}$-classes of a $\textbf{CS}$-diagonal principal factor then it is not an $\eta^{\ast}$-root. Some of these results are also expressed in terms of pseudovarieties of semigroups.