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

Abstract

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 $\text{CS}$-diagonal finite regular Rees matrix semigroup. We prove that, if $S$ is a $\text{CS}$-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 $\text{CS}$-diagonal. If certain elements of $S$ act in the special way on the $\text{R}$-classes of a $\text{CS}$-diagonal principal factor then it is not an ${\eta }^{\ast }$-root. Some of these results are also expressed in terms of pseudovarieties of semigroups.

References in corpus (1)

• Finite semigroups that are minimal for not being Malcev nilpotent

Cited by in corpus (2)

• Nilpotency and strong nilpotency for finite semigroups
• The rank of variants of nilpotent pseudovarieties