THE CONGRUENCE η* ON SEMIGROUPS
M. H. Shahzamanian · The Quarterly Journal of Mathematics · 2016
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 Mal'cev). 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.