Commutators in Completely Simple Semigroups

TL;DR

This paper characterizes the binary commutator in completely simple semigroups via Rees matrix representation and links nilpotency and solvability of regular semigroups to their simplicity and properties of their H-classes.

Contribution

It provides a new characterization of the binary commutator in completely simple semigroups and relates nilpotency and solvability of regular semigroups to their simplicity and H-class properties.

Findings

Characterization of the binary commutator using Rees matrix representation.

Regular semigroups are nilpotent if and only if they are simple with nilpotent H-classes.

Regular semigroups are solvable if and only if they are simple with solvable H-classes.

Abstract

We obtain a characterization of the binary commutator on completely simple semigroups, using their Rees matrix representation. Consequently, we prove that a regular semigroup is nilpotent (solvable) if and only if it is simple, and all its $\mathcal{H}$-classes are nilpotent (solvable) groups.