Solvable, nilpotent and supernilpotent semigroups with completely simple ideal and monoids

TL;DR

This paper explores how concepts of solvability, nilpotence, and supernilpotence from commutator theory apply to semigroups with completely simple ideals, providing characterizations that connect classical and modern algebraic notions.

Contribution

It characterizes semigroups with completely simple ideals that are solvable, nilpotent, or supernilpotent in terms of classical semigroup theory, linking modern commutator concepts with traditional structures.

Findings

Semigroups with completely simple ideals are solvable iff they are nilpotent extensions of completely simple semigroups with solvable subgroups.

Finite and eventually regular semigroups satisfy these characterizations.

Monoids are nilpotent in commutator sense iff they embed into a nilpotent group.

Abstract

Around 1980 commutator theory was generalized from groups to arbitrary algebras using the socalled term condition commutator. The semigroups that are abelian with respect to this commutator were classified by Warne (1994). We study what solvability, nilpotence, and supernilpotence in the sense of commutator theory mean for semigroups and how these notions relate to classical concepts in semigroup theory. We show that a semigroup with a completely simple ideal is solvable (left nilpotent or right nilpotent or supernilpotent) in the sense of commutator theory iff it is a nilpotent extension in the classical sense of semigroup theory of a completely simple semigroup with solvable (nilpotent) subgroups. These characterizations hold in particular for finite semigroups and for eventually regular semigroups, i.e., semigroups in which every element has some regular power. We also show that a monoid is (left and right) nilpotent in the sense of commutator theory iff it embeds into a nilpotent group.