Solvable and (super)nilpotent semigroups with completely simple ideal and (super)nilpotent monoids
Year: 2026
Type: article
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 [22]. 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.
Source: Semigroup Forum
Author: Peter Mayr
Institution: University of Colorado Boulder
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W7155072980