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

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.