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

TL;DR

The paper extends binary and higher arity term-condition commutators to semigroups and studies how solvability, nilpotence, and supernilpotence translate from groups to semigroups. It proves that if a semigroup $S$ has a completely simple ideal $K$, then $S$ is solvable iff $S$ is a nilpotent extension of $K$ and all subgroups of $K$ are solvable, and $S$ is left/right nilpotent or $d$-supernilpotent iff $S$ is such a nilpotent extension with all subgroups of $K$ nilpotent; these characterizations apply in particular to finite and eventually regular semigroups. For monoids, $S$ is cancellative and embeds into a nilpotent group iff it is nilpotent in the commutator sense. The work also provides reduction principles for commutators via Rees congruences and analyzes the relation between semigroup and subgroup commutators, unifying several prior results and connecting semigroup structure with classical group-theoretic notions.

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.

Theorems & Definitions (35)

• Example 1.1
• Example 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof