A description of a class of finite semigroups that are near to being Malcev nilpotent

TL;DR

This paper studies a class of finite semigroups called extbackslash B\ semigroups, where the property of being Malcev nilpotent can be understood through their associated graphs, revealing structural insights about their ideals and components.

Contribution

The paper introduces extbackslash B\ semigroups, characterizes their structure via associated graphs, and shows that finite monoids are extbackslash B\ if and only if they are nilpotent.

Findings

Finite monoids are extbackslash B\\ if and only if they are nilpotent.

extbackslash B\ semigroups have a largest nilpotent ideal.

The structure of extbackslash B\ semigroups can be described using their associated non-nilpotent graphs.

Abstract

In this paper we continue the investigations on the algebraic structure of a finite semigroup $S$ that is determined by its associated upper non-nilpotent graph $\mathcal{N}_{S}$. The vertices of this graph are the elements of $S$ and two vertices are adjacent if they generate a semigroup that is not nilpotent (in the sense of Malcev). We introduce a class of semigroups in which the Mal'cev nilpotent property lifts through ideal chains. We call this the class of \B\ semigroups. The definition is such that the global information that a semigroup is not nilpotent induces local information, i.e. some two-generated subsemigroups are not nilpotent. It turns out that a finite monoid (in particular, a finite group) is \B\ if and only if it is nilpotent. Our main result is a description of \B\ finite semigroups $S$ in terms of their associated graph ${\mathcal N}_{S}$. In particular, $S$ has a largest nilpotent ideal, say $K$, and $S/K$ is a 0-disjoint union of its connected components (adjoined with a zero) with at least two elements.