Not all nilpotent monoids are finitely related

TL;DR

This paper investigates the property of finite relatedness in nilpotent monoids, showing that all 4-nilpotent monoids are finitely related, but providing a counterexample at 5-nilpotent level, highlighting nuanced behavior in algebraic structures.

Contribution

It proves that every 4-nilpotent monoid is finitely related and presents the first example of a finitely related semigroup whose identity adjoinment results in a non-finitely related semigroup.

Findings

All 4-nilpotent monoids are finitely related.

A 5-nilpotent monoid can be not finitely related.

Examples of finitely related semigroups with non-finitely related substructures.

Abstract

A finite semigroup is finitely related (has finite degree) if its term functions are determined by a finite set of finitary relations. For example, it is known that all nilpotent semigroups are finitely related. A nilpotent monoid is a nilpotent semigroup with adjoined identity. We show that every $4$-nilpotent monoid is finitely related. We also give an example of a $5$-nilpotent monoid that is not finitely related. To our knowledge, this is the first example of a finitely related semigroup where adjoining an identity yields a semigroup which is not finitely related. We also provide examples of finitely related semigroups which have subsemigroups, homomorphic images, and in particular Rees quotients, that are not finitely related.