Non-finitely related and finitely related monoids

TL;DR

This paper advances the understanding of monoids by providing new criteria to determine when they are finitely or non-finitely related, including specific examples and a complete classification within a certain class.

Contribution

It introduces a new sufficient condition for non-finite relatedness, extends the collection of interlocking word-patterns, and completes the classification of finitely related monoids in a specific family.

Findings

The monoid M(ab^2a, a^2b^2) is non-finitely related.

The monoid M(a^2b^2) is finitely related.

The results complete the classification of finitely related monoids in a specific class.

Abstract

We transform the method of Glasson into a sufficient condition under which a monoid is non-finitely related, add a new member to the collection of interlocking word-patterns, and use it to show that the monoid $M(ab^2a, a^2b^2)$ is non-finitely related. We also give a sufficient condition under which a monoid is finitely related and use it show that $M(a^2b^2)$ is finitely related. Together with the results of Glasson this completes the description of all finitely related monoids among the monoids of the form $M(W)$ where every word ${\bf u} \in W$ depends on two variables and every variable occurs twice in $\bf u$.