The conjugacy problem for positive homogeneously presented monoids

TL;DR

This paper investigates the conjugacy problem in positive homogeneously presented monoids, providing solutions under certain algebraic conditions and extending methods to examples lacking the LCM property.

Contribution

It introduces new solutions to the conjugacy problem for monoids without the LCM condition, expanding on previous methods by Brieskorn and Saito.

Findings

Solved conjugacy problem for specific monoids without LCM

Extended existing methods to broader classes of monoids

Provided concrete examples illustrating the new solutions

Abstract

Let $M$ be a positive homogeneously presented monoid ${\langle L \mid R\,\rangle}_{mo}$. If $M$ satisfies the cancellation condition and carries certain particular elements similar to the \emph{fundamental elements} in Artin monoids, then the solvability of the conjugacy problem for $M$ implies that in the corresponding group ${\langle L \mid R\,\rangle}$. In addition to these conditions, if $M$ satisfies the LCM condition (i.e. any two elements $\alpha$ and $\beta$ in $M$ admit the left (resp.~right) least common multiple), then the solution to the conjugacy problem for $M$ is known. We will give two kinds of examples that do not satisfy only the LCM condition. For these examples, we will give a solution to the conjugacy problem by improving the method given by E. Brieskorn and K. Saito.