# Decidability and Independence of Conjugacy Problems in Finitely Presented Monoids

## TL;DR

This paper investigates the decidability and independence of various conjugacy notions in finitely presented monoids, revealing decidability results in polycyclic monoids and undecidability and independence in more general classes.

## Contribution

It introduces new results on the decidability and independence of conjugacy problems for different notions in finitely presented monoids, including polycyclic monoids and monoids with complete presentations.

## Key findings

- $p$-conjugacy is almost transitive in polycyclic monoids
- $c$-conjugacy is strictly included in $p$-conjugacy in polycyclic monoids
- Decidable conjugacy problems with linear complexity in certain classes

## Abstract

There have been several attempts to extend the notion of conjugacy from groups to monoids. The aim of this paper is study the decidability and independence of conjugacy problems for three of these notions (which we will denote by $\sim_p$, $\sim_o$, and $\sim_c$) in certain classes of finitely presented monoids. We will show that in the class of polycyclic monoids, $p$-conjugacy is "almost" transitive, $\sim_c$ is strictly included in $\sim_p$, and the $p$- and $c$-conjugacy problems are decidable with linear compexity. For other classes of monoids, the situation is more complicated. We show that there exists a monoid $M$ defined by a finite complete presentation such that the $c$-conjugacy problem for $M$ is undecidable, and that for finitely presented monoids, the $c$-conjugacy problem and the word problem are independent, as are the $c$-conjugacy and $p$-conjugacy problems.