# António Malheiro

## Assistant Professor, Mathematical Department

Faculdade de Ciências e Tecnologia UNL, Monte da Caparica, 2829-516 Caparica, Tel: (+351) 212948388 ext. 10832 (email)

Faculdade de Ciências e Tecnologia UNL, Monte da Caparica, 2829-516 Caparica, Tel: (+351) 212948388 ext. 10832 (email)

- Citation:
- Araújo, J., M. Kinyon, J. Konieczny, and A. Malheiro. "Decidability and Independence of Conjugacy Problems in Finitely Presented Monoids." (Submitted).

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.