# Decidability and Independence of Conjugacy Problems in Finitely Presented Monoids

## Publication Date

4-17-2018

## Document Type

Article

## Organizational Units

College of Natural Science and Mathematics, Mathematics

## Keywords

Conjugacy, Finitely presented monoids, Polycyclic monoids, Decision problem, Decidability, Independence, Complexity

## 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 ∼p" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">∼p, ∼o" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">∼o, and ∼c" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">∼c) in certain classes of finitely presented monoids. We will show that in the class of polycyclic monoids, *p*-conjugacy is “almost” transitive, ∼c" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">∼c is strictly included in ∼p" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">∼p, and the *p*- and *c*-conjugacy problems are decidable with linear complexity on a two-tape Turing Machine. 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. On other hand, we show that for finitely presented monoids, the *o*-conjugacy problem is reducible to the *c*-conjugacy problem.

## Publication Statement

Copyright held by author or publisher. User is responsible for all copyright compliance.

## Recommended Citation

Araújo, João, et al. “Decidability and Independence of Conjugacy Problems in Finitely Presented Monoids.” Theoretical Computer Science, vol. 731, 2018, pp. 88–98. doi: 10.1016/j.tcs.2018.04.002.