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, ∼o, and ∼c) in certain classes of finitely
presented monoids. We will show that in the class of polycyclic monoids,
p-conjugacy is "almost" transitive, ∼c is strictly included in
∼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.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1503.0091