Araújo, J., J. Konieczny, and A. Malheiro. "Conjugation in semigroups." J. Algebra. 403 (2014): 93-134. AbstractWebsite

The action of any group on itself by conjugation and the corresponding conjugacy relation play an important role in group theory. There have been several attempts to extend the notion of conjugacy to semigroups. In this paper, we present a new definition of conjugacy that can be applied to an arbitrary semigroup and it does not reduce to the universal relation in semigroups with a zero. We compare the new notion of conjugacy with existing definitions, characterize the conjugacy in various semigroups of transformations on a set, and count the number of conjugacy classes in these semigroups when the set is infinite.

Cain, A. J., and A. Malheiro. "Deciding conjugacy in sylvester monoids and other homogeneous monoids." Int. J. Algebra Comput.. 25 (2015): 899-915. AbstractWebsite

We give a combinatorial characterization of conjugacy in the sylvester monoid, showing that conjugacy is decidable for this monoid. We then prove that conjugacy is undecidable in general for homogeneous monoids and even for multihomogeneous monoids.