On the decidability of conjugacy for homogeneous monoids

The well known notion of conjugacy for groups can be generalized in several different ways to semigroups. In this talk we would like to present some of the known generalizations, and how they interact in the case of monoids defined by homogeneous presentations.  We give a combinatorial description of conjugacy in the sylvester monoid, showing that is decidable for this monoid. We then prove that conjugacy is undecidable in general for homogeneous monoids and even for multihomogeneous monoids.

The results were obtained in collaboration with João Araújo, Alan Cain, Michael Kinyon, and Janusz Konieczny.