Tue, 7 Mar 2023, 1:25 pm MST

The conjugacy equivalence relation in groups can be generalized to semigroups in various distinct ways. For example, in an inverse monoid $S$, one might define $a,b\in S$ to be conjugate if there exists $g\in S$ such that $g^{-1}ag = b$ and $gbg^{-1} = a$. There are other semigroup generalizations of conjugacy which do not involve inverses at all. In this talk I will survey the various generalizations, focusing particularly on a notion known as natural conjugacy, which, in inverse semigroups, coincides with the one defined above. Along the way, I will discuss connections with partial inner automorphisms and with the (universal algebra) center of an inverse semigroup.
The work described herein is joint with J. Araújo, W. Bentz, J. Konieczny, A. Malheiro, V. Mercier and D. Stanovský.