Conjugacy in inverse semigroups

Abstract

In a group $G$, elements $a$ and $b$ are conjugate if there exists $g\in G$ such that $g^{-1}ag=b$. This conjugacy relation, which plays an important role in group theory, can be extended in a natural way to inverse semigroups: for elements $a$ and $b$ in an inverse semigroup $S$, $a$ is conjugate to $b$, which we will write as $a{\sim }_{\mathrm{i}}b$, if there exists $g\in S^1$ such that $g^{-1}ag=b$ and $gb{g}^{-1}=a$. The purpose of this paper is to study the conjugacy ${\sim }_{\mathrm{i}}$ in several classes of inverse semigroups: symmetric inverse semigroups, free inverse semigroups, McAllister $P$-semigroups, factorizable inverse monoids, Clifford semigroups, the bicyclic monoid and stable inverse semigroups.

22 pages