Conjugacy in inverse semigroups (Q1999785)

The authors study the rather natural conjugacy relation \(\sim_i\), defined on an inverse semigroup \(S\) by \(a \sim_i b\) if there exists \(g \in S^1\) such that \(g^{-1}ag = b\) and \(gbg^{-1} = a\). Here \(g\) need not be a unit and \(g^{-1} \) is the usual inverse of \(g\). The body of the paper begins with a brief survey of various other notions of conjugacy that have appeared in the literature for general semigroups, followed by their relationship with \(\sim_i\) in the case of inverse semigroups. The remainder of the paper is devoted to calculating \(\sim_i\) explicitly in various well-known classes of inverse semigroups. For example, two elements of the symmetric inverse semigroup on a set \(X\) are \(\sim_i\)-related if and only if they are conjugate via a permutation of \(X\) (that is, by a unit) if \(X\) is finite, but for infinite \(X\) the situation is quite technical. Conjugation in free inverse semigroups has a transparent description in turns of Munn trees. The authors note that \textit{M. Sapir} introduced an equivalent conjugacy relation in [\url{http://mathoverflow.net/questions/52017/the-concept-conjugate-class-in-monoids}].