Answering Five Open Problems Involving Semigroup Conjugacy

TL;DR

This paper addresses five open problems in semigroup conjugacy, providing new constructions and proofs that clarify the properties and relationships of various conjugacy relations within semigroups.

Contribution

It offers novel examples and proofs that resolve longstanding open problems about semigroup conjugacy relations and their properties.

Findings

Six conjugacy relations are partition-covering.

An example of a semigroup where ~p is not transitive.

Construction of a semigroup with specific conjugacy properties.

Abstract

A semigroup conjugacy is an equivalence relation that equals group conjugacy when the semigroup is a group. In this note, we answer five open problems related to semigroup conjugacy. (Problem One) We say a conjugacy ~ is partition-covering if for every set X and every partition of the set, there exists a semigroup with universe X such that the partition gives the ~-conjugacy classes of the semigroup. We prove that six well-studied conjugacy relations -- ~o, ~c, ~n, ~p, ~p*, and ~tr -- are all partition-covering. (Problem Two) For two semigroup elements a and b in S, we say a ~p b if there exists u and v in S such that a=uv and b=vu. We give an example of a semigroup that is embeddable in a group for which ~p is not transitive. (Problem Three) We construct an infinite chain of first-order definable semigroup conjugacies. (Problem Four) We construct a semigroup for which ~o is a congruence and S\~o is not cancellative. (Problem Five) We construct a semigroup for which ~p is not transitive while, for each of the semigroup's variants, ~p is transitive.