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 \(\sim \) 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 \(\sim \) -conjugacy classes of the semigroup. We prove that six well-studied conjugacy relations – \(\sim _o\) , \(\sim _c\) , \(\sim _n\) , \(\sim _p\) , \(\sim _{p^*}\) , and \(\sim _{tr}\) – are all partition-covering. (Problem Two) For two semigroup elements \(a,b \in S\) , we say \(a \sim _p b\) if there exists \(u,v \in S\) such that \(a=uv\) and \(b=vu\) . We give an example of a semigroup that is embeddable in a group for which \(\sim _p\) is not transitive. (Problem Three) We construct an infinite chain of first-order definable semigroup conjugacies. (Problem Four) We construct a semigroup for which \(\sim _o\) is a congruence and \(S / \sim _o\) is not cancellative. (Problem Five) We construct a semigroup for which \(\sim _p\) is not transitive while, for each of the semigroup’s variants, \(\sim _p\) is transitive.
