2024/11/25 by Trevor Jack, Jack, Trevor
Computer Science · #Advanced Algebra and Logic #FOS: Mathematics #Group Theory (math.GR) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2411.16284
openalex publication_date 2024/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Sõ 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.