Answering Five Open Problems Involving Semigroup Conjugacy
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913585213472768 |
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| author | Jack, Trevor |
| author_facet | Jack, Trevor |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16284 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Answering Five Open Problems Involving Semigroup Conjugacy Jack, Trevor Group Theory 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. |
| title | Answering Five Open Problems Involving Semigroup Conjugacy |
| topic | Group Theory |
| url | https://arxiv.org/abs/2411.16284 |