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On three approaches to conjugacy in semigroups. (English) Zbl 1171.20035

There are some approaches for extensions of the notion of conjugation in the group theory to some kinds of semigroups. The authors introduce three such extensions which generalize some of the known ones and compare them. The first of them is the transitive closure of the relation \(x\sim y\Leftrightarrow\exists u,v\in S^1\) \((x=uv\,\&\,y=vu)\) in a semigroup \(S\). The second notion is defined for an inverse group-bound semigroup \(S\) and is the transitive closure of the relation \(x\approx y\Leftrightarrow\exists a\in S^1\) \((y=ax\vee x=ay)\). The third one is defined via characters of finite-dimensional representations: \(x\equiv y\) if for every finite-dimensional complex representation \(\varphi\) we have \(\chi_\varphi(x)=\chi_\varphi(y)\). The results show cases where some of these notions coincide.

MSC:

20M10 General structure theory for semigroups
20M15 Mappings of semigroups
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