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Four notions of conjugacy for abstract semigroups

Part of: Semigroups

Published online by Cambridge University Press:  20 September 2017

João Araújo ,
Michael Kinyon ,
Janusz Konieczny  and
António Malheiro
João Araújo
Affiliation:
Universidade Aberta, R. Escola Politécnica, 147, 1269-001 Lisboa, Portugal (jaraujo@uab.pt) and CEMAT-Ciências, Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, 1749–016, Lisboa, Portugal (jjaraujo@fc.ul.pt)
Michael Kinyon
Affiliation:
Department of Mathematics, University of Denver, 2360 S Gaylord St, Denver, CO 80208, USA (mkinyon@du.edu) and CEMAT-Ciências, Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, 1749–016, Lisboa, Portugal
Janusz Konieczny
Affiliation:
Department of Mathematics, University of Mary Washington, Fredericksburg, VA 22401, USA (jkoniecz@umw.edu)
António Malheiro
Affiliation:
Departamento de Matemática and Centro de Matemática e Aplicações (CMA), Faculdade de Ciências e Tecnologia (FCT), Universidade Nova de Lisboa (UNL), 2829-516 Caparica, Portugal (ajm@fct.unl.pt)
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Extract

The action of any group on itself by conjugation and the corresponding conjugacy relation play an important role in group theory. There have been many attempts to find notions of conjugacy in semigroups that would be useful in special classes of semigroups occurring in various areas of mathematics, such as semigroups of matrices, operator and topological semigroups, free semigroups, transition monoids for automata, semigroups given by presentations with prescribed properties, monoids of graph endomorphisms, etc. In this paper we study four notions of conjugacy for semigroups, their interconnections, similarities and dissimilarities. They appeared originally in various different settings (automata, representation theory, presentations, and transformation semigroups). Here we study them in full generality. The paper ends with a large list of open problems.

Keywords

conjugacysymmetric inverse semigroupsepigroups

MSC classification

Secondary: 20M07: Varieties and pseudovarieties of semigroups 20M20: Semigroups of transformations, etc. 20M15: Mappings of semigroups
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 147 , Issue 6 , December 2017 , pp. 1169 - 1214
Copyright
Copyright © Royal Society of Edinburgh 2017 

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