Four notions of conjugacy for abstract semigroups. Issue 6 (20th September 2017)

• Record Type: Journal Article
• Authors: Araújo, João; Kinyon, Michael; Konieczny, Janusz; Malheiro, António
• Is Part Of: Proceedings. Volume 147:Issue 6(2017)
• Page Start: 1169
• Page End: 1214
• Publication Date: 2017-09-20
• Subjects: conjugacy, -- symmetric inverse semigroups, -- epigroups; Primary 20M07, -- 20M20, -- 20M15; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0308210517000099
• Languages: English
• ISSNs: 0308-2105

Abstract

Abstract : 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.