[2301.04252] Conjugacy in Abstract Semigroups, Transformation and Diagram Monoids, and Conjugacy Growth Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Group Theory arXiv:2301.04252 (math) [Submitted on 11 Jan 2023 ( v1 ), last revised 28 Apr 2026 (this version, v3)] Title: Conjugacy in Abstract Semigroups, Transformation and Diagram Monoids, and Conjugacy Growth Authors: João Araújo , Wolfram Bentz , Michael Kinyon , Janusz Konieczny , António Malheiro , Valentin Mercier View a PDF of the paper titled Conjugacy in Abstract Semigroups, Transformation and Diagram Monoids, and Conjugacy Growth, by Jo\~ao Ara\'ujo and 5 other authors View PDF HTML (experimental) Abstract: We study conjugacy relations on semigroups and monoids, focusing on the relation $a \cfn b$, defined by the existence of $g,h \in S^1$ such that $ag = gb$, $bh = ha$, $hag = b$, and $gbh = a$. This notion emerged as one that yields particularly elegant results. The interplay between $\cfn$ and other standard conjugacy relations is analyzed, and some results on special classes of abstract semigroups are established. We then specialize to the case of transformation semigroups. A complete classification of $\cfn$-classes is obtained for the full transformation monoid $\mathcal{T}_n$, the symmetric inverse monoid $\mathcal{I}_n$, and the endomorphism monoid of $G$-sets, among others. We also investigate the natural conjugacy in diagram semigroups, including the partition monoid, the Brauer monoid, and the partial Brauer monoid. Finally, we investigate the conjugacy growth function in polycyclic monoids and obtain a precise asymptotic estimate. The paper concludes with some open problems. Comments: V.1: 80 pages. V.2: Vastly expanded. V.3: At the suggestion of the referee, we cut section 5 on partial inner automorphisms and spun it off into its own paper. We thus changed the title of this one and (again at the referee's request), drastically cut down the abstract Subjects: Group Theory (math.GR) MSC classes: 20E45, 20M10, 20M20 Cite as: arXiv:2301.04252 [math.GR] (or arXiv:2301.04252v3 [math.GR] for this version) https://doi.org/10.48550/arXiv.2301.04252 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Michael Kinyon [ view email ] [v1] Wed, 11 Jan 2023 00:21:15 UTC (63 KB) [v2] Sun, 9 Jun 2024 17:02:41 UTC (100 KB) [v3] Tue, 28 Apr 2026 21:12:34 UTC (69 KB) Full-text links: Access Paper: View a PDF of the paper titled Conjugacy in Abstract Semigroups, Transformation and Diagram Monoids, and Conjugacy Growth, by Jo\~ao Ara\'ujo and 5 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.GR < prev | next > new | recent | 2023-01 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from