{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/structural-aspects-of-semigroups-based-on","title":"Structural aspects of semigroups based on digraphs","arxiv_id":"1704.00937","date":"2017-04-04","proceeding":null,"authors":["James East","Maximilien Gadouleau","James D. Mitchell"],"abstract":"Given any digraph $D$ without loops or multiple arcs, there is a natural construction of a semigroup $\\langle D\\rangle$ of transformations. To every arc $(a,b)$ of $D$ is associated the idempotent transformation $(a\\to b)$ mapping $a$ to $b$ and fixing all vertices other than $a$. The semigroup $\\langle D\\rangle$ is generated by the idempotent transformations $(a\\to b)$ for all arcs $(a,b)$ of $D$. In this paper, we consider the question of when there is a transformation in $\\langle D\\rangle$ containing a large cycle, and, for fixed $k\\in \\mathbb N$, we give a linear time algorithm to verify if $\\langle D\\rangle$ contains a transformation with a cycle of length $k$. We also classify those digraphs $D$ such that $\\langle D\\rangle$ has one of the following properties: inverse, completely regular, commutative, simple, 0-simple, a semilattice, a rectangular band, congruence-free, is $\\mathscr{K}$-trivial or $\\mathscr{K}$-universal where $\\mathscr{K}$ is any of Green's $\\mathscr{H}$-, $\\mathscr{L}$-, $\\mathscr{R}$-, or $\\mathscr{J}$-relation, and when $\\langle D\\rangle$ has a left, right, or two-sided zero.","url_abs":"http://arxiv.org/abs/1704.00937v2","url_pdf":"http://arxiv.org/pdf/1704.00937v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"structural-aspects-of-semigroups-based-on","repo_url":"https://github.com/michaeljklein/enum-categorical-digraphs","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}