{ "id": "2407.10489", "version": "v1", "published": "2024-07-15T07:21:42.000Z", "updated": "2024-07-15T07:21:42.000Z", "title": "The growth of free inverse monoids", "authors": [ "Mark Kambites", "Carl-Fredrik Nyberg-Brodda", "Nóra Szakács", "Richard Webb" ], "comment": "18 pages. Comments welcome!", "categories": [ "math.GR", "math.RA" ], "abstract": "We compute the rate of exponential growth of the free inverse monoid of rank $r$ (and hence an upper bound on the corresponding rate for all $r$-generated inverse monoids and semigroups). This turns out to be an algebraic number strictly between the obvious bounds of $2r-1$ and $2r$, tending to $2r$ as the rank tends to infinity. We also find an explicit expression for the exponential growth rate of the number of idempotents, and prove that this tends to $\\sqrt{e(2k-1)}$ as $k \\to \\infty$.", "revisions": [ { "version": "v1", "updated": "2024-07-15T07:21:42.000Z" } ], "analyses": { "subjects": [ "20M18", "20M05", "05C05" ], "keywords": [ "free inverse monoid", "exponential growth rate", "upper bound", "generated inverse monoids", "explicit expression" ], "note": { "typesetting": "TeX", "pages": 18, "language": "en", "license": "arXiv", "status": "editable" } } }