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{
    "pk": 64923,
    "title": "Minimum degrees of finite rectangular bands, null semigroups, and variants of full transformation semigroups",
    "subtitle": null,
    "abstract": "For a positive integer \\(n\\), the full transformation semigroup \\({\\mathcal T}_n\\) consists of all self maps of the set \\(\\{1,\\ldots,n\\}\\) under composition. Any finite semigroup \\(S\\) embeds in some \\({\\mathcal T}_n\\), and the least such \\(n\\) is called the (minimum transformation) degree of \\(S\\) and denoted \\(\\mu(S)\\). We find degrees for various classes of finite semigroups, including rectangular bands, rectangular groups and null semigroups. The formulae we give involve natural parameters associated to integer compositions. Our results on rectangular bands answer a question of Easdown from 1992, and our approach utilises some results of independent interest concerning partitions/colourings of hypergraphs.\nAs an application, we prove some results on the degree of a variant \\({\\mathcal T}_n^a\\). (The variant \\(S^a=(S,\\star)\\) of a semigroup \\(S\\), with respect to a fixed element \\(a\\in S\\), has underlying set \\(S\\) and operation \\(x\\star y=xay\\).) It has been previously shown that \\(n\\leq \\mu({\\mathcal T}_n^a)\\leq 2n-r\\) if the sandwich element \\(a\\) has rank \\(r\\), and the upper bound of \\(2n-r\\) is known to be sharp if \\(r\\geq n-1\\). Here we show that \\(\\mu({\\mathcal T}_n^a)=2n-r\\) for \\(r\\geq n-6\\). In stark contrast to this, when \\(r=1\\), and the above inequality says \\(n\\leq\\mu({\\mathcal T}_n^a)\\leq 2n-1\\), we show that \\(\\mu({\\mathcal T}_n^a)/n\\to1\\) and \\(\\mu({\\mathcal T}_n^a)-n\\to\\infty\\) as \\(n\\to\\infty\\).\nAmong other results, we also classify the \\(3\\)-nilpotent subsemigroups of \\({\\mathcal T}_n\\), and calculate the maximum size of such a subsemigroup.\n \nMathematics Subject Classifications: 20M20, 20M15, 20M30, 05E16, 05C65\n \nKeywords: Transformation semigroup, transformation representation, semigroup variant, rectangular band, nilpotent semigroup, hypergraph",
    "language": "en",
    "license": {
        "name": "Creative Commons Attribution 4.0",
        "short_name": "CC BY 4.0",
        "text": "Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.",
        "url": "https://creativecommons.org/licenses/by/4.0"
    },
    "keywords": [
        {
            "word": "Transformation semigroup"
        },
        {
            "word": "transformation representation"
        },
        {
            "word": "semigroup variant"
        },
        {
            "word": "rectangular band"
        },
        {
            "word": "nilpotent semigroup"
        },
        {
            "word": "hypergraph"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/4z41x488",
    "frozenauthors": [
        {
            "first_name": "Peter",
            "middle_name": "J.",
            "last_name": "Cameron",
            "name_suffix": "",
            "institution": "Mathematical Institute, University of St Andrews, St Andrews, Fife KY16 9SS, U.K.",
            "department": ""
        },
        {
            "first_name": "James",
            "middle_name": "",
            "last_name": "East",
            "name_suffix": "",
            "institution": "Centre for Research in Mathematics and Data Science, Western Sydney University, Locked Bag 1797, Penrith NSW 2751, Australia",
            "department": ""
        },
        {
            "first_name": "Des",
            "middle_name": "",
            "last_name": "FitzGerald",
            "name_suffix": "",
            "institution": "School of Natural Sciences, University of Tasmania, Private Bag 37, nipaluna/Hobart 7001, Australia",
            "department": ""
        },
        {
            "first_name": "James",
            "middle_name": "D.",
            "last_name": "Mitchell",
            "name_suffix": "",
            "institution": "Mathematical Institute, University of St Andrews, St Andrews, Fife KY16 9SS, U.K.",
            "department": ""
        },
        {
            "first_name": "Luke",
            "middle_name": "",
            "last_name": "Pebody",
            "name_suffix": "",
            "institution": "",
            "department": ""
        },
        {
            "first_name": "Thomas",
            "middle_name": "",
            "last_name": "Quinn-Gregson",
            "name_suffix": "",
            "institution": "",
            "department": ""
        }
    ],
    "date_submitted": "2023-12-22T14:42:15Z",
    "date_accepted": "2023-12-22T14:42:15Z",
    "date_published": "2023-12-22T08:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64923/galley/49733/download/"
        }
    ]
}