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Computing maximal subsemigroups of a finite semigroup
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dc.contributor.author | Donoven, C. R. | |
dc.contributor.author | Mitchell, J. D. | |
dc.contributor.author | Wilson, W. A. | |
dc.date.accessioned | 2019-02-15T00:34:30Z | |
dc.date.available | 2019-02-15T00:34:30Z | |
dc.date.issued | 2018-07-01 | |
dc.identifier | 247458992 | |
dc.identifier | 49ec975c-2b16-4912-9f95-d814a318372b | |
dc.identifier | 85042216783 | |
dc.identifier.citation | Donoven , C R , Mitchell , J D & Wilson , W A 2018 , ' Computing maximal subsemigroups of a finite semigroup ' , Journal of Algebra , vol. 505 , pp. 559-596 . https://doi.org/10.1016/j.jalgebra.2018.01.044 | en |
dc.identifier.issn | 0021-8693 | |
dc.identifier.other | ArXiv: http://arxiv.org/abs/1606.05583v1 | |
dc.identifier.other | ORCID: /0000-0002-5489-1617/work/73700795 | |
dc.identifier.other | ORCID: /0000-0002-3382-9603/work/85855348 | |
dc.identifier.uri | https://hdl.handle.net/10023/17072 | |
dc.description | The third author wishes to acknowledge the support of his Carnegie Ph.D. Scholarship from the Carnegie Trust for the Universities of Scotland. | en |
dc.description.abstract | A proper subsemigroup of a semigroup is maximal if it is not contained in any other proper subsemigroup. A maximal subsemigroup of a finite semigroup has one of a small number of forms, as described in a paper of Graham, Graham, and Rhodes. Determining which of these forms arise in a given finite semigroup is difficult, and no practical mechanism for doing so appears in the literature. We present an algorithm for computing the maximal subsemigroups of a finite semigroup S given knowledge of the Green's structure of S, and the ability to determine maximal subgroups of certain subgroups of S, namely its group H-classes. In the case of a finite semigroup S represented by a generating set X, in many examples, if it is practical to compute the Green's structure of S from X, then it is also practical to find the maximal subsemigroups of S using the algorithm we present. In such examples, the time taken to determine the Green's structure of S is comparable to that taken to find the maximal subsemigroups. The generating set X for S may consist, for example, of transformations, or partial permutations, of a finite set, or of matrices over a semiring. Algorithms for computing the Green's structure of S from X include the Froidure–Pin Algorithm, and an algorithm of the second author based on the Schreier–Sims algorithm for permutation groups. The worst case complexity of these algorithms is polynomial in |S|, which for, say, transformation semigroups is exponential in the number of points on which they act. Certain aspects of the problem of finding maximal subsemigroups reduce to other well-known computational problems, such as finding all maximal cliques in a graph and computing the maximal subgroups in a group. The algorithm presented comprises two parts. One part relates to computing the maximal subsemigroups of a special class of semigroups, known as Rees 0-matrix semigroups. The other part involves a careful analysis of certain graphs associated to the semigroup S, which, roughly speaking, capture the essential information about the action of S on its J-classes. | |
dc.format.extent | 38 | |
dc.format.extent | 679060 | |
dc.language.iso | eng | |
dc.relation.ispartof | Journal of Algebra | en |
dc.subject | Algorithms | en |
dc.subject | Computational group theory | en |
dc.subject | Computational semigroup theory | en |
dc.subject | Maximal subsemigroups | en |
dc.subject | QA Mathematics | en |
dc.subject | Algebra and Number Theory | en |
dc.subject | DAS | en |
dc.subject.lcc | QA | en |
dc.title | Computing maximal subsemigroups of a finite semigroup | en |
dc.type | Journal article | en |
dc.contributor.institution | University of St Andrews. Pure Mathematics | en |
dc.contributor.institution | University of St Andrews. Centre for Interdisciplinary Research in Computational Algebra | en |
dc.identifier.doi | 10.1016/j.jalgebra.2018.01.044 | |
dc.description.status | Peer reviewed | en |
dc.date.embargoedUntil | 2019-02-15 | |
dc.identifier.url | https://arxiv.org/abs/1606.05583v4 | en |
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