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dc.contributor.authorDonoven, C. R.
dc.contributor.authorMitchell, J. D.
dc.contributor.authorWilson, W. A.
dc.date.accessioned2019-02-15T00:34:30Z
dc.date.available2019-02-15T00:34:30Z
dc.date.issued2018-07-01
dc.identifier247458992
dc.identifier49ec975c-2b16-4912-9f95-d814a318372b
dc.identifier85042216783
dc.identifier.citationDonoven , 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.2018.01.044en
dc.identifier.issn0021-8693
dc.identifier.otherArXiv: http://arxiv.org/abs/1606.05583v1
dc.identifier.otherORCID: /0000-0002-5489-1617/work/73700795
dc.identifier.otherORCID: /0000-0002-3382-9603/work/85855348
dc.identifier.urihttps://hdl.handle.net/10023/17072
dc.descriptionThe 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.abstractA 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.extent38
dc.format.extent679060
dc.language.isoeng
dc.relation.ispartofJournal of Algebraen
dc.subjectAlgorithmsen
dc.subjectComputational group theoryen
dc.subjectComputational semigroup theoryen
dc.subjectMaximal subsemigroupsen
dc.subjectQA Mathematicsen
dc.subjectAlgebra and Number Theoryen
dc.subjectDASen
dc.subject.lccQAen
dc.titleComputing maximal subsemigroups of a finite semigroupen
dc.typeJournal articleen
dc.contributor.institutionUniversity of St Andrews. Pure Mathematicsen
dc.contributor.institutionUniversity of St Andrews. Centre for Interdisciplinary Research in Computational Algebraen
dc.identifier.doi10.1016/j.jalgebra.2018.01.044
dc.description.statusPeer revieweden
dc.date.embargoedUntil2019-02-15
dc.identifier.urlhttps://arxiv.org/abs/1606.05583v4en


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