Show simple item record

Files in this item


Item metadata

dc.contributor.authorDonoven, C. R.
dc.contributor.authorMitchell, J. D.
dc.contributor.authorWilson, W. A.
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 .
dc.identifier.otherPURE: 247458992
dc.identifier.otherPURE UUID: 49ec975c-2b16-4912-9f95-d814a318372b
dc.identifier.otherScopus: 85042216783
dc.identifier.otherORCID: /0000-0002-5489-1617/work/73700795
dc.identifier.otherORCID: /0000-0002-3382-9603/work/85855348
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.relation.ispartofJournal of Algebraen
dc.rights© 2018, Elsevier, Inc. This work has been made available online in accordance with the publisher’s policies. This is the author created accepted version manuscript following peer review and as such may differ slightly from the final published version. The final published version of this work is available at
dc.subjectComputational group theoryen
dc.subjectComputational semigroup theoryen
dc.subjectMaximal subsemigroupsen
dc.subjectQA Mathematicsen
dc.subjectAlgebra and Number Theoryen
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.description.statusPeer revieweden

This item appears in the following Collection(s)

Show simple item record