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Computing maximal subsemigroups of a finite semigroup

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Date
01/07/2018
Author
Donoven, C. R.
Mitchell, J. D.
Wilson, W. A.
Keywords
Algorithms
Computational group theory
Computational semigroup theory
Maximal subsemigroups
QA Mathematics
Algebra and Number Theory
DAS
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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.
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
Publication
Journal of Algebra
Status
Peer reviewed
DOI
https://doi.org/10.1016/j.jalgebra.2018.01.044
ISSN
0021-8693
Type
Journal article
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 https://doi.org/10.1016/j.jalgebra.2018.01.044
Description
The third author wishes to acknowledge the support of his Carnegie Ph.D. Scholarship from the Carnegie Trust for the Universities of Scotland.
Collections
  • Centre for Interdisciplinary Research in Computational Algebra (CIRCA) Research
  • Pure Mathematics Research
  • University of St Andrews Research
URL
https://arxiv.org/abs/1606.05583v4
URI
http://hdl.handle.net/10023/17072

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