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 Hclasses. 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 wellknown 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 0matrix 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 Jclasses.
Original language  English 

Pages (fromto)  559596 
Number of pages  38 
Journal  Journal of Algebra 
Volume  505 
Early online date  15 Feb 2018 
DOIs  
Publication status  Published  1 Jul 2018 
Keywords
 Algorithms
 Computational group theory
 Computational semigroup theory
 Maximal subsemigroups
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GAP Package Semigroups
Mitchell, J. D. (Creator), Burrell, S. A. (Contributor), Delgado, M. (Contributor), East, J. (Contributor), EgriNagy, A. (Contributor), Ham, N. (Contributor), Jonusas, J. (Contributor), Pfeiffer, M. J. (Creator), Russell, C. (Contributor), Steinberg, B. (Contributor), Smith, F. L. (Contributor), Smith, J. (Contributor), Young, M. (Contributor) & Wilson, W. A. (Contributor), GitHub, 29 May 2018
https://semigroups.github.io/Semigroups
Dataset

GAP Package Digraphs
De Beule, J. (Creator), Jonusas, J. (Creator), Mitchell, J. D. (Creator), Young, M. (Creator) & Wilson, W. A. (Creator), GitHub, 26 Apr 2018
https://digraphs.github.io/Digraphs
Dataset