Computing finite semigroups
Abstract
Using a variant of Schreier's Theorem, and the theory of Green's relations, we show how to reduce the computation of an arbitrary subsemigroup of a finite regular semigroup to that of certain associated subgroups. Examples of semigroups to which these results apply include many important classes: transformation semigroups, partial permutation semigroups and inverse semigroups, partition monoids, matrix semigroups, and subsemigroups of finite regular Rees matrix and 0-matrix semigroups over groups. For any subsemigroup of such a semigroup, it is possible to, among other things, efficiently compute its size and Green's relations, test membership, factorize elements over the generators, find the semigroup generated by the given subsemigroup and any collection of additional elements, calculate the partial order of the D-classes, test regularity, and determine the idempotents. This is achieved by representing the given subsemigroup without exhaustively enumerating its elements. It is also possible to compute the Green's classes of an element of such a subsemigroup without determining the global structure of the semigroup.
Citation
East , J , Egri-Nagy , A , Mitchell , J D & Péresse , Y 2019 , ' Computing finite semigroups ' , Journal of Symbolic Computation , vol. 92 , pp. 110-155 . https://doi.org/10.1016/j.jsc.2018.01.002
Publication
Journal of Symbolic Computation
Status
Peer reviewed
ISSN
0747-7171Type
Journal article
Rights
© 2018, Elsevier Ltd. 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.jsc.2018.01.002
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