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Maximal subsemigroups of finite transformation and diagram monoids

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East_2018_JA_Maximalsubsemigroups_AAM.pdf (442.6Kb)
Date
15/06/2018
Author
East, James
Kumar, Jitender
Mitchell, James D.
Wilson, Wilf A.
Keywords
Maximal subsemigroups
Transformation semigroup
Diagram monoid
Permutation groups
Maximal subgroups
Graph
Maximal independent set
Partition monoid
QA Mathematics
T-NDAS
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Abstract
We describe and count the maximal subsemigroups of many well-known transformation monoids, and diagram monoids, using a new unified framework that allows the treatment of several classes of monoids simultaneously. The problem of determining the maximal subsemigroups of a finite monoid of transformations has been extensively studied in the literature. To our knowledge, every existing result in the literature is a special case of the approach we present. In particular, our technique can be used to determine the maximal subsemigroups of the full spectrum of monoids of order- or orientation-preserving transformations and partial permutations considered by I. Dimitrova, V. H. Fernandes, and co-authors. We only present details for the transformation monoids whose maximal subsemigroups were not previously known; and for certain diagram monoids, such as the partition, Brauer, Jones, and Motzkin monoids. The technique we present is based on a specialised version of an algorithm for determining the maximal subsemigroups of any finite semigroup, developed by the third and fourth authors, and available in the Semigroups package for GAP, an open source computer algebra system. This allows us to concisely present the descriptions of the maximal subsemigroups, and to clearly see their common features.
Citation
East , J , Kumar , J , Mitchell , J D & Wilson , W A 2018 , ' Maximal subsemigroups of finite transformation and diagram monoids ' , Journal of Algebra , vol. 504 , pp. 176-216 . https://doi.org/10.1016/j.jalgebra.2018.01.048
Publication
Journal of Algebra
Status
Peer reviewed
DOI
https://doi.org/10.1016/j.jalgebra.2018.01.048
ISSN
0021-8693
Type
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 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.2018.01.048
Description
The first author gratefully acknowledges the support of the Glasgow Learning, Teaching, and Research Fund in partially funding his visit to the third author in July, 2014. The second author wishes to acknowledge the support of research initiation grant [0076|2016] provided by BITS Pilani, Pilani. The fourth 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
URI
http://hdl.handle.net/10023/17110

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