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Maximal subsemigroups of the semigroup of all mappings on an infinite set

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Date
01/03/2015
Author
East, J.
Mitchell, James David
Péresse, Y.
Keywords
QA Mathematics
T-NDAS
BDC
R2C
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Abstract
In this paper we classify the maximal subsemigroups of the full transformation semigroup ΩΩ, which consists of all mappings on the infinite set Ω, containing certain subgroups of the symmetric group Sym (Ω) on Ω. In 1965 Gavrilov showed that there are five maximal subsemigroups of ΩΩ containing Sym (Ω) when Ω is countable, and in 2005 Pinsker extended Gavrilov's result to sets of arbitrary cardinality. We classify the maximal subsemigroups of ΩΩ on a set Ω of arbitrary infinite cardinality containing one of the following subgroups of Sym (Ω): the pointwise stabiliser of a non-empty finite subset of Ω, the stabiliser of an ultrafilter on Ω, or the stabiliser of a partition of Ω into finitely many subsets of equal cardinality. If G is any of these subgroups, then we deduce a characterisation of the mappings f, g ∈ ΩΩ such that the semigroup generated by G ∪ {f, g} equals ΩΩ.
Citation
East , J , Mitchell , J D & Péresse , Y 2015 , ' Maximal subsemigroups of the semigroup of all mappings on an infinite set ' , Transactions of the American Mathematical Society , vol. 367 , no. 3 , pp. 1911-1944 . https://doi.org/10.1090/S0002-9947-2014-06110-2
Publication
Transactions of the American Mathematical Society
Status
Peer reviewed
DOI
https://doi.org/10.1090/S0002-9947-2014-06110-2
ISSN
0002-9947
Type
Journal article
Rights
© 2014. American Mathematical Society. First published in Transactions of the American Mathematical Society 2014.
Collections
  • Centre for Interdisciplinary Research in Computational Algebra (CIRCA) Research
  • Pure Mathematics Research
  • University of St Andrews Research
URI
http://hdl.handle.net/10023/5793

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