Maximal subsemigroups of the semigroup of all mappings on an infinite set
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- by J. East, J. D. Mitchell and Y. Péresse PDF
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Abstract:
In this paper we classify the maximal subsemigroups of the full transformation semigroup $\Omega ^\Omega$, which consists of all mappings on the infinite set $\Omega$, containing certain subgroups of the symmetric group $\operatorname {Sym}(\Omega )$ on $\Omega$. In 1965 Gavrilov showed that there are five maximal subsemigroups of $\Omega ^\Omega$ containing $\operatorname {Sym}(\Omega )$ when $\Omega$ is countable, and in 2005 Pinsker extended Gavrilov’s result to sets of arbitrary cardinality.
We classify the maximal subsemigroups of $\Omega ^\Omega$ on a set $\Omega$ of arbitrary infinite cardinality containing one of the following subgroups of $\operatorname {Sym}(\Omega )$: the pointwise stabiliser of a non-empty finite subset of $\Omega$, the stabiliser of an ultrafilter on $\Omega$, or the stabiliser of a partition of $\Omega$ into finitely many subsets of equal cardinality. If $G$ is any of these subgroups, then we deduce a characterisation of the mappings $f,g\in \Omega ^\Omega$ such that the semigroup generated by $G\cup \{f,g\}$ equals $\Omega ^\Omega$.
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Additional Information
- J. East
- Affiliation: Centre for Research in Mathematics, School of Computing, Engineering and Mathematics, University of Western Sydney, Penrith, New South Wales 2751, Australia
- MR Author ID: 770366
- ORCID: 0000-0001-6112-9754
- J. D. Mitchell
- Affiliation: Mathematics Institute, University of Saint Andrews, St. Andrews, KY16 9SS, United Kingdom
- MR Author ID: 691066
- Y. Péresse
- Affiliation: Mathematics Institute, University of Saint Andrews, St. Andrews, KY16 9SS, United Kingdom
- Address at time of publication: School of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield AL10 9AB, United Kingdom
- Email: y.peresse@herts.ac.uk
- Received by editor(s): September 18, 2012
- Received by editor(s) in revised form: February 11, 2013
- Published electronically: November 18, 2014
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 1911-1944
- MSC (2010): Primary 20B30, 20B35, 20M20
- DOI: https://doi.org/10.1090/S0002-9947-2014-06110-2
- MathSciNet review: 3286503