Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 

 

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


Authors: J. East, J. D. Mitchell and Y. Péresse
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
Published electronically: November 18, 2014
MathSciNet review: 3286503
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 20B30, 20B35, 20M20

Retrieve articles in all journals with MSC (2010): 20B30, 20B35, 20M20


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

J. D. Mitchell
Affiliation: Mathematics Institute, University of Saint Andrews, St. Andrews, KY16 9SS, United Kingdom

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

DOI: https://doi.org/10.1090/S0002-9947-2014-06110-2
Received by editor(s): September 18, 2012
Received by editor(s) in revised form: February 11, 2013
Published electronically: November 18, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.