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

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Original languageEnglish
Number of pages34
Pages (from-to)1911-1944
JournalTransactions of the American Mathematical Society
Journal publication date18 Nov 2014
Volume367
Issue3
DOIs
Publication statusPublished - 18 Nov 2014

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.

Notes

This is the accepted manuscript of the following article: J. East, J. D. Mitchell and Y. Péresse, “Maximal subsemigroupsof the semigroup of all mappings on an infinite set”, Transactions of the American Mathematical Society, Vol. 367(3), November 2014. The final published version is available online at: http://www.ams.org/journals/tran/2015-367-03/S0002-9947-2014-06110-2/ © Copyright 2014 American Mathematical Society

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