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

J. East, James David Mitchell, Y. Péresse

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)

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.
Original languageEnglish
Pages (from-to)1911-1944
Number of pages34
JournalTransactions of the American Mathematical Society
Volume367
Issue number3
Early online date18 Nov 2014
DOIs
Publication statusPublished - 1 Mar 2015

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