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

East, J.; Mitchell, J.; Péresse, Y.

doi:10.1090/S0002-9947-2014-06110-2

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

Trans. Amer. Math. Soc. 367 (2015), 1911-1944 Request permission

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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