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

Abstract

In this paper we classify the maximal subsemigroups of the \emph{full transformation semigroup} ${\Omega }^{\Omega }$, which consists of all mappings on the infinite set $\Omega$, containing certain subgroups of the symmetric group $\sym(Ω)$ on $\Omega$. In 1965 Gavrilov showed that there are five maximal subsemigroups of ${\Omega }^{\Omega }$ containing $\sym(Ω)$ 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 $\sym(Ω)$: 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 ${\Omega }^{\Omega }$.

arXiv commentRevised according to comments by the referee, 29 pages, 11 figures, to appear in Trans. American Mathematical Society

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• Maximal Subsemigroups containing a particular semigroup