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

In this paper we classify the maximal subsemigroups of the \emph{full transformation semigroup} $Ω^Ω$, 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 $Ω^Ω$ 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 $Ω^Ω$ 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 $f,g\in Ω^Ω$ such that the semigroup generated by $G\cup \{f,g\}$ equals $Ω^Ω$.