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(\Omega) on $\Omega$. In 1965 Gavrilov showed that there are five maximal subsemigroups of $\Omega^\Omega$ containing \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 \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 $\Omega^\Omega$.Comment: Revised according to comments by the referee, 29 pages, 11 figures, to appear in Trans. American Mathematical Societ