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

TL;DR

This paper classifies the maximal subsemigroups of the full transformation semigroup on an infinite set that contain specific subgroups, extending previous results to arbitrary infinite cardinalities and providing characterizations of generated semigroups.

Contribution

It generalizes the classification of maximal subsemigroups containing certain subgroups to all infinite cardinalities and characterizes generating pairs for the entire semigroup.

Findings

Classified maximal subsemigroups containing specific subgroups for any infinite set.

Extended Gavrilov and Pinsker's results to arbitrary infinite cardinalities.

Provided criteria for when two mappings generate the full transformation semigroup.

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 $\Omega^\Omega$.