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

Abstract

In this paper we classify the maximal subsemigroups of the full transformation semigroup Ω Ω \Omega ^\Omega , which consists of all mappings on the infinite set Ω \Omega , containing certain subgroups of the symmetric group Sym ⁡ ( Ω ) \operatorname {Sym}(\Omega ) on Ω \Omega . In 1965 Gavrilov showed that there are five maximal subsemigroups of Ω Ω \Omega ^\Omega containing Sym ⁡ ( Ω ) \operatorname {Sym}(\Omega ) when Ω \Omega is countable, and in 2005 Pinsker extended Gavrilov’s result to sets of arbitrary cardinality. We classify the maximal subsemigroups of Ω Ω \Omega ^\Omega on a set Ω \Omega of arbitrary infinite cardinality containing one of the following subgroups of Sym ⁡ ( Ω ) \operatorname {Sym}(\Omega ) : the pointwise stabiliser of a non-empty finite subset of Ω \Omega , the stabiliser of an ultrafilter on Ω \Omega , or the stabiliser of a partition of Ω \Omega into finitely many subsets of equal cardinality. If G G is any of these subgroups, then we deduce a characterisation of the mappings f , g ∈ Ω Ω f,g\in \Omega ^\Omega such that the semigroup generated by G ∪ { f , g } G\cup \{f,g\} equals Ω Ω \Omega ^\Omega .