2011/04/11 by James East, East, J., James D. Mitchell +3 · 1 citation
Computer Science · Mathematics · #20M20 #Advanced Topology and Set Theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Rings and Algebras (math.RA) #semigroups and automata theory
paper · doi:10.48550/arxiv.1104.2011
openalex publication_date 2011/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we classify the maximal subsemigroups of the full transformation semigroup ΩΩ, 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 ΩΩ 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 ΩΩ 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∈ ΩΩ such that the semigroup generated by G∪ \f,g\ equals ΩΩ.