A finite interval in the subsemigroup lattice of the full transformation monoid

Abstract

In this paper we describe a portion of the subsemigroup lattice of the full transformation semigroup Ω Ω , which consists of all mappings on the countable infinite set Ω. Gavrilov showed that there are five maximal subsemigroups of Ω Ω containing the symmetric group $\operatorname {Sym}(\varOmega )$ . The portion of the subsemigroup lattice of Ω Ω which we describe is that between the intersection of these five maximal subsemigroups and Ω Ω . We prove that there are only 38 subsemigroups in this interval, in contrast to the $2^{2^{\aleph_{0}}}$ subsemigroups between $\operatorname {Sym}(\varOmega )$ and Ω Ω .