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

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 .