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 \emph{full transformation semigroup} ${\mathrm{\Omega }}^{\mathrm{\Omega }}$, which consists of all mappings on the countable infinite set $\mathrm{\Omega }$. Gavrilov showed that there are five maximal subsemigroups of ${\mathrm{\Omega }}^{\mathrm{\Omega }}$ containing the symmetric group $\text{sym}(\mathrm{\Omega })$. The portion of the subsemigroup lattice of ${\mathrm{\Omega }}^{\mathrm{\Omega }}$ which we describe is that between the intersection of these five maximal subsemigroups and ${\mathrm{\Omega }}^{\mathrm{\Omega }}$. We prove that there are only 38 subsemigroups in this interval, in contrast to the $2^{2^{{\mathrm{\aleph }}_0}}$ subsemigroups between $\text{sym}(\mathrm{\Omega })$ and ${\mathrm{\Omega }}^{\mathrm{\Omega }}$.