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 \emph{full transformation semigroup} $\Omega^\Omega$, which consists of all mappings on the countable infinite set $\Omega$. Gavrilov showed that there are five maximal subsemigroups of $\Omega^\Omega$ containing the symmetric group $\sym(\Omega)$. The portion of the subsemigroup lattice of $\Omega^\Omega$ which we describe is that between the intersection of these five maximal subsemigroups and $\Omega^\Omega$. We prove that there are only 38 subsemigroups in this interval, in contrast to the $2^{2^{\aleph_0}}$ subsemigroups between $\sym(\Omega)$ and $\Omega^\Omega$.