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 $\sym(Ω)$. 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 $\sym(Ω)$ and ${\Omega }^{\Omega }$.

12 pages, slightly updated according to referee's comments