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

TL;DR

This paper characterizes a specific finite interval within the subsemigroup lattice of the full transformation monoid on a countably infinite set, revealing a surprisingly small number of subsemigroups compared to the total possible.

Contribution

It identifies and describes a finite sublattice interval in the otherwise vast lattice of subsemigroups of the full transformation monoid.

Findings

38 subsemigroups in the interval, significantly fewer than the total possible.

The interval is between the intersection of five maximal subsemigroups and the full monoid.

Contrasts with the exponential number of subsemigroups in the entire lattice.

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 $\Omega^\Omega$.