The Largest Subsemilattices of the Semigroup of Transformations on a Finite Set

TL;DR

This paper determines the maximum size of subsemilattices within the semigroup of all transformations on a finite set and characterizes those of maximum size, revealing their structure and count.

Contribution

It proves the maximum size of subsemilattices in T(X) and classifies all subsemilattices reaching this size, linking them to idempotent semilattices.

Findings

Maximum subsemilattice size is 2^{n-1}

Exactly n subsemilattices reach this size

Each maximum subsemilattice is isomorphic to the idempotent semilattice of the symmetric inverse semigroup

Abstract

Let T(X) be the semigroup of full transformations on a finite set X with n elements. We prove that every subsemilattice of T(X) has at most 2^{n-1} elements and that there are precisely n subsemilattices of size exactly 2^{n-1}, each isomorphic to the semilattice of idempotents of the symmetric inverse semigroup on a set with n-1 elements.