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

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.