Universality of the lattice of transformation monoids

Abstract

The set of all transformation monoids on a fixed set of infinite cardinality \lambda, equipped with the order of inclusion, forms a complete algebraic lattice Mon(\lambda) with 2^{\lambda} compact elements. We show that this lattice is universal with respect to closed sublattices, i.e., the closed sublattices of Mon(\lambda) are, up to isomorphism, precisely the complete algebraic lattices with at most 2^{\lambda} compact elements.