Universality of the lattice of transformation monoids

Abstract

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