Topological representation of lattice homomorphisms

Wallman [13] proved that if L is a distributive lattice with 0 and 1, then there is a T1-space with a base (for closed subsets) being a homomorphic image of L. We show that this theorem can be extended over homomorphisms. More precisely: if NLat denotes the category of normal and distributive lattices with 0 and 1 and homomorphisms, and Comp denotes the category of compact Hausdorff spaces and continuous mappings, then there exists a contravariant functor Ult:NLat→Comp. When restricted to the subcategory of Boolean lattices this functor coincides with a well-known Stone functor which realizes the Stone Duality. The functor W carries monomorphisms into surjections. However, it does not carry epimorphisms into injections. The last property makes a difference with the Stone functor. Some applications to topological constructions are given as well.