Topological Representation of lattice homomorphisms

Wallman proved that if L is a distributive lattice with 0 and 1, then there is a T 1 -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 Lat 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 W : Lat → Comp.
When restricted to the subcategory of Boolean lattices this functor coin-
cides 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.