On Maps between Stone-Cech Compactifications Induced by Lattice Homomorphisms

Broverman has shown that if X and Y are Tychonoff spaces and t : Z(Y) -> Z(X) is a lattice homomorphism between the lattices of their zero-sets, then there is a continuous map tau : beta X -> beta Y induced by t. In this note we expound this idea and supplement Broverman's results by first showing that this phenomenon holds in the category of completely regular frames. Among results we obtain, which were not considered by Broverman, are necessary and sufficient conditions (in terms of properties of the map t) for the induced map tau to be (i) the inclusion of a subspace, (ii) surjective, and (iii) irreducible. We show that if X and Y are pseudocompact then t pulls back z-ultrafilters to z-ultrafilters if and only if cl(beta X) t(Z) = tau(<-)[cl(beta Y)Z] for every Z is an element of Z(Y) if and only if t is sigma-homomorphism.