On Maps between Stone--Čech Compactifications Induced by Lattice Homomorphisms

Themba Dube

Broverman has shown that if $X$ and $Y$ are Tychonoff spaces and $t\colon Z(Y)\to Z(X) $is a lattice homomorphism between the lattices of their zero-sets, then there is a continuous map $\tau\colon\beta X\to\beta Y$ 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 $\operatorname{cl}_{\beta X}t(Z)=\tau^{\leftarrow}[\operatorname{cl}_{\beta Y}Z]$ for every $Z\in Z(Y)$ if and only if t is $\sigma$-homomorphism.