When Max_d(G) is zero-dimensional

TL;DR

This paper investigates conditions under which the space of maximal d-subgroups of certain lattice-ordered groups, especially W-objects, is zero-dimensional, extending previous classifications of related subgroup spaces.

Contribution

It provides a classification of when the space of maximal d-subgroups of W-objects is zero-dimensional, generalizing known results for $C(X)$ functions.

Findings

Characterization of zero-dimensional maximal d-subgroup spaces

Extension of classification from $C(X)$ to W-objects

Identification of conditions for clopen $ ext{π}$-bases in subgroup spaces

Abstract

This article is a continuation of [6] where a classification of when the space of minimal prime subgroups of a given lattice-ordered group equipped with the inverse topology has a clopen $\pi$-base. For nice $\ell$-groups, (e.g. W-objects) this occurs precisely when the space of maximal $d$-subgroups (qua the hull kernel topology) has a clopen $\pi$-base. It occurred to us that presently there is no classification of when the space of maximal $d$-subgroups of a W-object is zero-dimensional, except for the case of the $C(X)$, the real-valued continuous functions on a topological space $X$, considered in [5].