When Max_d(G) is zero-dimensional

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 $\mathrm{\ell }$-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].