# Article

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Keywords:
lattice-ordered group; minimal prime subgroup; maximal $d$-subgroup; archimedean $l$-group; $\bold {W}$
Summary:
It is our aim to contribute to the flourishing collection of knowledge centered on the space of minimal prime subgroups of a given lattice-ordered group. Specifically, we are interested in the inverse topology. In general, this space is compact and $T_1$, but need not be Hausdorff. In 2006, W. Wm. McGovern showed that this space is a boolean space (i.e. a compact zero-dimensional and Hausdorff space) if and only if the $l$-group in question is weakly complemented. A slightly weaker topological property than having a base of clopen subsets is having a clopen $\pi$-base. Recall that a $\pi$-base is a collection of nonempty open subsets such that every nonempty open subset of the space contains a member of the $\pi$-base; obviously, a base is a $\pi$-base. In what follows we classify when the inverse topology on the space of prime subgroups has a clopen $\pi$-base.
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