## When $$\mathrm{Min}(G)^{-1}$$ has a clopen $$\pi$$-base.(English)Zbl 07332743

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 [J. Pure Appl. Algebra 205, No. 2, 243–265 (2006; Zbl 1095.13025)], 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.

### MSC:

 54Gxx Peculiar topological spaces

Zbl 1095.13025
Full Text:

### References:

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