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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

Citations:

Zbl 1095.13025
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References:

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