Geodesic
When ${\rm Min}(G)^{-1}$ has a clopen $\pi $-base
Mathematica Bohemica, Tome 146 (2021) no. 1, pp. 69-89
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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.
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.
DOI : 10.21136/MB.2020.0114-18
Classification : 06F15, 06F20
Keywords: lattice-ordered group; minimal prime subgroup; maximal $d$-subgroup; archimedean $l$-group; $\bold {W}$ 
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Lafuente-Rodriguez, Ramiro; McGovern, Warren Wm. When ${\rm Min}(G)^{-1}$ has a clopen $\pi $-base. Mathematica Bohemica, Tome 146 (2021) no. 1, pp. 69-89. doi: 10.21136/MB.2020.0114-18

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