When ${\rm Min}(G)^{-1}$ has a clopen $øldpi$-base

Abstract

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 $øldpi$-base. Recall that a $øldpi$-base is a collection of nonempty open subsets such that every nonempty open subset of the space contains a member of the $øldpi$-base; obviously, a base is a $øldpi$-base. In what follows we classify when the inverse topology on the space of prime subgroups has a clopen $øldpi$-base.

Keywords

• lattice-ordered group
• minimal prime subgroup
• maximal $d$-subgroup
• archimedean $l$-group
• $\bold{w}$