When ${\rm Min}(G)^{-1}$ has a clopen $øldpi$-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. M...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Institute of Mathematics of the Czech Academy of Science
2021-04-01
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| Series: | Mathematica Bohemica |
| Subjects: | |
| Online Access: | http://mb.math.cas.cz/full/146/1/mb146_1_6.pdf |
| 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 $øldpi$-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. |
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| ISSN: | 0862-7959 2464-7136 |