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...
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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 |
| _version_ | 1818577253064245248 |
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| author | Ramiro Lafuente-Rodriguez Warren Wm. McGovern |
| author_facet | Ramiro Lafuente-Rodriguez Warren Wm. McGovern |
| author_sort | Ramiro Lafuente-Rodriguez |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-16T06:26:58Z |
| format | Article |
| id | doaj.art-25a70972a54944feb99bf6367e888b6e |
| institution | Directory Open Access Journal |
| issn | 0862-7959 2464-7136 |
| language | English |
| last_indexed | 2024-12-16T06:26:58Z |
| publishDate | 2021-04-01 |
| publisher | Institute of Mathematics of the Czech Academy of Science |
| record_format | Article |
| series | Mathematica Bohemica |
| spelling | doaj.art-25a70972a54944feb99bf6367e888b6e2022-12-21T22:40:58ZengInstitute of Mathematics of the Czech Academy of ScienceMathematica Bohemica0862-79592464-71362021-04-011461698910.21136/MB.2020.0114-18MB.2020.0114-18When ${\rm Min}(G)^{-1}$ has a clopen $øldpi$-baseRamiro Lafuente-RodriguezWarren Wm. McGovernIt 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.http://mb.math.cas.cz/full/146/1/mb146_1_6.pdf lattice-ordered group minimal prime subgroup maximal $d$-subgroup archimedean $l$-group $\bold{w}$ |
| spellingShingle | Ramiro Lafuente-Rodriguez Warren Wm. McGovern When ${\rm Min}(G)^{-1}$ has a clopen $øldpi$-base Mathematica Bohemica lattice-ordered group minimal prime subgroup maximal $d$-subgroup archimedean $l$-group $\bold{w}$ |
| title | When ${\rm Min}(G)^{-1}$ has a clopen $øldpi$-base |
| title_full | When ${\rm Min}(G)^{-1}$ has a clopen $øldpi$-base |
| title_fullStr | When ${\rm Min}(G)^{-1}$ has a clopen $øldpi$-base |
| title_full_unstemmed | When ${\rm Min}(G)^{-1}$ has a clopen $øldpi$-base |
| title_short | When ${\rm Min}(G)^{-1}$ has a clopen $øldpi$-base |
| title_sort | when rm min g 1 has a clopen oldpi base |
| topic | lattice-ordered group minimal prime subgroup maximal $d$-subgroup archimedean $l$-group $\bold{w}$ |
| url | http://mb.math.cas.cz/full/146/1/mb146_1_6.pdf |
| work_keys_str_mv | AT ramirolafuenterodriguez whenrmming1hasaclopenøldpibase AT warrenwmmcgovern whenrmming1hasaclopenøldpibase |