| Title:
|
Productivity of the Zariski topology on groups (English) |
| Author:
|
Dikranjan, D. |
| Author:
|
Toller, D. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
54 |
| Issue:
|
2 |
| Year:
|
2013 |
| Pages:
|
219-237 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper investigates the productivity of the Zariski topology $\mathfrak Z_G$ of a group $G$. If $\mathcal G = \{G_i\mid i\in I\}$ is a family of groups, and $G = \prod_{i\in I}G_i$ is their direct product, we prove that $\mathfrak Z_G\subseteq \prod_{i\in I}\mathfrak Z_{G_i}$. This inclusion can be proper in general, and we describe the doubletons $\mathcal G = \{G_1,G_2\}$ of abelian groups, for which the converse inclusion holds as well, i.e., $\mathfrak Z_G = \mathfrak Z_{G_1}\times \mathfrak Z_{G_2}$. If $e_2\in G_2$ is the identity element of a group $G_2$, we also describe the class $\Delta$ of groups $G_2$ such that $G_1\times \{e_{2}\}$ is an elementary algebraic subset of ${G_1\times G_2}$ for every group $G_1$. We show among others, that $\Delta$ is stable under taking finite products and arbitrary powers and we describe the direct products that belong to $\Delta$. In particular, $\Delta$ contains arbitrary direct products of free non-abelian groups. (English) |
| Keyword:
|
Zariski topology |
| Keyword:
|
(elementary, additively) algebraic subset |
| Keyword:
|
$\delta$-word |
| Keyword:
|
universal word |
| Keyword:
|
verbal function |
| Keyword:
|
(semi) $\mathfrak Z$-productive pair of groups |
| Keyword:
|
direct product |
| MSC:
|
20E22 |
| MSC:
|
20F70 |
| MSC:
|
20K25 |
| MSC:
|
20K45 |
| MSC:
|
22A05 |
| MSC:
|
57M07 |
| idZBL:
|
Zbl 1274.20038 |
| idMR:
|
MR3067705 |
| . |
| Date available:
|
2013-06-25T12:51:53Z |
| Last updated:
|
2015-07-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143271 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
|
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| . |
