Partitions of groups into sparse subsets
A subset \(A\) of a group \(G\) is called sparse if, for every infinite subset \(X\) of \(G\), there exists a finite subset \(F\subset X\), such that \(\bigcap_{x\in F} xA\) is finite. We denote by \(\eta(G)\) the minimal cardinal such that \(G\) can be partitioned in \(\eta(G)\) sparse subsets. If...
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| Дата: | 2018 |
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| Формат: | Стаття |
| Мова: | Англійська |
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Lugansk National Taras Shevchenko University
2018
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/695 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543265285734400 |
|---|---|
| author | Protasov, Igor |
| author_facet | Protasov, Igor |
| author_sort | Protasov, Igor |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-04-04T09:42:12Z |
| description | A subset \(A\) of a group \(G\) is called sparse if, for every infinite subset \(X\) of \(G\), there exists a finite subset \(F\subset X\), such that \(\bigcap_{x\in F} xA\) is finite. We denote by \(\eta(G)\) the minimal cardinal such that \(G\) can be partitioned in \(\eta(G)\) sparse subsets. If \(|G| > (\kappa^+)^{\aleph_0}\) then \(\eta(G) > \kappa\), if \(|G| \leqslant \kappa^+\) then \(\eta(G) \leqslant \kappa\). We show also that \(cov(A) \geqslant cf|G|\) for each sparse subset \(A\) of an infinite group \(G\), where \(cov(A)=\min\{|X|: G = XA\}.\) |
| first_indexed | 2025-12-02T15:32:08Z |
| format | Article |
| id | admjournalluguniveduua-article-695 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:32:08Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-6952018-04-04T09:42:12Z Partitions of groups into sparse subsets Protasov, Igor partition of a group, sparse subset of a group 03E75, 20F99, 20K99 A subset \(A\) of a group \(G\) is called sparse if, for every infinite subset \(X\) of \(G\), there exists a finite subset \(F\subset X\), such that \(\bigcap_{x\in F} xA\) is finite. We denote by \(\eta(G)\) the minimal cardinal such that \(G\) can be partitioned in \(\eta(G)\) sparse subsets. If \(|G| > (\kappa^+)^{\aleph_0}\) then \(\eta(G) > \kappa\), if \(|G| \leqslant \kappa^+\) then \(\eta(G) \leqslant \kappa\). We show also that \(cov(A) \geqslant cf|G|\) for each sparse subset \(A\) of an infinite group \(G\), where \(cov(A)=\min\{|X|: G = XA\}.\) Lugansk National Taras Shevchenko University 2018-04-04 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/695 Algebra and Discrete Mathematics; Vol 13, No 1 (2012) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/695/228 Copyright (c) 2018 Algebra and Discrete Mathematics |
| spellingShingle | partition of a group sparse subset of a group 03E75 20F99 20K99 Protasov, Igor Partitions of groups into sparse subsets |
| title | Partitions of groups into sparse subsets |
| title_full | Partitions of groups into sparse subsets |
| title_fullStr | Partitions of groups into sparse subsets |
| title_full_unstemmed | Partitions of groups into sparse subsets |
| title_short | Partitions of groups into sparse subsets |
| title_sort | partitions of groups into sparse subsets |
| topic | partition of a group sparse subset of a group 03E75 20F99 20K99 |
| topic_facet | partition of a group sparse subset of a group 03E75 20F99 20K99 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/695 |
| work_keys_str_mv | AT protasovigor partitionsofgroupsintosparsesubsets |