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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| Datum: | 2018 |
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| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Lugansk National Taras Shevchenko University
2018
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| Schlagworte: | |
| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/695 |
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| Назва журналу: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Zusammenfassung: | 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 \(cov(A)=\min\{|X|: G = XA\}.\) |
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