The dynamical look at the subsets of a group
We consider the action of a group $G$ on the family $\mathcal{P}(G)$ of all subsets of $G$ by the right shifts $A\mapsto Ag$ and give the dynamical characterizations of thin, $n$-thin, sparse and scattered subsets. For $n\in\mathbb{N}$, a subset $A$ of a group $G$ is called $n$-thin if $g_0A\cap\do...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Universitat Politècnica de València
2015-10-01
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| Series: | Applied General Topology |
| Subjects: | |
| Online Access: | http://polipapers.upv.es/index.php/AGT/article/view/3584 |
| _version_ | 1819056265447342080 |
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| author | Igor V. Protasov Sergii Slobodianiuk |
| author_facet | Igor V. Protasov Sergii Slobodianiuk |
| author_sort | Igor V. Protasov |
| collection | DOAJ |
| description | We consider the action of a group $G$ on the family $\mathcal{P}(G)$ of all subsets of $G$ by the right shifts $A\mapsto Ag$ and give the dynamical characterizations of thin, $n$-thin, sparse and scattered subsets.
For $n\in\mathbb{N}$, a subset $A$ of a group $G$ is called $n$-thin if $g_0A\cap\dots\cap g_nA$ is finite for all distinct $g_0,\dots,g_n\in G$.
Each $n$-thin subset of a group of cardinality $\aleph_0$ can be partitioned into $n$ $1$-thin subsets but there is a $2$-thin subset in some Abelian group of cardinality $\aleph_2$ which cannot be partitioned into two $1$-thin subsets. We eliminate the gap between $\aleph_0$ and $\aleph_2$ proving that each $n$-thin subset of an Abelian group of cardinality $\aleph_1$ can be partitioned into $n$ $1$-thin subsets. |
| first_indexed | 2024-12-21T13:20:40Z |
| format | Article |
| id | doaj.art-baca831a682d42398815dfdf415683ba |
| institution | Directory Open Access Journal |
| issn | 1576-9402 1989-4147 |
| language | English |
| last_indexed | 2024-12-21T13:20:40Z |
| publishDate | 2015-10-01 |
| publisher | Universitat Politècnica de València |
| record_format | Article |
| series | Applied General Topology |
| spelling | doaj.art-baca831a682d42398815dfdf415683ba2022-12-21T19:02:35ZengUniversitat Politècnica de ValènciaApplied General Topology1576-94021989-41472015-10-0116221722410.4995/agt.2015.35843064The dynamical look at the subsets of a groupIgor V. Protasov0Sergii SlobodianiukKyiv UniversityWe consider the action of a group $G$ on the family $\mathcal{P}(G)$ of all subsets of $G$ by the right shifts $A\mapsto Ag$ and give the dynamical characterizations of thin, $n$-thin, sparse and scattered subsets. For $n\in\mathbb{N}$, a subset $A$ of a group $G$ is called $n$-thin if $g_0A\cap\dots\cap g_nA$ is finite for all distinct $g_0,\dots,g_n\in G$. Each $n$-thin subset of a group of cardinality $\aleph_0$ can be partitioned into $n$ $1$-thin subsets but there is a $2$-thin subset in some Abelian group of cardinality $\aleph_2$ which cannot be partitioned into two $1$-thin subsets. We eliminate the gap between $\aleph_0$ and $\aleph_2$ proving that each $n$-thin subset of an Abelian group of cardinality $\aleph_1$ can be partitioned into $n$ $1$-thin subsets.http://polipapers.upv.es/index.php/AGT/article/view/3584Thinsparse and scatterad subsets of a grouprecurrent pointchromatic number of a graph. |
| spellingShingle | Igor V. Protasov Sergii Slobodianiuk The dynamical look at the subsets of a group Applied General Topology Thin sparse and scatterad subsets of a group recurrent point chromatic number of a graph. |
| title | The dynamical look at the subsets of a group |
| title_full | The dynamical look at the subsets of a group |
| title_fullStr | The dynamical look at the subsets of a group |
| title_full_unstemmed | The dynamical look at the subsets of a group |
| title_short | The dynamical look at the subsets of a group |
| title_sort | dynamical look at the subsets of a group |
| topic | Thin sparse and scatterad subsets of a group recurrent point chromatic number of a graph. |
| url | http://polipapers.upv.es/index.php/AGT/article/view/3584 |
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