Coarse selectors of groups
For a group \(G\), \(\mathcal{F}_G\) denotes the set of all non-empty finite subsets of \(G\). We extend the finitary coarse structure of \(G\) from \(G\times G\) to \(\mathcal{F}_G\times \mathcal{F}_G\) and say that a macro-uniform mapping \(f\colon \mathcal{F}_G \to \mathcal{F}_G\) (resp. \(f\colo...
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| Date: | 2023 |
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| Format: | Article |
| Language: | English |
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Lugansk National Taras Shevchenko University
2023
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| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2127 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543247850012672 |
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| author | Protasov, I. |
| author_facet | Protasov, I. |
| author_sort | Protasov, I. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2023-10-30T03:22:37Z |
| description | For a group \(G\), \(\mathcal{F}_G\) denotes the set of all non-empty finite subsets of \(G\). We extend the finitary coarse structure of \(G\) from \(G\times G\) to \(\mathcal{F}_G\times \mathcal{F}_G\) and say that a macro-uniform mapping \(f\colon \mathcal{F}_G \to \mathcal{F}_G\) (resp. \(f\colon [G]^2 \to G\)) is a finitary selector (resp. 2-selector) of \(G\) if \(f(A)\in A\) for each \(A\in \mathcal{F}_G\) (resp. \( A \in [G]^2 \)). We prove that a group \(G\) admits a finitary selector if and only if \(G\) admits a 2-selector and if and only if \(G\) is a finite extension of an infinite cyclic subgroup or \(G\) is countable and locally finite. We use this result to characterize groups admitting linear orders compatible with finitary coarse structures. |
| first_indexed | 2025-12-02T15:31:01Z |
| format | Article |
| id | admjournalluguniveduua-article-2127 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:31:01Z |
| publishDate | 2023 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-21272023-10-30T03:22:37Z Coarse selectors of groups Protasov, I. finitary coarse structure, Cayley graph, selector 20F69, 54C65 For a group \(G\), \(\mathcal{F}_G\) denotes the set of all non-empty finite subsets of \(G\). We extend the finitary coarse structure of \(G\) from \(G\times G\) to \(\mathcal{F}_G\times \mathcal{F}_G\) and say that a macro-uniform mapping \(f\colon \mathcal{F}_G \to \mathcal{F}_G\) (resp. \(f\colon [G]^2 \to G\)) is a finitary selector (resp. 2-selector) of \(G\) if \(f(A)\in A\) for each \(A\in \mathcal{F}_G\) (resp. \( A \in [G]^2 \)). We prove that a group \(G\) admits a finitary selector if and only if \(G\) admits a 2-selector and if and only if \(G\) is a finite extension of an infinite cyclic subgroup or \(G\) is countable and locally finite. We use this result to characterize groups admitting linear orders compatible with finitary coarse structures. Lugansk National Taras Shevchenko University 2023-10-12 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2127 10.12958/adm2127 Algebra and Discrete Mathematics; Vol 35, No 2 (2023) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2127/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2127/1093 Copyright (c) 2023 Algebra and Discrete Mathematics |
| spellingShingle | finitary coarse structure Cayley graph selector 20F69 54C65 Protasov, I. Coarse selectors of groups |
| title | Coarse selectors of groups |
| title_full | Coarse selectors of groups |
| title_fullStr | Coarse selectors of groups |
| title_full_unstemmed | Coarse selectors of groups |
| title_short | Coarse selectors of groups |
| title_sort | coarse selectors of groups |
| topic | finitary coarse structure Cayley graph selector 20F69 54C65 |
| topic_facet | finitary coarse structure Cayley graph selector 20F69 54C65 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2127 |
| work_keys_str_mv | AT protasovi coarseselectorsofgroups |