Algebra in the Stone-\(\check{C}\)ech compactification: applications to topologies on groups
For every discrete group \(G\), the Stone-\(\check{C}\)ech compactification \(\beta G\) of \(G\) has a natural structure of compact right topological semigroup. Assume that \(G\) is endowed with some left invariant topology \(\Im\) and let \(\overline{\tau}\) be the set of all ultrafilters on \(G\)...
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| Date: | 2018 |
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| Format: | Article |
| Language: | English |
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Lugansk National Taras Shevchenko University
2018
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| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/771 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543436142804992 |
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| author | Protasov, I. V. |
| author_facet | Protasov, I. V. |
| author_sort | Protasov, I. V. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-04-04T08:31:48Z |
| description | For every discrete group \(G\), the Stone-\(\check{C}\)ech compactification \(\beta G\) of \(G\) has a natural structure of compact right topological semigroup. Assume that \(G\) is endowed with some left invariant topology \(\Im\) and let \(\overline{\tau}\) be the set of all ultrafilters on \(G\) converging to the unit of \(G\) in \(\Im\). Then \(\overline{\tau}\) is a closed subsemigroup of \(\beta G\). We survey the results clarifying the interplays between the algebraic properties of \(\overline{\tau}\) and the topological properties of \((G,\Im)\) and apply these results to solve some open problems in the topological group theory.The paper consists of 13 sections: Filters on groups, Semigroup of ultrafilters, Ideals, Idempotents, Equations, Continuity in \(\beta G\) and \(G^*\), Ramsey-like ultrafilters, Maximality, Refinements, Resolvability, Potential compactness and ultraranks, Selected open questions. |
| first_indexed | 2026-02-08T07:59:40Z |
| format | Article |
| id | admjournalluguniveduua-article-771 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2026-02-08T07:59:40Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-7712018-04-04T08:31:48Z Algebra in the Stone-\(\check{C}\)ech compactification: applications to topologies on groups Protasov, I. V. Stone-\(\check{C}\)ech compactification, product of ultrafilters, idempotents, ideals, maximality, resolvability, extremal disconnectedness 22A05, 22A15, 22A20, 05A18, 54A35, 54D80 For every discrete group \(G\), the Stone-\(\check{C}\)ech compactification \(\beta G\) of \(G\) has a natural structure of compact right topological semigroup. Assume that \(G\) is endowed with some left invariant topology \(\Im\) and let \(\overline{\tau}\) be the set of all ultrafilters on \(G\) converging to the unit of \(G\) in \(\Im\). Then \(\overline{\tau}\) is a closed subsemigroup of \(\beta G\). We survey the results clarifying the interplays between the algebraic properties of \(\overline{\tau}\) and the topological properties of \((G,\Im)\) and apply these results to solve some open problems in the topological group theory.The paper consists of 13 sections: Filters on groups, Semigroup of ultrafilters, Ideals, Idempotents, Equations, Continuity in \(\beta G\) and \(G^*\), Ramsey-like ultrafilters, Maximality, Refinements, Resolvability, Potential compactness and ultraranks, Selected open questions. 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/771 Algebra and Discrete Mathematics; Vol 8, No 1 (2009) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/771/301 Copyright (c) 2018 Algebra and Discrete Mathematics |
| spellingShingle | Stone-\(\check{C}\)ech compactification product of ultrafilters idempotents ideals maximality resolvability extremal disconnectedness 22A05 22A15 22A20 05A18 54A35 54D80 Protasov, I. V. Algebra in the Stone-\(\check{C}\)ech compactification: applications to topologies on groups |
| title | Algebra in the Stone-\(\check{C}\)ech compactification: applications to topologies on groups |
| title_full | Algebra in the Stone-\(\check{C}\)ech compactification: applications to topologies on groups |
| title_fullStr | Algebra in the Stone-\(\check{C}\)ech compactification: applications to topologies on groups |
| title_full_unstemmed | Algebra in the Stone-\(\check{C}\)ech compactification: applications to topologies on groups |
| title_short | Algebra in the Stone-\(\check{C}\)ech compactification: applications to topologies on groups |
| title_sort | algebra in the stone-\(\check{c}\)ech compactification: applications to topologies on groups |
| topic | Stone-\(\check{C}\)ech compactification product of ultrafilters idempotents ideals maximality resolvability extremal disconnectedness 22A05 22A15 22A20 05A18 54A35 54D80 |
| topic_facet | Stone-\(\check{C}\)ech compactification product of ultrafilters idempotents ideals maximality resolvability extremal disconnectedness 22A05 22A15 22A20 05A18 54A35 54D80 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/771 |
| work_keys_str_mv | AT protasoviv algebrainthestonecheckcechcompactificationapplicationstotopologiesongroups |