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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| Дата: | 2018 |
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| Формат: | Стаття |
| Мова: | English |
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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/771 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua: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 |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
Stone-\(\check{C}\)ech compactification product of ultrafilters idempotents ideals maximality resolvability extremal disconnectedness 22A05 22A15 22A20 05A18 54A35 54D80 |
| 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 |
| topic_facet |
Stone-\(\check{C}\)ech compactification product of ultrafilters idempotents ideals maximality resolvability extremal disconnectedness 22A05 22A15 22A20 05A18 54A35 54D80 |
| format |
Article |
| author |
Protasov, I. V. |
| author_facet |
Protasov, I. V. |
| author_sort |
Protasov, I. V. |
| title |
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_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_sort |
algebra in the stone-\(\check{c}\)ech compactification: applications to topologies on groups |
| 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. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/771 |
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AT protasoviv algebrainthestonecheckcechcompactificationapplicationstotopologiesongroups |
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2024-04-12T06:26:17Z |
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2024-04-12T06:26:17Z |
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1796109162619338752 |