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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| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2018
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| Теми: | |
| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/771 |
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| Назва журналу: | Algebra and Discrete Mathematics |
Репозитарії
Algebra and Discrete Mathematics| Резюме: | 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. |
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