Algebra in superextensions of groups, II: cancelativity and centers
Given a countable group \(X\) we study the algebraic structure of itssuperextension \(\lambda(X)\). This is a right-topological semigroup consisting of all maximal linked systems on \(X\) endowed with the operation \(\mathcal A\circ\mathcal B=\{C\subset X:\{x\in X:x^{-1}C\in\mathcal B\}\in\ma...
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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/823 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-8232018-03-22T09:57:42Z Algebra in superextensions of groups, II: cancelativity and centers Banakh, Taras Gavrylkiv, Volodymyr Superextension, right-topological semigroup, cancelable element, topological center, algebraic center 20M99, 54B20 Given a countable group \(X\) we study the algebraic structure of itssuperextension \(\lambda(X)\). This is a right-topological semigroup consisting of all maximal linked systems on \(X\) endowed with the operation \(\mathcal A\circ\mathcal B=\{C\subset X:\{x\in X:x^{-1}C\in\mathcal B\}\in\mathcal A\}\) that extends the group operation of \(X\). We show that the subsemigroup \(\lambda^\circ(X)\) of free maximal linked systems contains an open dense subset of right cancelable elements. Also we prove that the topological center of \(\lambda(X)\) coincides with the subsemigroup \(\lambda^\bullet(X)\) of all maximal linked systems with finite support. This result is applied to show that the algebraic center of \(\lambda(X)\) coincides with the algebraic center of \(X\) provided \(X\) is countably infinite. On the other hand, for finite groups \(X\) of order \(3\le|X|\le5\) the algebraic center of \(\lambda(X)\) is strictly larger than the algebraic center of \(X\). Lugansk National Taras Shevchenko University 2018-03-22 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/823 Algebra and Discrete Mathematics; Vol 7, No 4 (2008) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/823/353 Copyright (c) 2018 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
Superextension right-topological semigroup cancelable element topological center algebraic center 20M99 54B20 |
| spellingShingle |
Superextension right-topological semigroup cancelable element topological center algebraic center 20M99 54B20 Banakh, Taras Gavrylkiv, Volodymyr Algebra in superextensions of groups, II: cancelativity and centers |
| topic_facet |
Superextension right-topological semigroup cancelable element topological center algebraic center 20M99 54B20 |
| format |
Article |
| author |
Banakh, Taras Gavrylkiv, Volodymyr |
| author_facet |
Banakh, Taras Gavrylkiv, Volodymyr |
| author_sort |
Banakh, Taras |
| title |
Algebra in superextensions of groups, II: cancelativity and centers |
| title_short |
Algebra in superextensions of groups, II: cancelativity and centers |
| title_full |
Algebra in superextensions of groups, II: cancelativity and centers |
| title_fullStr |
Algebra in superextensions of groups, II: cancelativity and centers |
| title_full_unstemmed |
Algebra in superextensions of groups, II: cancelativity and centers |
| title_sort |
algebra in superextensions of groups, ii: cancelativity and centers |
| description |
Given a countable group \(X\) we study the algebraic structure of itssuperextension \(\lambda(X)\). This is a right-topological semigroup consisting of all maximal linked systems on \(X\) endowed with the operation \(\mathcal A\circ\mathcal B=\{C\subset X:\{x\in X:x^{-1}C\in\mathcal B\}\in\mathcal A\}\) that extends the group operation of \(X\). We show that the subsemigroup \(\lambda^\circ(X)\) of free maximal linked systems contains an open dense subset of right cancelable elements. Also we prove that the topological center of \(\lambda(X)\) coincides with the subsemigroup \(\lambda^\bullet(X)\) of all maximal linked systems with finite support. This result is applied to show that the algebraic center of \(\lambda(X)\) coincides with the algebraic center of \(X\) provided \(X\) is countably infinite. On the other hand, for finite groups \(X\) of order \(3\le|X|\le5\) the algebraic center of \(\lambda(X)\) is strictly larger than the algebraic center of \(X\). |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/823 |
| work_keys_str_mv |
AT banakhtaras algebrainsuperextensionsofgroupsiicancelativityandcenters AT gavrylkivvolodymyr algebrainsuperextensionsofgroupsiicancelativityandcenters |
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2024-04-12T06:25:51Z |
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2024-04-12T06:25:51Z |
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1796109212511633408 |