Algebra in superextensions of groups, I: zeros and commutativity
Given a group \(X\) we study the algebraic structure of its superextension \(\lambda(X)\). This is a right-topological semigroup consisting of all maximal linked systems on \(X\) endowed with the operation \(\mathcal A\circ\mathcal B=\{C\subset X:\{x\in X:x^{-1}C\in\mathcal B\}\in\mathcal A\}\) t...
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| Datum: | 2018 |
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| Sprache: | Englisch |
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Lugansk National Taras Shevchenko University
2018
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| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/815 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543437074989056 |
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| author | T. Banakh, T. Gavrylkiv, V. Nykyforchyn, O. |
| author_facet | T. Banakh, T. Gavrylkiv, V. Nykyforchyn, O. |
| author_sort | T. Banakh, T. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-03-22T09:42:02Z |
| description | Given a group \(X\) we study the algebraic structure of its superextension \(\lambda(X)\). 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 characterize right zeros of \(\lambda(X)\) as invariant maximal linked systems on \(X\) and prove that \(\lambda(X)\) has a right zero if and only if each element of \(X\) has odd order. On the other hand, the semigroup \(\lambda(X)\) contains a left zero if and only if it contains a zero if and only if \(X\) has odd order \(|X|\le5\). The semigroup \(\lambda(X)\) is commutative if and only if \(|X|\le4\). We finish the paper with a complete description of the algebraic structure of the semigroups \(\lambda(X)\) for all groups \(X\) of cardinality \(|X|\le5\). |
| first_indexed | 2026-02-08T07:59:46Z |
| format | Article |
| id | admjournalluguniveduua-article-815 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2026-02-08T07:59:46Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-8152018-03-22T09:42:02Z Algebra in superextensions of groups, I: zeros and commutativity T. Banakh, T. Gavrylkiv, V. Nykyforchyn, O. Superextension, right-topological semigroup 20M99, 54B20 Given a group \(X\) we study the algebraic structure of its superextension \(\lambda(X)\). 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 characterize right zeros of \(\lambda(X)\) as invariant maximal linked systems on \(X\) and prove that \(\lambda(X)\) has a right zero if and only if each element of \(X\) has odd order. On the other hand, the semigroup \(\lambda(X)\) contains a left zero if and only if it contains a zero if and only if \(X\) has odd order \(|X|\le5\). The semigroup \(\lambda(X)\) is commutative if and only if \(|X|\le4\). We finish the paper with a complete description of the algebraic structure of the semigroups \(\lambda(X)\) for all groups \(X\) of cardinality \(|X|\le5\). 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/815 Algebra and Discrete Mathematics; Vol 7, No 3 (2008) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/815/345 Copyright (c) 2018 Algebra and Discrete Mathematics |
| spellingShingle | Superextension right-topological semigroup 20M99 54B20 T. Banakh, T. Gavrylkiv, V. Nykyforchyn, O. Algebra in superextensions of groups, I: zeros and commutativity |
| title | Algebra in superextensions of groups, I: zeros and commutativity |
| title_full | Algebra in superextensions of groups, I: zeros and commutativity |
| title_fullStr | Algebra in superextensions of groups, I: zeros and commutativity |
| title_full_unstemmed | Algebra in superextensions of groups, I: zeros and commutativity |
| title_short | Algebra in superextensions of groups, I: zeros and commutativity |
| title_sort | algebra in superextensions of groups, i: zeros and commutativity |
| topic | Superextension right-topological semigroup 20M99 54B20 |
| topic_facet | Superextension right-topological semigroup 20M99 54B20 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/815 |
| work_keys_str_mv | AT tbanakht algebrainsuperextensionsofgroupsizerosandcommutativity AT gavrylkivv algebrainsuperextensionsofgroupsizerosandcommutativity AT nykyforchyno algebrainsuperextensionsofgroupsizerosandcommutativity |