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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| Date: | 2018 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2018
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| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/815 |
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| Journal Title: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Summary: | 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 \(|X|\le5\). |
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