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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| 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/823 |
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| Journal Title: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Summary: | 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 \(X\). |
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