Algebra in superextensions of groups, I: zeros and commutativity

• Datum: 2018
• Hauptverfasser: T. Banakh, T., Gavrylkiv, V., Nykyforchyn, O.
• Format: Artikel
• Sprache: English
• Veröffentlicht: Lugansk National Taras Shevchenko University 2018
• Schlagworte: Superextension; right-topological semigroup; 20M99; 54B20
• Online Zugang: https://admjournal.luguniv.edu.ua/index.php/adm/article/view/815
• Назва журналу: Algebra and Discrete Mathematics

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 \(|X|\le5\).