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Algebra and Discrete Mathematics, 2008, Issue 4, Pages 1–14
(Mi adm174)
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This article is cited in 8 scientific papers (total in 8 papers)
RESEARCH ARTICLE
Algebra in superextensions of groups, II: cancelativity and centers
Taras Banakha, Volodymyr Gavrylkivb a Ivan Franko National University of Lviv,
Universytetska 1, 79000, Ukraine
b Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine
Abstract:
Given a countable 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 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 $X$ provide $X$ 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 $X$.
Keywords:
Superextension, right-topological semigroup, cancelable element, topological center, algebraic center.
Received: 14.02.2008 Revised: 25.08.2008
Citation:
Taras Banakh, Volodymyr Gavrylkiv, “Algebra in superextensions of groups, II: cancelativity and centers”, Algebra Discrete Math., 2008, no. 4, 1–14
Linking options:
https://www.mathnet.ru/eng/adm174 https://www.mathnet.ru/eng/adm/y2008/i4/p1
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