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Algebra and Discrete Mathematics, 2008, выпуск 3, страницы 1–29
(Mi adm166)
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Эта публикация цитируется в 13 научных статьях (всего в 13 статьях)
RESEARCH ARTICLE
Algebra in superextensions of groups, I: zeros and commutativity
T. Banakha, V. Gavrylkivb, O. Nykyforchynb a Ivan Franko National University of Lviv,
Universytetska 1, 79000, Ukraine
b Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine
Аннотация:
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 $|X|\le 5$. The semigroup $\lambda(X)$ is commutative if and only if $|X|\le 4$. We finish the paper with a complete description of the algebraic structure of the semigroups $\lambda(X)$ for all groups $X$ of cardinality $|X|\le 5$.
Ключевые слова:
Superextension, right-topological semigroup.
Поступила в редакцию: 14.02.2008 Исправленный вариант: 14.10.2008
Образец цитирования:
T. Banakh, V. Gavrylkiv, O. Nykyforchyn, “Algebra in superextensions of groups, I: zeros and commutativity”, Algebra Discrete Math., 2008, no. 3, 1–29
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/adm166 https://www.mathnet.ru/rus/adm/y2008/i3/p1
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Страница аннотации: | 139 | PDF полного текста: | 104 | Первая страница: | 1 |
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