Algebra in superextensions of groups, I: zeros and commutativity
Algebra and discrete mathematics, no. 3 (2008), pp. 1-29
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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 $|X|\le 5$.
Keywords:
Superextension, right-topological semigroup.
@article{ADM_2008_3_a0,
author = {T. Banakh and V. Gavrylkiv and O. Nykyforchyn},
title = {Algebra in superextensions of groups, {I:} zeros and commutativity},
journal = {Algebra and discrete mathematics},
pages = {1--29},
publisher = {mathdoc},
number = {3},
year = {2008},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ADM_2008_3_a0/}
}
T. Banakh; V. Gavrylkiv; O. Nykyforchyn. Algebra in superextensions of groups, I: zeros and commutativity. Algebra and discrete mathematics, no. 3 (2008), pp. 1-29. http://geodesic.mathdoc.fr/item/ADM_2008_3_a0/