Algebra in superextensions of groups, I: zeros and commutativity

Details Algebra in superextensions of groups, I: zeros and commutativity Algebra in superextensions of groups, I: zeros and commutativity

2008-02-13

arxiv.org/abs/0802.1853v1

Algebra Discrete Math. (2008), no.3, 1-29

Mathematics

General Topology

Scientific paper

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 $|X|\le5$.

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