群的超扩张中的代数，I：零点与交换性
Algebra in superextensions of groups, I: zeros and commutativity
摘要：
给定一个群$X$，研究了它的超扩张$λ(X)$的代数结构.这是一个右拓扑半群，它由$X$上的所有极大链系组成，并赋予$$\mathcal A\circ\mathcal B=\{C\subset X:\{x\in X:x^{-1}C\in\mathcal B\}\in\mathcal A\}$$运算，该运算扩展了$X$的群运算。本文将$λ(X)$的右零点刻画为$X$上的不变极大联结系，并证明了$λ(X)$有右零点当且仅当$X$的每个元素都是奇数阶.另一方面，半群$λ(X)$包含左零当且仅当它包含零当且仅当$X$有奇数阶$|X|\le5$。半群$λ(X)$是交换的当且仅当$|X|\le4$。最后，我们完成了一个完整的描述的代数结构的半群$λ(X)$的所有群体$X$的基数$|X|\le5$。
英文摘要：
Given a group $X$ we study the algebraic structure of its superextension $λ(X)$. 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 $λ(X)$ as invariant maximal linked systems on $X$ and prove that $λ(X)$ has a right zero if and only if each element of $X$ has odd order. On the other hand, the semigroup $λ(X)$ contains a left zero if and only if it contains a zero if and only if $X$ has odd order $|X|\le5$. The semigroup $λ(X)$ is commutative if and only if $|X|\le4$. We finish the paper with a complete description of the algebraic structure of the semigroups $λ(X)$ for all groups $X$ of cardinality $|X|\le5$.
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