群的超扩张代数，II：可消性与中心
Algebra in superextension of groups, II: cancelativity and centers
摘要：
给定一个可数群$X$，研究其超扩张$λ(X)$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。证明了自由极大链系的子半群$λ^\circ(X)$包含右可消元的开稠密子集。证明了$λ(X)$的拓扑中心与所有有限支撑极大链系的子半群$λ^\bullet(X)$重合。这个结果被应用于证明$λ(X)$的代数中心与$X$的代数中心重合，只要$X$是可数无穷的。另一方面，对于$3\le|X|\le5$阶的有限群$X$，$λ(X)$的代数中心严格大于$X$的代数中心。
英文摘要：
Given a countable 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 show that the subsemigroup $λ^\circ(X)$ of free maximal linked systems contains an open dense subset of right cancelable elements. Also we prove that the topological center of $λ(X)$ coincides with the subsemigroup $λ^\bullet(X)$ of all maximal linked systems with finite support. This result is applied to show that the algebraic center of $λ(X)$ coincides with the algebraic center of $X$ provided $X$ is countably infinite. On the other hand, for finite groups $X$ of order $3\le|X|\le5$ the algebraic center of $λ(X)$ is strictly larger than the algebraic center of $X$.
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