Algebra in superextension of groups, II: cancelativity and centers

Abstract

Given a countable 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 show that the subsemigroup $\lambda^\circ(X)$ of free maximal linked systems contains an open dense subset of right cancelable elements. Also we prove that the topological center of $\lambda(X)$ coincides with the subsemigroup $\lambda^\bullet(X)$ of all maximal linked systems with finite support. This result is applied to show that the algebraic center of $\lambda(X)$ 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 $\lambda(X)$ is strictly larger than the algebraic center of $X$.