Algebra in superextensions 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 \(X\).