Algebra in superextensions of groups, II: cancelativity and centers

Abstract

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

A∘B={C⊂X:{x∈X:x−1C∈B}∈A}

that extends the group operation of X. We show that the subsemigroup λ∘(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∙(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≤|X|≤5 the algebraic center of λ(X) is strictly larger than the algebraic center of X.