Algebra in superextensions of groups, I: zeros and commutativity

Abstract

Given a 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 characterize right zeros of \(\lambda(X)\) as invariant maximal linked systems on \(X\) and prove that \(\lambda(X)\) has a right zero if and only if each element of \(X\) has odd order. On the other hand, the semigroup \(\lambda(X)\) contains a left zero if and only if it contains a zero if and only if \(X\) has odd order \(|X|\le5\). The semigroup \(\lambda(X)\) is commutative if and only if \(|X|\le4\). We finish the paper with a complete description of the algebraic structure of the semigroups \(\lambda(X)\) for all groups \(X\) of cardinality \(|X|\le5\).