Algebra in superextensions of groups, I: zeros and commutativity

Abstract

Given a 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 characterize right zeros of λ(X) as invariant maximal linked systems on X and prove that λ(X) has a right zero if and only if each element of X has odd order. On the other hand, the semigroup λ(X) contains a left zero if and only if it contains a zero if and only if X has odd order |X|≤5. The semigroup λ(X) is commutative if and only if |X|≤4. We finish the paper with a complete description of the algebraic structure of the semigroups λ(X) for all groups X of cardinality |X|≤5.