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 $$\mathcal A\circ\mathcal B=\{C\subset X:\{x\in X:x^{-1}C\in\mathcal B\}\in\mathcal A\}$$ that extends the group operation of $X$. We characterize right zeros of $λ(X)$ 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 $|X|\le5$. The semigroup $λ(X)$ is commutative if and only if $|X|\le4$. We finish the paper with a complete description of the algebraic structure of the semigroups $λ(X)$ for all groups $X$ of cardinality $|X|\le5$.