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 $|X|\le 5$. The semigroup $\lambda (X)$ is commutative if and only if $|X|\le 4$. We finish the paper with a complete description of the algebraic structure of the semigroups $\lambda (X)$ for all groups $X$ of cardinality $|X|\le 5$.

Cited by in corpus (2)

• Characterizing compact Clifford semigroups that embed into convolution and functor-semigroups
• Automorphism groups of superextensions of finite monogenic semigroups