On representation of semigroups of inclusion hyperspaces

Abstract

Given a group $X$ we study the algebraic structure of the compact right-topological semigroup $G(X)$ consisting of inclusion hyperspaces on $X$. This semigroup contains the semigroup $\lambda(X)$ of maximal linked systems as a closed subsemigroup. We construct a faithful representation of the semigroups $G(X)$ and $\lambda(X)$ in the semigroup $\mathsf P(X)^{\mathsf P(X)}$ of all self-maps of the power-set $\mathsf P(X)$. Using this representation we prove that each minimal left ideal of $\lambda(X)$ is topologically isomorphic to a minimal left ideal of the semigroup $\mathsf{pT}^{\mathsf{pT}}$, where by $\mathsf{pT}$ we denote the family of pretwin subsets of $X$.