On representation of semigroups of inclusion hyperspaces

Abstract

Given a group $X$ we study the algebraic structure of the compactright-topological semigroup $G(X)$ consisting of inclusionhyperspaces 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 $mathsf P(X)^{mathsf P(X)}$ ofall self-maps of the power-set $mathsf P(X)$. Using thisrepresentation we prove that each minimal left ideal of$lambda(X)$ is topologically isomorphic to a minimal left idealof the semigroup $ pT^{pT}$, where by $pT$ we denote thefamily of pretwin subsets of $X$.