On representation of semigroups of inclusion hyperspaces

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 $\mathbf P(X)^{\mathbf P(X)}$ of all self-maps of the power-set $\mathbf P(X)$. Using this representation we prove that each minimal left ideal of $\lambda(X)$ is topologically isomorphic to a minimal left ideal of the semigroup $\mathbf{pT}^{\mathbf{pT}}$, where by $\mathbf{pT}$ we denote the family of pretwin subsets of $X$