On representation of semigroups of inclusion hyperspaces - Gavrylkiv, V.M. (Carpathian Mathematical Publications, 2010)
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 $λ(X)$ of maximal linked systems as a closed subsemigroup. We construct a faithful representation of the semigroups $G(X)$ and $λ(X)$ 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 $λ(X)$ 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 $X$.