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Given a group X we study the algebraic structure of the compact right-topological semigroup λ(X) consisting of all maximal linked systems on X. This semigroup contains the semigroup β(X) of ultrafilters as a closed subsemigroup. We construct a faithful representation of the semigroup λ(X) in the semigroup ${\left(X\right)}^{\left(X\right)}$ of all self-maps of the power-set (X) and show that the image of λ(X) in ${\left(X\right)}^{\left(X\right)}$ coincides with the semigroup $En{d}_{\lambda }\left(\left(X\right)\right)$ of all functions f: (X) → (X) that are equivariant, monotone and symmetric in the sense...