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Algebra in superextensions of groups, IV: representation theory

Given a group X we study the algebraic structure of the compact right-topological semigroup λ(X) consisting of 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 P(X)P(X) of all self-maps of the power-set P(X) and show that the image of λ(X) in P(X)P(X) coincides with the semigroup λ(X, P(X)) of all functions f : P(X) → P(X) that are equivariant, monotone and symmetric in the sense that f(X \ A) = X \ f(A) for all A ⊂ X. Using this representation we describe the minimal ideal K(λ(X)) and minimal left ideals of the superextension λ(X) of each somewhat commutative group X, where a group X is somewhat commutative if the product xy of any two elements x, y ∈ X belongs to the semigroups generated by the set {yx, yx−1, y−1x, y−1x−1}. We prove that for each commutative group X (admitting no epimorphism onto the quasi-cyclic group C2∞ ) each minimal left ideal of the supserextension λ(X) is alegebraically (and topologically) isomorphic to the product