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abelian group Amer Aronszajn artinian ring assume Aut G automorphism Banach space Boolean lattice Borel bounded bundle closed commutative component congruence contains converges convex Corollary countable defined Definition denote element embedded equivalent equivariant exists extension F-algebra fibration finite follows function G-manifold G-orbit given groups over F Heegaard splitting Hence Hilbert space homomorphism homotopy hypergroup idempotent identity implies induced infinite injective irreducible isomorphism isotoxal kernel lattice Lemma Let F Let G linear locally compact manifold Math matrix metric metric space minimal morphism multiplication neighborhood neighbornet nonzero normal normal subgroup notation obtain operator orbit spaces polynomial positive integer prime pro-affine algebraic groups proof of Theorem Proposition prove quadratic R-module regular representation result right R-module ring satisfies semigroup semiprime sequence smooth stratified map subgroup subset Suppose symmetric Theorem theory tilings topological trivial vector zero