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abelian group abelian operator group algebra analytic approximately finite arbitrary assume automorphism Axiom belongs bounded closed closure relations coefficients commutative complete condition conjugate consequence consider constant contains continued fraction coordinate system Corollary corresponding curve cycles defined definition deformation deformation retract denote direct sum elements equation equivalent exists factor finite number follows formula function fundamental given Hence homology homomorphism homotopy identity implies infinite integral intersection inverse isomorphic left-annulet Lemma linear mapping Math Mathematical matrix maximal metric necessary and sufficient neighborhood normal subgroup obtain open set parametric partially ordered partially ordered set Peano space positive number primary abelian operator prime ideal properties prove real numbers representation ring satisfies scalar curvature semi-local ring sequence solution subset summable Suppose tangent tensor Theorem theory topological abelian group topological space transformation unique values vector zero