Ordered abelian groups over a CW complex

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If X is a CW complex, one can assign to each point of X an ordered abelian group of finite rank whose subset of positive elements depends continuously on the points of X. A locally trivial bundle which arises in this way we denote by E(X). In the present work we establish a topological classification of such bundles in terms of the first cohomology group of X with coefficients in the ring Z₂. This result has an amazing application in the theory of characteristic classes of foliations on compact manifolds.