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We give a description of the closure of the natural affine continuous function representation of K0(R) for any exchange ring R. This goal is achieved by extending the results of Goodearl and Handelman, about metric completions of dimension groups, to a more general class of pre-ordered groups, which includes K0 of exchange rings. As a consequence, the results about K + 0 of regular rings, which the author gave in an earlier paper, can be extended to a wider class of rings, which includes C∗-algebras of real rank zero, among others. Also, the framework of pre-ordered groups developed here allows other potential applications.

Metric completions of ordered groups and 𝐾₀ of exchange rings