Dimension group

Summary

A dimension group is a partially ordered abelian group that is directed, unperforated, and has the Riesz interpolation property. By the Effros–Handelman–Shen theorem, the countable dimension groups are exactly the inductive limits of groups $\Z^n$ with the coordinatewise order. The countable ones are exactly the ordered $K_0$-groups of separable AF algebras, and they carry the invariant in Elliott's classification of AF algebras.