I . BACKGROUND We review the basic definitions and concepts relevant to classifying algebras by means of ordered abelian groups. All rings, algebras, and modules dealt with in this section are assumed to be unital, as are all ring and algebra maps. A pre-ordered abelian group is an abelian group G equipped with a particular pre-order relation (i.e., a reflexive, transitive relation) which is translation-invariant (that is, x y implies x + z y + z). The positive cone of G is the set G+ = {x e G I x 0}. A partially ordered abelian group is an abelian group equipped with a particular translation-invariant partial order relation. Given pre-ordered abelian groups G and H, a positive homomorphism from G to H is any group homomorphism f : G H such that f(G+) c H+. Equivalently, a group homomorphism f: G H is positive if and only if f preserves the pre-ordering. An order-unit in a pre-ordered abelian group G is an element u 0 such that for any x e G there is a positive integer n for which x nu. Given pre-ordered abelian groups G and H and order-units u e G and v e H, a normalized positive homomorphism from (G,u) to (H,v) is any positive homomorphism f: G » H such that f(u) = v. By the category of pre-ordered abelian groups with order-unit, we mean the category whose objects are all pairs (G,u), where G is a pre-ordered abelian group and u is an order-unit in G, and whose morphisms are all normalized positive homomorphisms between these objects. Throughout the paper, we shall denote this category by *P. Given any objects (G,u) and (H,v) in *P, we shall use the phrase "map from (G,u) to (H,v)" only to refer to morphisms in *P. Given a ring R with unit, the Grothendieck group KQ(R) is an abelian group consisting of expressions [A] - [B] where A and B are finitely generated projective right R-modules. Two •Received by the Editor November 26,1986. One of us (D.E.H.) would like to acknowledge very valuable discussions with D. Bures at an early stage in this research. The research of K.R.G. was done partially under a grant from the National Science Foundation (USA) and partially while he held a research fellowship from the Alexander von Humboldt Foundation at the University of Passau (Germany), and that of D.E.H. was partially supported by a grant from the Natural Science and Engineering Research Council (Canada). 1
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