Liftings of Diagrams of Semilattices by Diagrams of Dimension Groups

There are various ways to obtain distributive semilattices from other mathematical objects. Two of them are the following; we refer to Section 1 for more precise definitions. A dimension group is a directed, unperforated partially ordered abelian group with the interpolation property, see also K.R. Goodearl [6]. With a dimension group G we can associate its semilattice of compact ( = finitely generated) ideals Idc G. Because of the interpolation property the positive cone G + has the refinement property, thus the compact ideal semilattice IdcG is distributive. A ring R is locally matricial over a field K, if it is isomorphic to a direct limit of finite products of full matricial rings Mn(K). If R is locally matricial, or, more generally, regular (in von Neumann’s sense), then its semilattice IdcR of compact ideals is distributive, see, for example, K.R. Goodearl [5]. These two different contexts are related as follows. With a locally matricial ring R, we can associate its (partially ordered) Grothendieck group K0(R). It turns out that K0(R) is a dimension group, and the following relation holds: Idc R ∼= Idc(K0(R)), (0.1)