The positive cone G+ of a partially ordered abelian group G is a refinement monoid if and only if G is an interpolation group, the latter meaning that for any elements a0, a1, b0, b1 of G such that ai ≤ bj for all i, j<2, there exists an element x of G such that ai ≤ x ≤ bj for all i, j<2.
The class of isomorphism types of Boolean algebras, endowed with the addition defined by
denotes the isomorphism type of X), is a conical refinement monoid.
Hans Dobbertin proved in 1983 that any conical refinement monoid with at most ℵ1 elements is measurable.
He raised there the problem whether any conical refinement monoid is measurable.
For a ring (with unit) R, denote by FP(R) the class of finitely generated projective right R-modules.
Then the set V(R) of all isomorphism types of members of FP(R), endowed with the addition defined by
In addition, if R is von Neumann regular, then V(R) is a refinement monoid.
We say that V(R) encodes the nonstable K-theory of R. For example, if R is a division ring, then the members of FP(R) are exactly the finite-dimensional right vector spaces over R, and two vector spaces are isomorphic if and only if they have the same dimension.
A matricial algebra over a field F is a finite product of rings of the form
For any locally matricial algebra R, V(R) is the positive cone of a so-called dimension group.
[4] Elliott proved in 1976 that the positive cone of any countable direct limit of simplicial groups is isomorphic to V(R), for some locally matricial ring R.[5] Finally, Goodearl and Handelman proved in 1986 that the positive cone of any dimension group with at most ℵ1 elements is isomorphic to V(R), for some locally matricial ring R (over any given field).
[6] Wehrung proved in 1998 that there are dimension groups with order-unit whose positive cone cannot be represented as V(R), for a von Neumann regular ring R.[2] The given examples can have any cardinality greater than or equal to ℵ2.
Whether any conical refinement monoid with at most ℵ1 (or even ℵ0) elements can be represented as V(R) for R von Neumann regular is an open problem.