Abstract

We characterize, in terms of elementary properties, the abelian monoids which are direct limits of finite direct sums of monoids of the form (ℤ / n ℤ) ⊔ {0} (where 0 is a new zero element), for positive integers n. The key properties are the Riesz refinement property and the requirement that each element x has finite order, that is, (n + 1)x = x  for some positive integer n. Such monoids are necessarily semilattices of abelian groups, and part of our approach yields a characterization of the Riesz refinement property among semilattices of abelian groups. Further, we describe the monoids in question as certain submonoids of direct products Λ × G  for semilattices Λ and torsion abelian groups G. When applied to the monoids V (A) appearing in the non-stable K-theory of C*-algebras, our results yield characterizations of the monoids V (A) for C* inductive limits A of sequences of finite direct products of matrix algebras over Cuntz algebras

. In particular, this completely solves the problem of determining the range of the invariant in the unital case of Rørdam’s classification of inductive limits of the above type.