We characterize, in terms of elementary properties, the abelian monoids
which are direct limits of finite direct sums of monoids of the form ðZ=nZÞ t f0g (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, ðn þ 1Þx ¼ 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 L G for semilattices L and torsion abelian groups G.
When applied to the monoids VðAÞ appearing in the non-stable K-theory of C*-algebras,
our results yield characterizations of the monoids VðAÞ for C* inductive limits A of sequences
of finite direct products of matrix algebras over Cuntz algebras On. 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