We give an elementary characterization of those (abelian) semigroups M that are direct limits of countable sequences of finite direct products of monoids of the form C∪{0} for monogenic groups C. This characterization involves the Riesz refinement property together with lattice-theoretical properties of the collection of subgroups of M, and it makes it possible to express M
Nearby in the stack
as a certain submonoid of a direct product
S×G
, where
S
is a distributive semilattice with zero and
G
is an abelian group. When applied to the monoids
V(A)
appearing in the nonstable K-theory of C*-algebras, our results yield a full description of
V(A)
for C*-inductive limits
A
of finite products of full matrix algebras over either Cuntz algebras
O_n
, where
2≤n<∞
, or corners of
O_∞
by projections, thus extending to the case including
O_∞
earlier work by the authors together with K.R. Goodearl.