  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 $C\cup\{0\}$ 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 $M$ as a certain submonoid of a direct product $S\times G$, 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 $2\leq n<\infty$, or corners of $O\_{\infty}$ by
projections, thus extending to the case including $O\_{\infty}$ earlier work by
the authors together with K.R. Goodearl.
