2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72116We characterize, in terms of elementary properties, the abelian monoids which are direct limits of finite direct sums of monoids of the form $(Z/nZ)\sqcup\{0\}$ (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 $Λ\times G$ 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 $O_n$. 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.Operator Algebras20M17; 46L35; 06A12; 06F05text