We characterize, in terms of elementary properties, the abelian monoids which are direct limits of finite direct sums of monoids of the form $(Z/nZ)\cup\{0\}$ (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 $\Lambda\times G$ for semilattices $\Lambda$ 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 R\o rdam's classification of inductive limits of the above type.

Infinite Abelian groups

Semilattices of groups and inductive limits of Cuntz algebras

Some topological groups with, and some without, proper dense subgroups

Direct Sums of Self-Small Mixed Groups

A ring is said to be normal if all of its idempotents are central. It is proved that a mixed group A with a normal endomorphism ring contains a pure fully invariant subgroup G ⊕ B, the endomorphism ring of a group G is commutative, and a subgroup B is not always distinguished by a direct summand in A. We describe separable, coperiodic, and other groups with normal endomorphism rings. Also we consider Abelian groups in which the square of the Lie bracket of any two endomorphisms is the zero endomorphism. It is proved that every central invariant subgroup of a group is fully invariant iff the endomorphism ring of the group is commutative.

Abelian groups with normal endomorphism rings

. It is shown that a knice subgroup with cardinality Ni, of a torsion-free completely decomposable abelian group, is again completely de- composable. Any torsion-free abelian fc-group of cardinality H„ has balanced projective dimension s}. G(s*) is the subgroup generated by the set{x G G(s): J2pepi\x\p ~ sp) 's unbounded}. Two height sequences (sp) and (tp)are said to be equivalent if YlPep \sp ~ ?p\ 18 finite.

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Infinite Abelian groups

Citations

Semilattices of groups and inductive limits of Cuntz algebras

Some topological groups with, and some without, proper dense subgroups

Direct Sums of Self-Small Mixed Groups

Abelian groups with normal endomorphism rings

On torsion-free abelian $k$-groups

Related Papers (5)

Finite Rank Torsion Free Abelian Groups and Rings

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