29,626 research outputs found
Algebraic Entropies, Hopficity and co-Hopficity of Direct Sums of Abelian Groups
Necessary and sufficient conditions to ensure that the direct sum of two Abelian groups with zero entropy is again of zero entropy are still unknown; interestingly the same problem is also unresolved for direct sums of Hopfian and co-Hopfian groups. We obtain sufficient conditions in some situations by placing restrictions on the homomorphisms between the groups. There are clear similarities between the various cases but there is not a simple duality involved
Thresholds for zero-sums with small cross numbers in abelian groups
For an additive group the sequence of
elements of is a zero-sum sequence if .
The cross number of is defined to be the sum , where
denotes the order of in . Call good if it contains a
zero-sum subsequence with cross number at most 1. In 1993, Geroldinger proved
that if is abelian then every length sequence of its
elements is good, generalizing a 1989 result of Lemke and Kleitman that had
proved an earlier conjecture of Erd\H{o}s and Lemke. In 1989 Chung re-proved
the Lemke and Kleitman result by applying a theorem of graph pebbling, and in
2005, Elledge and Hurlbert used graph pebbling to re-prove and generalize
Geroldinger's result. Here we use probabilistic theorems from graph pebbling to
derive a sharp threshold version of Geroldinger's theorem for abelian groups of
a certain form. Specifically, we prove that if are (not
necessarily distinct) primes and has the form then there is a function (which we specify
in Theorem 4) with the following property: if as
then the probability that is good in tends
to 1, while if then that probability tends to 0
Semilattices of groups and inductive limits of Cuntz algebras
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
- …