49,199 research outputs found
On the Olson and the Strong Davenport constants
A subset of a finite abelian group, written additively, is called
zero-sumfree if the sum of the elements of each non-empty subset of is
non-zero. We investigate the maximal cardinality of zero-sumfree sets, i.e.,
the (small) Olson constant. We determine the maximal cardinality of such sets
for several new types of groups; in particular, -groups with large rank
relative to the exponent, including all groups with exponent at most five.
These results are derived as consequences of more general results, establishing
new lower bounds for the cardinality of zero-sumfree sets for various types of
groups. The quality of these bounds is explored via the treatment, which is
computer-aided, of selected explicit examples. Moreover, we investigate a
closely related notion, namely the maximal cardinality of minimal zero-sum
sets, i.e., the Strong Davenport constant. In particular, we determine its
value for elementary -groups of rank at most , paralleling and building
on recent results on this problem for the Olson constant
Inverse zero-sum problems and algebraic invariants
In this article, we study the maximal cross number of long zero-sumfree
sequences in a finite Abelian group. Regarding this inverse-type problem, we
formulate a general conjecture and prove, among other results, that this
conjecture holds true for finite cyclic groups, finite Abelian p-groups and for
finite Abelian groups of rank two. Also, the results obtained here enable us to
improve, via the resolution of a linear integer program, a result of W. Gao and
A. Geroldinger concerning the minimal number of elements with maximal order in
a long zero-sumfree sequence of a finite Abelian group of rank two.Comment: 17 pages, to appear in Acta Arithmetic
Zero-sum problems with congruence conditions
For a finite abelian group and a positive integer , let denote the smallest integer such that
every sequence over of length has a nonempty zero-sum
subsequence of length . We determine for all when has rank at most two and, under mild
conditions on , also obtain precise values in the case of -groups. In the
same spirit, we obtain new upper bounds for the Erd{\H o}s--Ginzburg--Ziv
constant provided that, for the -subgroups of , the Davenport
constant is bounded above by . This
generalizes former results for groups of rank two
The system of sets of lengths in Krull monoids under set addition
Let be a Krull monoid with class group and suppose that each class
contains a prime divisor. Then every element has a factorization into
irreducible elements, and the set of all possible factorization
lengths is the set of lengths of . We consider the system of all sets of lengths, and we characterize
(in terms of the class group ) when is additively closed
under set addition.Comment: Revista Matem{\'a}tica Iberoamericana, to appea
Inverse zero-sum problems II
Let be an additive finite abelian group. A sequence over is called a
minimal zero-sum sequence if the sum of its terms is zero and no proper
subsequence has this property. Davenport's constant of is the maximum of
the lengths of the minimal zero-sum sequences over . Its value is well-known
for groups of rank two. We investigate the structure of minimal zero-sum
sequences of maximal length for groups of rank two. Assuming a well-supported
conjecture on this problem for groups of the form , we
determine the structure of these sequences for groups of rank two. Combining
our result and partial results on this conjecture, yields unconditional results
for certain groups of rank two.Comment: new version contains results related to Davenport's constant only;
other results will be described separatel
Consequences of the existence of Auslander-Reiten triangles with applications to perfect complexes for self-injective algebras
In a k-linear triangulated category (where k is a field) we show that the
existence of Auslander-Reiten triangles implies that objects are determined, up
to shift, by knowing dimensions of homomorphisms between them. In most cases
the objects themselves are distinguished by this information, a conclusion
which was also reached under slightly different hypotheses in a theorem of
Jensen, Su and Zimmermann. The approach is to consider bilinear forms on
Grothendieck groups which are analogous to the Green ring of a finite group.
We specialize to the category of perfect complexes for a self-injective
algebra, for which the Auslander-Reiten quiver has a known shape. We
characterize the position in the quiver of many kinds of perfect complexes,
including those of lengths 1, 2 and 3, rigid complexes and truncated projective
resolutions. We describe completely the quiver components which contain
projective modules. We obtain relationships between the homology of complexes
at different places in the quiver, deducing that every self-injective algebra
of radical length at least 3 has indecomposable perfect complexes with
arbitrarily large homology in any given degree. We find also that homology
stabilizes away from the rim of the quiver. We show that when the algebra is
symmetric, one of the forms considered earlier is Hermitian, and this allows us
to compute its values knowing them only on objects on the rim of the quiver.Comment: 27 page
- …