2,410 research outputs found
Davenport constant for semigroups II
Let be a finite commutative semigroup. The Davenport constant
of , denoted , is defined to be the least
positive integer such that every sequence of elements in
of length at least contains a proper subsequence
() with the sum of all terms from equaling the sum of all terms
from . Let be a prime power, and let \F_q[x] be the ring of
polynomials over the finite field \F_q. Let be a quotient ring of
\F_q[x] with 0\neq R\neq \F_q[x]. We prove that where denotes
the multiplicative semigroup of the ring , and denotes
the group of units in .Comment: In press in Journal of Number Theory. arXiv admin note: text overlap
with arXiv:1409.1313 by other author
The universal zero-sum invariant and weighted zero-sum for infinite abelian groups
Let be an abelian group, and let be the free commutative
monoid with basis . For , define the
universal zero-sum invariant to be the smallest
integer such that every sequence over of length has a
subsequence in . Let be the submonoid of consisting of all zero-sum sequences over , and let be
the set consisting of all minimal zero-sum subsequences over . In this
paper, we show that except for a few special classes of groups, there always
exists a proper subset of such that . Furthermore, in the setting of finite cyclic
groups, we discuss the distributions of all minimal sets by determining their
intersections.
By connecting the universal zero-sum invariant with weights, we make a study
of zero-sum problems in the setting of {\sl infinite} abelian groups. The
universal zero-sum invariant with weights set
of homomorphisms of groups is introduced for all abelian groups. The
weighted Davenport constant (being an special form of the
universal invariant with weights) is also investigated for infinite abelian
groups. Among other results, we obtain the necessary and sufficient conditions
such that in terms of the weights set when
is finite. In doing this, by using the Neumann Theorem on Cover Theory
for groups we establish a connection between the existence of a finite cover of
an abelian group by cosets of some given subgroups of , and the
finiteness of weighted Davenport constant.Comment: 31 page
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