The universal zero-sum invariant and weighted zero-sum for infinite abelian groups

Abstract

Let $G$ be an abelian group, and let $\mathcal F (G)$ be the free commutative monoid with basis $G$. For $\Omega \subset \mathcal F (G)$, define the universal zero-sum invariant ${\mathsf d}_{\Omega}(G)$ to be the smallest integer $\ell$ such that every sequence $T$ over $G$ of length $\ell$ has a subsequence in $\Omega$. The invariant ${\mathsf d}_{\Omega}(G)$ unifies many classical zero-sum invariants. Let $\mathcal B (G)$ be the submonoid of $\mathcal F (G)$ consisting of all zero-sum sequences over $G$, and let $\mathcal A (G)$ be the set consisting of all minimal zero-sum sequences over $G$. In this paper, we show that except for a few special classes of groups, there always exists a proper subset $\Omega$ of $\mathcal A (G)$ such that ${\mathsf d}_{\Omega}(G)={\rm D}(G)$. 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 ${\mathsf d}_{\Omega; \Psi}(G)$ with weights set $\Psi$ of homomorphisms of groups is introduced for all abelian groups. The weighted Davenport constant ${\rm D}_{\Psi}(G)$ (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 ${\rm D}_{\Psi}(G)<\infty$ in terms of the weights set $\Psi$ when $|\Psi|$ 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 $G$ by cosets of some given subgroups of $G$, and the finiteness of weighted Davenport constant.