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

Abstract

Let $G$ be an abelian group, and let $\mathcal F (G)$ be the free commutative monoid with basis $G$, and $\mathcal A (G)$ the set consisting of all minimal zero-sum subsequences over $G$. For any subset $Ω\subset \mathcal F (G)$, we define the universal zero-sum invariant ${\mathsf d}_Ω(G)$ as the minimal positive integer $\ell$ such that every sequence $T$ over $G$ of length $\ell$ contains a subsequence lying in $Ω$. The classical Davenport constant ${\rm D}(G)$ for $G$ can also be written as ${\mathsf d}_{\mathcal A (G)}(G)$. We give a complete classification of all finite abelian groups for which $\mathcal A(G)$ is a minimal set to represent the Davenport constant. We also investigate the weighted Davenport constant over abelian groups (which may be infinite). Let $F$ and $G$ be abelian groups, and let $Ψ\subseteq \mathrm{Hom}(F,G)$ denote a weight set. We reinterpret the weighted Davenport constant $D_Ψ(G)$ in terms of coverings of Cartesian powers $F^n$ by kernels of induced homomorphisms arising from tuples in $Ψ^n$; these homomorphisms are naturally linked to coproducts in the category of abelian groups. This motivates the notion of kernel-cover compactness, a property characterizing when such kernel coverings admit finite subcovers. We establish a correspondence between weighted zero-sum invariants and kernel-cover structures, where the bound $D_Ψ(G)\le n$ is equivalent to a canonical kernel-cover property on $F^n$. We further study finite reduction phenomena for infinite weight sets and provide sufficient conditions ensuring uniform kernel-cover compactness. The present work constitutes a follow-up to [G. Wang, Comm. Algebra, 2025].