Representation of zero-sum invariants by sets of zero-sum sequences over a finite abelian group

Abstract

Let G be an additive finite abelian group. For a sequence T over G and $g\in G$, let $\mathrm {v}_{g}(T)$ denote the multiplicity of g in T. Let $\mathcal {B}(G)$ denote the set of all zero-sum sequences over G. For $\Omega \subset \mathcal {B}(G)$, let $\mathsf {d}_{\Omega }(G)$ be the smallest integer t such that every sequence S over G of length $|S|\ge t$ has a subsequence in $\Omega$. The invariant $\mathsf {d}_{\Omega }(G)$ was formulated recently in [3] to take a unified look at zero-sum invariants, it led to the first results there, and some open problems were formulated as well. In this paper, we make some further study on $\mathsf {d}_{\Omega }(G)$. Let ${\mathsf {q}}'(G)$ be the smallest integer t such that every sequence S over G of length $|S|\ge t$ has two nonempty zero-sum subsequences, say $T_{1}$ and $T_{2}$, having different forms, i.e., $\mathrm {v}_{g}(T_{1})\ne \mathrm {v}_{g}(T_{2})$ for some $g\in G$. Let ${\mathsf {q}}(G)$ be the smallest integer t such that

$$\begin{aligned} \bigcap _{{\mathsf {d}}_{\Omega }(G)=t}\Omega =\emptyset . \end{aligned}$$

The invariants ${\mathsf {q}}(G)$ and ${\mathsf {q}}'(G)$ were also introduced in [3]. We prove, among other results, that ${\mathsf {q}}(G)={\mathsf {q}}'(G)$ in fact.