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

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.

设G是一个可加的有限阿贝尔群。对于序列Ť超过ģ和\（克\ G中\） ，让\（\ mathrm {V} _ {G}（T）\）表示的多重克在Ť。让\（\ mathcal {B}（G）\）表示设定在所有零和序列的G ^。对于\(\Omega \subset \mathcal {B}(G)\)，让\(\mathsf {d}_{\Omega }(G)\)是最小的整数t使得每个序列S超过G的长度\(|S|\ge t\)在\(\Omega \) 中有一个子序列。不变量\(\mathsf {d}_{\Omega }(G)\)最近在 [3] 中被公式化以统一看待零和不变量，它导致了那里的第一个结果，一些开放问题被公式化为嗯。在本文中，我们进一步研究了\(\mathsf {d}_{\Omega }(G)\)。让\（{\ mathsf {Q}}'（G）\）是最小的整数吨，使得每个序列š超过ģ长度的\（| S | \ GE吨\）有两个非空零和子序列，说\ (T_{1}\)和\(T_{2}\)，具有不同的形式，即\(\mathrm {v}_{g}(T_{1})\ne \mathrm {v}_{g }(T_{2})\)对于某些\(g\in G\)。让\({\mathsf {q}}(G)\)是最小的整数t使得

不变量\({\mathsf {q}}(G)\)和\({\mathsf {q}}'(G)\)也在 [3] 中引入。我们证明，除其他结果外，\({\mathsf {q}}(G)={\mathsf {q}}'(G)\)事实上。