On generalized Narkiewicz constants of finite abelian groups

For finite abelian groups $G$, we introduce some generalized zero-sum invariants $\mathsf D^N(G)$, $\eta ^N(G)$, and $\mathsf s^N(G)$. For example, $\mathsf D^N(G)$ is the smallest integer $t$ such that every sequence $S=g_1\cdot \ldots \cdot g_{t}$ over $G\setminus \{0\}$ of length $t$ has two zero-sum subsequences $T_1=\prod _{i\in I}g_i$ and $T_2=\prod _{j\in J}g_j$ such that $\prod _{k\in I\cap J}g_k$ is not zero-sum, where $I,J$ are distinct subsets of $[1,t]$. These invariants have close connection with Narkiewicz constant and significant applications in factorization theory. We are the first to systematically study these three invariants.