Let $\mathbf{H}$ be the cartesian product of a family of finite abelian groups indexed by a finite set $\Omega$. A given poset (i.e., partially ordered set) $\mathbf{P}=(\Omega,\preccurlyeq_{\mathbf{P}})$ gives rise to a poset metric on $\mathbf{H}$, which further leads to a partition $\mathcal{Q}(\mathbf{H},\mathbf{P})$ of $\mathbf{H}$. We prove that if $\mathcal{Q}(\mathbf{H},\mathbf{P})$ is Fourier-reflexive, then its dual partition $\Lambda$ coincides with the partition of $\hat{\mathbf{H}}$ induced by $\mathbf{\overline{P}}$, the dual poset of $\mathbf{P}$, and moreover, $\mathbf{P}$ is necessarily hierarchical. This result establishes a conjecture proposed by Gluesing-Luerssen in \cite{4}. We also show that with some other assumptions, $\Lambda$ is finer than the partition of $\hat{\mathbf{H}}$ induced by $\mathbf{\overline{P}}$. In addition, we give some necessary and sufficient conditions for $\mathbf{P}$ to be hierarchical, and for the case that $\mathbf{P}$ is hierarchical, we give an explicit criterion for determining whether two codewords in $\hat{\mathbf{H}}$ belong to the same block of $\Lambda$. We prove these results by relating the involved partitions with certain family of polynomials, a generalized version of which is also proposed and studied to generalize the aforementioned results.

翻译：Let\ mathbf{H}$ 是一个数量有限的ABL组家族的卡通产值 。 以一定的 美元制成的 美元 。 给定的 mathbf{P} (Omega,\ preccurlyeq\\mathbf{P}} 美元 产生一个 $\ mathbf{H} 美元 的配置度量 。 这进一步导致一个部分的 P&mb 美元 。 (mathb{H} 以一定的 美元制成的分区 。 (mathb}) 美元 。 (mathb} h} 美元 美元 。 ) 如果 $\ mathbffff} 美元制成的配置值 。 (f) 由 $\f_ 美元制成的 美元制成的配置值 。 (f) 由 美元制成的 美元制成的 美元制成的 。 (mab) 美元 和 美元制成的基底的基底的计算结果 。