Characterization of complementing pairs of \((\mathbb{Z}_{\geq 0})^{n \ast}\) (Q6643130)

In this article, the authors characterize the \((Z_{\ge0})^{n}\)-pairs for all \(n\ge1\), and show that every \((Z_{\ge0})^{n}\)-pair is characterized by a weighted tree, if it is primitive or in other words it is not a Cartesian product of \((Z_{\ge0})^{p}\)-pair and a \((Z_{\ge0})^{q}\)-pair of lower dimensions. Te authors establish their results in the forms of two theorems \(1.1\) and \(1.2\). In order to proof their findings, some related results are also discussed and recorded as theorems and lemmas, which are also useful to the researchers.