Characterization of complementing pairs of $({\mathbb{Z}}_{\ge 0}{)}^n$

Let $A,B,C$ be subsets of an abelian group $G$. A pair $(A,B)$ is called a $C$-pair if $A,B\subset C$ and $C$ is the direct sum of $A$ and $B$. The $({\mathbb{Z}}_{\ge 0})$-pairs are characterized by de Bruijn in 1950 and the $(\Z_{\geq 0})^2$-pairs are characterized by Niven in 1971. In this paper, we characterize the $({\mathbb{Z}}_{\ge 0}{)}^n$-pairs for all $n\ge 1$. We show that every $(\Z_{\geq 0})^n$-pair is characterized by a weighted tree if it is primitive, that is, it is not a Cartesian product of a $({\mathbb{Z}}_{\ge 0}{)}^p$-pair and a $(\Z_{\geq 0})^q$-pair of lower dimensions.