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

Abstract

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