Additive systems and a theorem of de Bruijn

Abstract: This paper gives a complete proof of a theorem of de Bruijn that classifies additive systems for the nonnegative integers, that is, families $\mca = (A_i){i\in I}$ of sets of nonnegative integers, each set containing 0, such that every nonnegative integer can be written uniquely in the form $\sum{i\in I} a_i$ with $a_i \in A_i$ for all $i$ and $a_i \neq 0$ for only finitely many $i$. All indecomposable additive systems are determined.