Additive systems and a theorem of de Bruijn

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$a_i \in A_i$forall$i$and$a_i \neq 0$foronlyfinitelymany$i$. All indecomposable additive systems are determined.