Additive systems and a theorem of de Bruijn

TL;DR

This paper provides a complete proof of de Bruijn's theorem classifying additive systems for nonnegative integers, identifying all indecomposable systems and their unique representations.

Contribution

It offers a comprehensive proof of de Bruijn's classification theorem and determines all indecomposable additive systems for nonnegative integers.

Findings

Complete proof of de Bruijn's theorem

Classification of all indecomposable additive systems

Unique representation of nonnegative integers

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.