Limits and decomposition of de Bruijn's additive systems

TL;DR

This paper classifies the fundamental building blocks of additive systems for nonnegative integers, exploring their limits, decompositions, and stability, building on de Bruijn's foundational work from 1956.

Contribution

It provides a complete classification of indecomposable additive systems and analyzes their limits and stability properties, extending de Bruijn's theory.

Findings

Classification of uncontractable additive systems

Analysis of limits and stability of additive systems

Extension of de Bruijn's construction to new cases

Abstract

An additive system for the nonnegative integers is a family (A_i)_{i\in I} of sets of nonnegative integers with 0 \in A_i for all i \in I 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. In 1956, de Bruijn proved that every additive system is constructed from an infinite sequence (g_i)_{i \in \N} of integers with g_i \geq 2 for all i, or is a contraction of such a system. This paper gives a complete classification of the "uncontractable" or "indecomposable" additive systems, and also considers limits and stability of additive systems.