Limits and Decomposition of de Bruijn’s Additive Systems

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 $a_i \ne 0$ for only finitely many i. In 1956, de Bruijn proved that every additive system is constructed from an infinite sequence $(g_i )_{i \in \mathbf N}$ of integers with $g_i \ge 2$ for all i or is a contraction of such a system. This paper discusses limits and the stability of additive systems and also describes the “uncontractable” or “indecomposable” additive systems.

Keywords

2010 Mathematics Subject Classification: