Interpretations of Linear Orderings in Presburger Arithmetic

TL;DR

This paper investigates when linear orderings can be interpreted within Presburger Arithmetic, providing complete criteria for interpretability in two dimensions and exploring implications for automatic orderings.

Contribution

It establishes necessary and sufficient conditions for interpreting linear orderings in Presburger Arithmetic, especially a complete criterion for two-dimensional cases.

Findings

Complete interpretability criterion for n=2

Conditions for interpretability depend on interpretation dimension

Relevance to automatic orderings and self-interpretations

Abstract

Presburger Arithmetic $\mathop{\mathbf{PrA}}\nolimits$ is the true theory of natural numbers with addition. We consider linear orderings interpretable in Presburger Arithmetic and establish various necessary and sufficient conditions for interpretability depending on dimension $n$ of interpretation. We note this problem is relevant to the interpretations of Presburger Arithmetic in itself, as well as the characterization of automatic orderings. For $n=2$ we obtain the complete criterion of interpretability.