Linear Orders in Presburger Arithmetic

TL;DR

This paper characterizes the linear orders definable in Presburger arithmetic as those embeddable into lexicographic orders on integer tuples, providing a precise structural description.

Contribution

It establishes a complete characterization of first-order definable linear orders in Presburger arithmetic via embeddability into lexicographic orders.

Findings

Linear orders definable in Presburger arithmetic are exactly those embeddable into lexicographic orders on .

The result provides a structural understanding of definable linear orders.

This characterization aids in understanding the complexity of definable orders in Presburger arithmetic.

Abstract

We prove the linear orders first-order definable in the standard model $(\ZZ;<,+)$ of Presburger arithmetic are exactly those that are $(\ZZ;<,+)$-definably embeddable into the lexicographic ordering on $\ZZ^n$ for some $n$.