Linear orders in Presburger arithmetic

Fedor Pakhomov; Alexander Zapryagaev

doi:10.1093/jigpal/jzag045

July 13, 2026Logic Journal of IGPLOpen Access

Linear orders in Presburger arithmetic

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Key Points

The study aims to characterize linear orders that are first-order definable in the framework of Presburger arithmetic.
Proving that linear orders in the standard model $(\mathbb{Z};<,+)$ are definably embeddable into lexicographic orderings.
Utilizing properties of first-order definability for modeling these linear orders.
Analyzing the relationship between definability and embeddability in the context of relation structures.
Identified that linear orders in Presburger arithmetic are exactly those that can be embedded into $\mathbb{Z}^{n}$ in a lexicographic manner for some $n$.
Provided a comprehensive framework for understanding the relationship between linear orders and Definability.

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Implication

This research proves exact criteria for linear orders in Presburger arithmetic, defining embeddability in lexicographic terms.

Abstract

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

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Cite This Study

Pakhomov et al. (2026) studied this question.

synapsesocial.com/papers/6a5873ed2b46c88ba9ad03b6 https://doi.org/10.1093/jigpal/jzag045

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Linear orders in Presburger arithmetic

Key Points

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Implication

Abstract

Discussion

Cite This Study

Linear orders in Presburger arithmetic

Key Points

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Implication

Abstract

Discussion

Cite This Study

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