Definable groups in models of Presburger Arithmetic and G^{00}

TL;DR

This paper investigates the structure of groups definable in Presburger arithmetic, establishing their abelian-by-finite nature and characterizing bounded groups as quotients of lattice structures.

Contribution

It proves that all definable groups in Presburger arithmetic are abelian-by-finite and classifies bounded definable groups as quotients of integer lattices.

Findings

Definable groups are abelian-by-finite.

Bounded definable groups are isomorphic to quotients of (Z,+)^n by a lattice.

Provides a structural classification of groups in Presburger models.

Abstract

This paper is devoted to understand groups definable in Presburger arithmetic. We prove the following theorems: Theorem 1. Every group definable in a model of Presburger Arithmetic is abelian-by-finite. Theorem 2. Every bounded group definable in a model (Z,+,<) of Presburger Arithmetic is definably isomorphic to (Z, +)^{n} mod out by a lattice.