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 abelian group definable in a model $(\mathbb{Z},+,<)$ of Presburger Arithmetic is definably isomorphic to ${(\mathbb{Z},+)}^n$ mod out by a lattice.

中文翻译：

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Presburger算法模型中的可定义组

本文致力于了解在Presburger算法中可定义的组。我们证明以下定理：

定理1.在Presburger算法的模型中可定义的每个组都是abelian-by-finite。

定理2.在模型中可定义的每个有界阿贝尔群 $（\mathbb{ž}，+，<）$ Presburger算术的绝对同构 ${（\mathbb{ž}，+）}^{ñ}$ 通过格子修改出来。