We give axioms for a class of ordered structures, called truncated ordered abelian groups (TOAG’s) carrying an addition. TOAG’s come naturally from ordered abelian groups with a 0 and a \(+\), but the addition of a TOAG is not necessarily even a cancellative semigroup. The main examples are initial segments \([0, \tau ]\) of an ordered abelian group, with a truncation of the addition. We prove that any model of these axioms (i.e. a truncated ordered abelian group) is an initial segment of an ordered abelian group. We define Presburger TOAG’s, and give a criterion for a TOAG to be a Presburger TOAG, and for two Presburger TOAG’s to be elementarily equivalent, proving analogues of classical results on Presburger arithmetic. Their main interest for us comes from the model theory of certain local rings which are quotients of valuation rings valued in a truncation [0, a] of the ordered group \({\mathbb {Z}}\) or more general ordered abelian groups, via a study of these truncations without reference to the ambient ordered abelian group. The results are used essentially in a forthcoming paper (D’Aquino and Macintyre, The model theory of residue rings of models of Peano Arithmetic: The prime power case, 2021, arXiv:2102.00295) in the solution of a problem of Zilber about the logical complexity of quotient rings, by principal ideals, of nonstandard models of Peano arithmetic.

我们给出一类有序结构的公理，称为带加法的截断有序阿贝尔群（TOAG）。TOAG自然来自具有0和\（+ \）的有序阿贝尔群，但添加TOAG甚至不一定是可加半群。主要示例是初始段\（[0，\ tau] \）一个有序的阿贝尔群，加上一个截断的加法。我们证明这些公理的任何模型（即截断的有序阿贝尔群）都是有序阿贝尔群的初始段。我们定义了Presburger TOAG，并给出了将TOAG定义为Presburger TOAG的标准，并给出了两个Presburger TOAG基本相等的标准，证明了Presburger算法的经典结果的相似性。他们对我们的主要兴趣来自某些局部环的模型理论，该局部环是有序组\（{\ mathbb {Z}} \）的截断[0，a ]上的估值环的商 或更多个一般的有序阿贝尔群，通过研究这些截断而无需参考周围有序的阿贝尔群。该结果基本上用在即将发表的论文（D'Aquino和Macintyre，《皮亚诺算术模型的残环模型理论：主要力量案例》，2021年，arXiv：2102.00295）中，解决了关于逻辑的齐尔伯问题根据主要理想，Peano算术的非标准模型的商环的复杂度。