A dichotomy for T $T$ -convex fields with a monomial group

IF 0.4 4区数学 Q4 LOGIC

Mathematical Logic Quarterly Pub Date : 2023-11-21 DOI:10.1002/malq.202300017

Elliot Kaplan, Christoph Kesting

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引用次数: 0

Abstract

We prove a dichotomy for o-minimal fields $R$ , expanded by a $T$ -convex valuation ring (where $T$ is the theory of $R$ ) and a compatible monomial group. We show that if $T$ is power bounded, then this expansion of $R$ is model complete (assuming that $T$ is), it has a distal theory, and the definable sets are geometrically tame. On the other hand, if $R$ defines an exponential function, then the natural numbers are externally definable in our expansion, precluding any sort of model-theoretic tameness.

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具有单项式群的t凸域的二分类

我们证明了0 -极小域R$\mathcal {R}$的二分类，该二分类由一个T-凸值环(其中T是R$\mathcal {R}$的理论)和一个相容单群展开。我们证明了如果T是幂有界的，那么R$\mathcal {R}$的展开式是模型完备的(假设T是)，它有一个远端理论，并且可定义集是几何上驯服的。另一方面，如果R$\mathcal {R}$定义了一个指数函数，那么在我们的展开中自然数是外部可定义的，排除了任何类型的模型理论驯服性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematical Logic Quarterly 数学-数学

CiteScore

0.60

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Mathematical Logic Quarterly publishes original contributions on mathematical logic and foundations of mathematics and related areas, such as general logic, model theory, recursion theory, set theory, proof theory and constructive mathematics, algebraic logic, nonstandard models, and logical aspects of theoretical computer science.