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

Pub Date : 2023-11-21 DOI:10.1002/malq.202300017
Elliot Kaplan, Christoph Kesting
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Abstract

We prove a dichotomy for o-minimal fields R $\mathcal {R}$ , expanded by a T $T$ -convex valuation ring (where T $T$ is the theory of R $\mathcal {R}$ ) and a compatible monomial group. We show that if T $T$ is power bounded, then this expansion of R $\mathcal {R}$ is model complete (assuming that T $T$ is), it has a distal theory, and the definable sets are geometrically tame. On the other hand, if R $\mathcal {R}$ 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。
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