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A tetrachotomy for expansions of the rea...
Hieronymi, Philipp...

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A tetrachotomy for expansions of the real ordered additive group by Hieronymi, Philipp ( Author )
Lemonjar Open Access
Australian National University
10-08-2023
Let $\mathcal{R}$ be an expansion of the ordered real additive group. When $\mathcal{R}$ is o-minimal, it is known that either $\mathcal{R}$ defines an ordered field isomorphic to $(\mathbb{R},<,+,\cdot)$ on some open subinterval $I\subseteq \mathbb{R}$, or $\mathcal{R}$ is a reduct of an ordered vector space. We say $\mathcal{R}$ is field-type if it satisfies the former condition. In this paper, we prove a more general result for arbitrary expansions of $(\mathbb{R},<,+)$. In particular, we show that for expansions that do not define dense $\omega$-orders (we call these type A expansions), an appropriate version of Zilber's principle holds. Among other things we conclude that in a type A expansion that is not field-type, every continuous definable function $[0,1]^m \to \mathbb{R}^n$ is locally affine outside a nowhere dense set.
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English
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MYR 0.01
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http://arxiv.org/abs/1709.03150
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