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02 November 2024
 
  » arxiv » arxiv.0706.0182

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The structure on the real field generated by the standard part map on an o-minimal expansion of a real closed field
Jana Maříková ;
Date 1 Jun 2007
AbstractLet R be a sufficiently saturated o-minimal expansion of a real closed field, let O be the convex hull of the rationals in R, and let st: O^n o mathbb{R}^n be the standard part map. For X subseteq R^n define st(X):=st(X cap O^n). We let mathbb{R}_{ind} be the structure with underlying set mathbb{R} and expanded by all sets of the form st(X), where X subseteq R^{n} is definable in R and n=1,2,.... We show that the subsets of mathbb{R}^n that are definable in mathbb{R}_{ind} are exactly the finite unions of sets of the form st(X) setminus st(Y), where X,Y subseteq R^n are definable in R. A consequence of the proof is a partial answer to a question by Hrushovski, Peterzil and Pillay about the existence of measures with certain invariance properties on the lattice of bounded definable sets in R^n.
Source arXiv, arxiv.0706.0182
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