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Linear O-minimal structures

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A linearly ordered structure\(\mathcal{M} = (M,< , \cdot \cdot \cdot )\) is called o-minimal if every definable subset ofM is a finite union of points and intervals. Such an\(\mathcal{M}\) is aCF structure if, roughly said, every definable family of curves is locally a one-parameter family. We prove that if\(\mathcal{M}\) is aCF structure which expands an (interval in an) ordered group, then it is elementary equivalent to a reduct of an (interval in an) ordered vector space. Along the way we prove several quantifier-elimination results for expansions and reducts of ordered vector spaces.

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The research for this article was begun when the authors were at Berkeley during the logic year at the Mathematical Science Research Institute. It was completed at McGill University. The research was supported by grants from NSERC and FCAR. JL would, as always, like to thank Alistair.

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Loveys, J., Peterzil, Y. Linear O-minimal structures. Israel J. Math. 81, 1–30 (1993).

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