Abstract
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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[vdD] L. van den Dries,Tame topology and σ-minimal structures, preliminary version, 1991.
[HL] E. Hrushovski and J. Loveys,Weakly minimal groups of bounded exponent, in preparation.
[KPS] J. Knight, A. Pillay and C. Steinhorn,Definable sets in ordered structures II, Trans. Amer. Math. Soc.295 (1986), 593–605.
[P1] Y. Peterzil,Some definability questions in structures over the reals and in general o-minimal structures, Ph.D. thesis, University of California, Berkeley, 1991.
[P2] Y. Peterzil,Zilber’s Conjecture for some o-minimal structures over the reals, to appear in Ann. Pure Appl. Logic.
[P3] Y. Peterzil,Constructing a group-interval in o-minimal structures, preprint.
[Pi] A. Pillay,On groups and fields definable in o-minimal structures, J. Pure Appl. Algebra53 (1988), 239–255.
[PSS] A. Pillay, P. Scowcroft and C. Steinhorn,Between groups and rings, Rocky Mountain J.9 (1989), 871–885.
[PS] A. Pillay and C. Steinhorn,Definable sets in ordered structures I, Trans. Amer. Math. Soc.295, (1986), 565–592.
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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). https://doi.org/10.1007/BF02761295
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DOI: https://doi.org/10.1007/BF02761295