Weakly o-minimal structures and real closed fields
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- by Dugald Macpherson, David Marker and Charles Steinhorn
- Trans. Amer. Math. Soc. 352 (2000), 5435-5483
- DOI: https://doi.org/10.1090/S0002-9947-00-02633-7
- Published electronically: April 13, 2000
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Abstract:
A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures were introduced by Dickmann, and they arise in several contexts. We here prove several fundamental results about weakly o-minimal structures. Foremost among these, we show that every weakly o-minimal ordered field is real closed. We also develop a substantial theory of definable sets in weakly o-minimal structures, patterned, as much as possible, after that for o-minimal structures.References
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Bibliographic Information
- Dugald Macpherson
- Affiliation: Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, U.K.
- MR Author ID: 224239
- Email: pmthdm@amsta.leeds.ac.uk
- David Marker
- Affiliation: Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607
- Email: marker@math.uic.edu
- Charles Steinhorn
- Affiliation: Department of Mathematics, Vassar College, Poughkeepsie, New York 12604
- Email: steinhorn@vassar.edu
- Received by editor(s): April 24, 1998
- Published electronically: April 13, 2000
- Additional Notes: The second author’s research was partially supported by NSF grant DMS-9626856, and the third author’s was partially supported by NSF grants DMS-9401723 and DMS-9704869, and SERC grant GR/H57097
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 5435-5483
- MSC (2000): Primary 03C60, 03C64
- DOI: https://doi.org/10.1090/S0002-9947-00-02633-7
- MathSciNet review: 1781273