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Weakly o-minimal structures and real closed fields


Authors: Dugald Macpherson, David Marker and Charles Steinhorn
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
Published electronically: April 13, 2000
MathSciNet review: 1781273
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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.

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Additional Information

Dugald Macpherson
Affiliation: Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, U.K.
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

DOI: https://doi.org/10.1090/S0002-9947-00-02633-7
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
Article copyright: © Copyright 2000 American Mathematical Society