Weakly o-minimal structures and real closed fields

Macpherson, Dugald; Marker, David; Steinhorn, Charles

doi:10.1090/S0002-9947-00-02633-7

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ISSN 1088-6850(online) ISSN 0002-9947(print)

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
Posted: 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.

1. A. Friedman, On the mean value theorem, Bull. Res. Council Israel. Sect. A. 6 (1956), 47–49. MR 0087716 (19,398c)
2. R. Arefiev, On monotonicity for weakly o-minimal structures, preprint.
3. Y. Baisalov and B. Poizat, Paires de structures o-minimales, J. Symbolic Logic, 63 (1998) 570-578. MR 89m:03063
4. Peter J. Cameron, Some treelike objects, Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155–183. MR 891613 (89a:05009), http://dx.doi.org/10.1093/qmath/38.2.155
5. Gregory Cherlin and Max A. Dickmann, Real closed rings. II. Model theory, Ann. Pure Appl. Logic 25 (1983), no. 3, 213–231. MR 730855 (86j:03033), http://dx.doi.org/10.1016/0168-0072(83)90019-2
6. M. A. Dickmann, Elimination of quantifiers for ordered valuation rings, J. Symbolic Logic 52 (1987), no. 1, 116–128. MR 877859 (88b:03045), http://dx.doi.org/10.2307/2273866
7. Lou van den Dries, Remarks on Tarski’s problem concerning (𝑅,+,⋅,𝑒𝑥𝑝), Logic colloquium ’82 (Florence, 1982) Stud. Logic Found. Math., vol. 112, North-Holland, Amsterdam, 1984, pp. 97–121. MR 762106 (86g:03052), http://dx.doi.org/10.1016/S0049-237X(08)71811-1
8. Lou van den Dries, 𝑇-convexity and tame extensions. II, J. Symbolic Logic 62 (1997), no. 1, 14–34. MR 1450511 (98h:03048), http://dx.doi.org/10.2307/2275729
9. Lou van den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998. MR 1633348 (99j:03001)
10. Lou van den Dries and Adam H. Lewenberg, 𝑇-convexity and tame extensions, J. Symbolic Logic 60 (1995), no. 1, 74–102. MR 1324502 (96a:03048), http://dx.doi.org/10.2307/2275510
11. Angus Macintyre, Kenneth McKenna, and Lou van den Dries, Elimination of quantifiers in algebraic structures, Adv. in Math. 47 (1983), no. 1, 74–87. MR 689765 (84f:03028), http://dx.doi.org/10.1016/0001-8708(83)90055-5
12. Deirdre Haskell and Dugald Macpherson, Cell decompositions of 𝐶-minimal structures, Ann. Pure Appl. Logic 66 (1994), no. 2, 113–162. MR 1262433 (95d:03059), http://dx.doi.org/10.1016/0168-0072(94)90064-7
13. B. Herwig, H. D. Macpherson, G. Martin, A. Nurtazin, and J. K. Truss, On $\aleph_0\/$ -categorical weakly o-minimal theories, preprint.
14. Julia F. Knight, Anand Pillay, and Charles Steinhorn, Definable sets in ordered structures. II, Trans. Amer. Math. Soc. 295 (1986), no. 2, 593–605. MR 833698 (88b:03050b), http://dx.doi.org/10.1090/S0002-9947-1986-0833698-1
15. F-V. Kuhlmann, Abelian groups with contractions I, Contemporary Math. 171 (1994) 217-241. MR 95:03079
16. Franz-Viktor Kuhlmann, Abelian groups with contractions. II. Weak 𝑜-minimality, Abelian groups and modules (Padova, 1994) Math. Appl., vol. 343, Kluwer Acad. Publ., Dordrecht, 1995, pp. 323–342. MR 1378210 (97g:03044)
17. Michael C. Laskowski, Vapnik-Chervonenkis classes of definable sets, J. London Math. Soc. (2) 45 (1992), no. 2, 377–384. MR 1171563 (93d:03039), http://dx.doi.org/10.1112/jlms/s2-45.2.377
18. Saunders MacLane, The universality of formal power series fields, Bull. Amer. Math. Soc. 45 (1939), 888–890. MR 0000610 (1,102c), http://dx.doi.org/10.1090/S0002-9904-1939-07110-3
19. Dugald Macpherson and Charles Steinhorn, On variants of o-minimality, Ann. Pure Appl. Logic 79 (1996), no. 2, 165–209. MR 1396850 (97e:03050), http://dx.doi.org/10.1016/0168-0072(95)00037-2
20. L. Mathews, D. Phil dissertation, University of Oxford 1992.
21. Anand Pillay, An introduction to stability theory, Oxford Logic Guides, vol. 8, The Clarendon Press Oxford University Press, New York, 1983. MR 719195 (85i:03104)
22. Anand Pillay and Charles Steinhorn, Definable sets in ordered structures. I, Trans. Amer. Math. Soc. 295 (1986), no. 2, 565–592. MR 833697 (88b:03050a), http://dx.doi.org/10.1090/S0002-9947-1986-0833697-X
23. Alexander Prestel, Lectures on formally real fields, Lecture Notes in Mathematics, vol. 1093, Springer-Verlag, Berlin, 1984. MR 769847 (86h:12013)
24. Paulo Ribenboim, Théorie des valuations, Deuxième édition multigraphiée. Séminaire de Mathématiques Supérieures, No. 9 (Été, vol. 1964, Les Presses de l’Université de Montréal, Montreal, Que., 1968 (French). MR 0249425 (40 #2670)

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