Archive for Mathematical Logic

, Volume 55, Issue 5–6, pp 735–748 | Cite as

Definable choice for a class of weakly o-minimal theories

  • Michael C. Laskowski
  • Christopher S. ShawEmail author


Given an o-minimal structure \({\mathcal M}\) with a group operation, we show that for a properly convex subset U, the theory of the expanded structure \({\mathcal M}'=({\mathcal M},U)\) has definable Skolem functions precisely when \({\mathcal M}'\) is valuational. As a corollary, we get an elementary proof that the theory of any such \({\mathcal M}'\) does not satisfy definable choice.


Weakly o-minimal Skolem functions Definable choice 

Mathematics Subject Classification



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  1. 1.
    Baisalov, Y., Poizat, B.: Paires de structures o-minimales. J. Symb. Logic 63(2), 570–578 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cherlin, G., Dickmann, M.A.: Real closed rings I: residue rings of rings of continuous functions. Fund. Math. 126, 147–183 (1986)MathSciNetzbMATHGoogle Scholar
  3. 3.
    van den Dries, L.: \({T}\)-convexity and tame extensions II. J. Symb. Logic 62(1), 14–34 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    van den Dries, L.: Algebraic theories with definable Skolem functions. J. Symb. Logic 49(2), 625–629 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    van den Dries, L.: Dense pairs of o-minimal structures. Fund. Math. 157, 61–78 (1998)MathSciNetzbMATHGoogle Scholar
  6. 6.
    van den Dries, L.: Tame Topology and O-Minimal Structures. London Mathematical Society Lecture Note Series, vol. 248. Cambridge University Press, Cambridge (1998)CrossRefzbMATHGoogle Scholar
  7. 7.
    van den Dries, L., Lewenberg, A.H.: \({T}\)-convexity and tame extensions. J. Symb. Logic 60(1), 74–102 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Eleftheriou, P., Hasson, A., Keren, G.: On weakly o-minimal non valuational expansions of ordered groups. preprint (2016)Google Scholar
  9. 9.
    Keren, G.: Definable compactness in weakly o-minimal structures. M.Sc. thesis, Ben-Gurion University of the Negev (2014)Google Scholar
  10. 10.
    Knight, J., Pillay, A., Steinhorn, C.: Definable sets and ordered structures II. Trans. Am. Math. Soc. 295, 593–605 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    MacPherson, D., Marker, D., Steinhorn, C.: Weakly o-minimal structures and real closed fields. Trans. Am. Math. Soc. 352(12), 5435–5483 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Marker, D.: Omitting types in o-minimal theories. J. Symb. Logic 51(1), 63–74 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Pillay, A., Steinhorn, C.: Definable sets in ordered structures I. Trans. Am. Math. Soc. 295, 565–592 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Pillay, A., Steinhorn, C.: Definable sets in ordered structures III. Trans. Am. Math. Soc. 309, 469–476 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Shaw, C.: Weakly o-minimal structures and Skolem functions. Ph.D. thesis, University of Maryland (2008)Google Scholar
  16. 16.
    Wencel, R.: Weakly o-minimal nonvaluational structures. Ann. Pure Appl. Logic 154(3), 139–162 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA
  2. 2.Department of Science and MathematicsColumbia College ChicagoChicagoUSA

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