Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T11:13:22.587Z Has data issue: false hasContentIssue false
  • English
  • Français

ON DEFINABLE SKOLEM FUNCTIONS IN WEAKLY O-MINIMAL NONVALUATIONAL STRUCTURES

Published online by Cambridge University Press:  09 January 2018

PANTELIS E. ELEFTHERIOU ,
ASSAF HASSON  and
GIL KEREN
PANTELIS E. ELEFTHERIOU
Affiliation:
ZUKUNFTSKOLLEG AND DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF KONSTANZ BOX 216, 78457 KONSTANZ, GERMANYE-mail:[email protected]
ASSAF HASSON
Affiliation:
DEPARTMENT OF MATHEMATICS BEN GURION UNIVERSITY OF THE NEGEV BE’ER SHEVA, ISRAELE-mail:[email protected]
GIL KEREN
Affiliation:
DEPARTMENT OF MATHEMATICS BEN GURION UNIVERSITY OF THE NEGEV BE’ER SHEVA, ISRAELE-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that all known examples of weakly o-minimal nonvaluational structures have no definable Skolem functions. We show, however, that such structures eliminate imaginaries up to definable families of cuts. Along the way we give some new examples of weakly o-minimal nonvaluational structures.

Keywords

03C64weakly o-minimal structuresdefinable Skolem functionso-minimal traces
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 4 , December 2017 , pp. 1482 - 1495
Copyright
Copyright © The Association for Symbolic Logic 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Baisalov, Y. and Poizat, B., Paires de structures o-minimales, this Journal, vol. 63 (1998), no. 2, pp. 570–578.Google Scholar
Bar Yehuda, E., On reducts of the canonical o-minimal completion of weakly o-minimal non-valuational structures, M.Sc. thesis, Ben Gurion University of the Negev, Israel, 2016.Google Scholar
Cherlin, G. and Dickmann, M. A., Real closed rings II. Model theory. Annals of Pure and Applied Logic, vol. 25 (1983), no. 3, pp. 213–231.Google Scholar
Hasson, A. and Onshuus, A., Embedded o-minimal structures. Bulletin of the London Mathematical Society, vol. 42 (2010), no. 1, pp. 64–74.Google Scholar
Laskowski, M. C. and Shaw, C., Definable choice for a class of weakly o-minimal theories. Archive for Mathematical Logic, vol. 55 (2016), no. 5, pp. 735–748.Google Scholar
Macpherson, D., Marker, D., and Steinhorn, C., Weakly o-minimal structures and real closed fields. Transactions of the American Mathematical Society, vol. 352 (2000), no. 12, pp. 5435–5483 (electronic).Google Scholar
Marker, D., Omitting types in o-minimal theories, this Journal, vol. 51 (1986), no. 1, pp. 63–74.Google Scholar
Miller, C. and Starchenko, S., A growth dichotomy for o-minimal expansions of ordered groups. Transactions of the American Mathematical Society, vol. 350 (1998), no. 9, pp. 3505–3521.Google Scholar
van den Dries, L, Tame Topology and o-minimal Structures, Cambridge University Press, Cambridge, 1998.Google Scholar
van den Dries, L, Dense pairs of o-minimal structures. Fundamenta Mathematicae, vol. 157 (1998), no. 1, pp. 61–78.Google Scholar
van den Dries, L, Limit sets in o-minimal structures, O-minimal Structures, Proceedings of the RAAG Summer School Lisbon 2003 (Edmundo, M., Richardson, D., and Wilkie, A., editors), Lecture Notes in Real Algebraic and Analytic Geometry, Cuvillier Verlag, 2005.Google Scholar
Wencel, R., Weakly o-minimal nonvaluational structures. Annals of Pure and Applied Logic, vol. 154 (2008), no. 3, pp. 139–162.CrossRefGoogle Scholar
Wencel, R., On expansions of weakly o-minimal non-valuational structures by convex predicates. Fundamenta Mathematicae, vol. 202 (2009), no. 2, pp. 147–159.Google Scholar
Wencel, R., On the strong cell decomposition property for weakly o-minimal structures. Mathematical Logic Quarterly, vol. 59 (2013), no. 6, pp. 452–470.Google Scholar
Wilkie, A. J., Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. Journal of the American Mathematical Society, vol. 9 (1996), no. 4, pp. 1051–1094.Google Scholar