Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T19:32:53.275Z Has data issue: false hasContentIssue false
  • English
  • Français

DISTAL AND NON-DISTAL PAIRS

Published online by Cambridge University Press:  21 March 2017

PHILIPP HIERONYMI  and
TRAVIS NELL
PHILIPP HIERONYMI
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 1409 WEST GREEN STREET URBANA, IL 61801, USAE-mail: [email protected]: http://www.math.uiuc.edu/∼phierony
TRAVIS NELL
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 1409 WEST GREEN STREET URBANA, IL 61801, USAE-mail: [email protected]: http://www.math.uiuc.edu/∼tnell2
Get access Rights & Permissions [Opens in a new window]

Abstract

The aim of this note is to determine whether certain non-o-minimal expansions of o-minimal theories which are known to be NIP, are also distal. We observe that while tame pairs of o-minimal structures and the real field with a discrete multiplicative subgroup have distal theories, dense pairs of o-minimal structures and related examples do not.

Keywords

Primary 03C64Secondary 03C45DistalNIPexpansions of o-minimal structures
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 1 , March 2017 , pp. 375 - 383
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

Belegradek, O. and Zilber, B., The model theory of the field of reals with a subgroup of the unit circle . Journal of the London Mathematical Society (2), vol. 78 (2008), no. 3, pp. 563–579.CrossRefGoogle Scholar
Berenstein, A., Dolich, A., and Onshuus, A., The independence property in generalized dense pairs of structures, this Journal, vol. 76 (2011), no. 2, pp. 391–404.Google Scholar
Boxall, G., NIP for some pair-like theories . Archive for Mathematical Logic, vol. 50 (2011), no. 3–4, pp. 353–359.CrossRefGoogle Scholar
Chernikov, A. and Simon, P., Externally definable sets and dependent pairs . Israel Journal of Mathematics, vol. 194 (2013), no. 1, pp. 409–425.CrossRefGoogle Scholar
Dolich, A., Goodrick, J., and Lippel, D., Dp-minimality: basic facts and examples . Notre Dame Journal of Formal Logic, vol. 52 (2011), no. 3, pp. 267–288.CrossRefGoogle Scholar
Dolich, A., Miller, C., and Steinhorn, C., Expansions of o-minimal structures by dense independent sets , Annals of Pure and Applied Logic, vol. 167, (2016) no. 8, pp. 684–706.CrossRefGoogle Scholar
Eleftheriou, P. E., Günaydın, A., and Hieronymi, P., Structure theorems in tame expansions of o-minimal structures by a dense predicate, 2015, preprint.Google Scholar
Günaydin, A. and Hieronymi, P., Dependent pairs, this Journal, vol. 76 (2011), no. 2, pp. 377–390.Google Scholar
Günaydin, A. and Hieronymi, P., The real field with the rational points of an elliptic curve . Fundamenta Mathematicae, vol. 211 (2011), no. 1, pp. 15–40.CrossRefGoogle Scholar
Hieronymi, P., Defining the set of integers in expansions of the real field by a closed discrete set . Proceedings of the American Mathematical Society, vol. 138 (2010), no. 6, pp. 2163–2168.CrossRefGoogle Scholar
Hieronymi, P., The real field with an irrational power function and a dense multiplicative subgroup . Journal of London Mathematical Society (2), vol. 83 (2011), no. 1, pp. 153–167.CrossRefGoogle Scholar
Miller, C., Tameness in expansions of the real field , Logic Colloquium ’01, Lecture Notes in Logic, vol. 20, Association of Symbolic Logic, Urbana, IL, 2005, pp. 281–316.CrossRefGoogle Scholar
Shelah, S., Stability, the f.c.p., and superstability; model theoretic properties of formulas in first order theory . Annals of Mathematical Logic, vol. 3 (1971), no. 3, pp. 271–362.CrossRefGoogle Scholar
Simon, P., Distal and non-distal NIP theories . Annals of Pure and Applied Logic, vol. 164 (2013), no. 3, pp. 294–318.CrossRefGoogle Scholar
Simon, P., A guide to NIP theories, Lecture Notes in Logic, vol. 44, Cambridge University Press, Cambridge, 2015.CrossRefGoogle Scholar
van den Dries, L., The field of reals with a predicate for the powers of two . Manuscripta Mathematica, vol. 54 (1985), no. 1–2, pp. 187–195.CrossRefGoogle Scholar
van den Dries, L., Dense pairs of o-minimal structures . Fundamenta Mathematicae, vol. 157 (1998), no. 1, pp. 61–78.CrossRefGoogle Scholar
van den Dries, L. and Günaydin, A., The fields of real and complex numbers with a small multiplicative group . Proceedings of the London Mathematical Society (3), vol. 93 (2006), no. 1, pp. 43–81.CrossRefGoogle Scholar
van den Dries, L. and Lewenberg, A. H., T-convexity and tame extensions, this Journal, vol. 60 (1995), no. 1, pp. 74–102.Google Scholar