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DP-MINIMAL VALUED FIELDS

Published online by Cambridge University Press:  21 March 2017

FRANZISKA JAHNKE ,
PIERRE SIMON  and
ERIK WALSBERG
FRANZISKA JAHNKE
Affiliation:
INSTITUT FÜR MATHEMATISCHE LOGIK UND GRUNDLAGENFORSCHUNG WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER EINSTEINSTR. 62, 48149 MÜNSTER GERMANYE-mail: franziska.jahnke@uni-muenster.de
PIERRE SIMON
Affiliation:
CNRS UMR 5208, INSTITUT CAMILLE JORDAN UNIV LYON, UNIVERSITÉ CLAUDE BERNARD LYON 1 43 BLVD. F-69622 VILLEURBANNE CEDEX FRANCEE-mail: simon@math.univ-lyon1.fr
ERIK WALSBERG
Affiliation:
INSTITUTE OF MATHEMATICS THE HEBREW UNIVERSITY OF JERUSALEM GIVAT RAM, JERUSALEM 91904, ISRAELE-mail: erikw@math.ucla.edu
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Abstract

We show that dp-minimal valued fields are henselian and give classifications of dp-minimal ordered abelian groups and dp-minimal ordered fields without additional structure.

Keywords

dp-minimalityvalued fieldsNIP
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 1 , March 2017 , pp. 151 - 165
Copyright
Copyright © The Association for Symbolic Logic 2017 

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References

REFERENCES

Aschenbrenner, M., Dolich, A., Haskell, D., Macpherson, D., and Starchenko, S., Vapnik-Chervonenkis density in some theories without the independence property , Transaction of the American Mathematical Society, (2016), 5889–5949.Google Scholar
Aschenbrenner, M., Dolich, A., Haskell, D., Macpherson, D., and Starchenko, S., Vapnik-Chervonenkis density in some theories without the independence property , Notre Dame Journal of Formal Logic, vol. 54 (2013), no. 3, 4, pp. 311–363.CrossRefGoogle Scholar
Chernikov, A. and Simon, P., Henselian valued fields and inp-minimality, preprint, 2015.Google Scholar
Cluckers, R. and Halupczok, I., Quantifier elimination in ordered abelian groups . Confluentes Mathematici, vol. 3 (2011), no. 4, pp. 587–615.CrossRefGoogle Scholar
Engler, A. J. and Prestel, A., Valued Fields, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005.Google Scholar
Goodrick, J., A monotonicity theorem for dp-minimal densely ordered groups, this Journal, vol. 75 (2010), no. 1, pp. 221–238.Google Scholar
Guingona, V., On vc-minimal fields and dp-smallness . Archive for Mathematical Logic, vol. 53 (2014), no. 5, 6, pp. 503–517.CrossRefGoogle Scholar
Jahnke, F. and Koenigsmann, J., Uniformly defining p-henselian valuations . Annals of Pure and Applied Logic, vol. 166 (2015), no. 7, 8, pp. 741–754.Google Scholar
Johnson, W., On dp-minimal fields, preprint, 2015.Google Scholar
Kaplan, I., Scanlon, T., and Wagner, F. O., Artin-Schreier extensions in NIP and simple fields . Israel Journal of Mathematics, vol. 185 (2011), pp. 141–153.Google Scholar
Kudaĭbergenov, K. Zh., Weakly quasi-o-minimal models . Turkish Journal of Mathematics, vol. 13 (2010), no. 1, pp. 156–168.Google Scholar
Macintyre, A., McKenna, K., and van den Dries, L., Elimination of quantifiers in algebraic structures . Advances in Mathematics, vol. 47 (1983), no. 1, pp. 74–87.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), pp. 5435–5483.Google Scholar
Onshuus, A. and Usvyatsov, A., On dp-minimality, strong dependence and weight, this Journal, vol. 76 (2011), no. 3, pp. 737–758.Google Scholar
Prestel, A. and Delzell, C. N., Mathematical Logic and Model Theory, Universitext, Springer, 2011.CrossRefGoogle Scholar
Prestel, A. and Ziegler, M., Model-theoretic methods in the theory of topological fields . Journal für die Reine und Angewandte Mathematik, vol. 299 (1978), no. 300, pp. 318–341.Google Scholar
Simon, P., On dp-minimal ordered structures, this Journal, vol. 76 (2011), pp. 448–460.Google Scholar
Simon, P., Dp-minimality: Invariant types and dp-rank, this Journal, vol. 79 (2014), pp. 1025–1045.Google Scholar
Simon, P., A Guide to NIP Theories, Lecture Notes in Logic. Cambridge University Press, Cambridge, 2015.CrossRefGoogle Scholar