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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.



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[1] 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), 58895949.
[2] 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. 311363.
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[6] Goodrick, J., A monotonicity theorem for dp-minimal densely ordered groups, this Journal, vol. 75 (2010), no. 1, pp. 221238.
[7] Guingona, V., On vc-minimal fields and dp-smallness . Archive for Mathematical Logic, vol. 53 (2014), no. 5, 6, pp. 503517.
[8] Jahnke, F. and Koenigsmann, J., Uniformly defining p-henselian valuations . Annals of Pure and Applied Logic, vol. 166 (2015), no. 7, 8, pp. 741754.
[9] Johnson, W., On dp-minimal fields, preprint, 2015.
[10] Kaplan, I., Scanlon, T., and Wagner, F. O., Artin-Schreier extensions in NIP and simple fields . Israel Journal of Mathematics, vol. 185 (2011), pp. 141153.
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[12] Macintyre, A., McKenna, K., and van den Dries, L., Elimination of quantifiers in algebraic structures . Advances in Mathematics, vol. 47 (1983), no. 1, pp. 7487.
[13] 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. 54355483.
[14] Onshuus, A. and Usvyatsov, A., On dp-minimality, strong dependence and weight, this Journal, vol. 76 (2011), no. 3, pp. 737758.
[15] Prestel, A. and Delzell, C. N., Mathematical Logic and Model Theory, Universitext, Springer, 2011.
[16] 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. 318341.
[17] Simon, P., On dp-minimal ordered structures, this Journal, vol. 76 (2011), pp. 448460.
[18] Simon, P., Dp-minimality: Invariant types and dp-rank, this Journal, vol. 79 (2014), pp. 10251045.
[19] Simon, P., A Guide to NIP Theories, Lecture Notes in Logic. Cambridge University Press, Cambridge, 2015.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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