Additivity of the dp-rank
Authors:
Itay Kaplan, Alf Onshuus and Alexander Usvyatsov
Journal:
Trans. Amer. Math. Soc. 365 (2013), 5783-5804
MSC (2010):
Primary 03C45, 03C98; Secondary 05D99, 68R05
DOI:
https://doi.org/10.1090/S0002-9947-2013-05782-0
Published electronically:
July 1, 2013
MathSciNet review:
3091265
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank cannot be reduced to the study of its dp-minimal types, and we discuss the possible relations between dp-rank and VC-density.
- 1. Hans Adler, Strong theories, burden, and weight, preprint (2007).
- 2. Matthias Aschenbrenner, Alfred Dolich, Dierdre Haskell, Dugald MacPherson, and Sergei Starchenko, Vapnik-chervonenkis density in some theories without the independence property, arXiv:1109.5438 (2011).
- 3. Vincent Guingona and Cameron Donnay Hill, Local dp-rank and vc-density over indiscernible sequences, arXiv:1108.2554 (2011).
- 4. Tapani Hyttinen, Remarks on structure theorems for 𝜔₁-saturated models, Notre Dame J. Formal Logic 36 (1995), no. 2, 269–278. MR 1345748, https://doi.org/10.1305/ndjfl/1040248458
- 5. Itay Kaplan and Pierre Simon, Unpublished notes (2011).
- 6. Itay Kaplan and Alexander Usvyatsov, Strict independence in dependent theories, preprint (2011).
- 7. Alf Onshuus and Alexander Usvyatsov, On dp-minimality, strong dependence and weight, J. Symbolic Logic 76 (2011), no. 3, 737–758. MR 2849244, https://doi.org/10.2178/jsl/1309952519
- 8. Alf Onshuus and Alexander Usvyatsov, Thorn orthogonality and domination in unstable theories, Fund. Math. 214 (2011), no. 3, 241–268. MR 2845725, https://doi.org/10.4064/fm214-3-3
- 9. S. Shelah, Classification theory and the number of nonisomorphic models, 2nd ed., Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, 1990. MR 1083551
- 10. Saharon Shelah, Classification theory for elementary classes with the dependence property—a modest beginning, Sci. Math. Jpn. 59 (2004), no. 2, 265–316. Special issue on set theory and algebraic model theory. MR 2062198
- 11. Saharon Shelah, Dependent first order theories, continued, Israel J. Math. 173 (2009), 1–60. MR 2570659, https://doi.org/10.1007/s11856-009-0082-1
- 12. -, Strongly dependent theories, arXiv:0504197 (2009).
- 13. Pierre Simon, On dp-minimal ordered structures, J. Symbolic Logic 76 (2011), no. 2, 448–460. MR 2830411, https://doi.org/10.2178/jsl/1305810758
- 14. Alexander Usvyatsov, On generically stable types in dependent theories, J. Symbolic Logic 74 (2009), no. 1, 216–250. MR 2499428, https://doi.org/10.2178/jsl/1231082310
-saturated models, Notre Dame J. Formal Logic 36 (1995), no. 2, 269-278. Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 03C45, 03C98, 05D99, 68R05
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Additional Information
Itay Kaplan
Affiliation:
Mathematischen Instituts, Universität Münster, Einsteinstraße 62, 48149 Münster, Germany
Alf Onshuus
Affiliation:
Departamento de Matemáticas, Universidad de los Andes, Cra 1 No. 18A-10, Edificio H, Bogotá, 111711, Colombia
Alexander Usvyatsov
Affiliation:
Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal
DOI:
https://doi.org/10.1090/S0002-9947-2013-05782-0
Keywords:
Dp-rank,
dp-minimality,
VC-density
Received by editor(s):
September 7, 2011
Received by editor(s) in revised form:
December 21, 2011
Published electronically:
July 1, 2013
Additional Notes:
The third author was supported by FCT grant SFRH/BPD/34893/2007 and FCT research project PTDC/MAT/101740/2008.
We thank the organizers of the meeting “Model theory: Around Valued Fields and Dependent Theories” at Oberwolfach in January 2010. Special thanks to Sergei Starchenko for valuable discussions during and after the meeting. We would also like to thank John Goodrick, Alfred Dolich, Pierre Simon and the anonymous referee for some very helpful remarks and comments on earlier drafts of this paper.
Article copyright:
© Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
