Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T05:55:29.993Z Has data issue: false hasContentIssue false
  • English
  • Français

AN INDEPENDENCE THEOREM FOR NTP2 THEORIES

Published online by Cambridge University Press:  17 April 2014

ITAÏ BEN YAACOV  and
ARTEM CHERNIKOV
ITAÏ BEN YAACOV
Affiliation:
UNIVERSITÉ CLAUDE BERNARD – LYON 1, INSTITUT CAMILLE JORDAN, CNRS UMR 5208, 43 BOULEVARD DU 11 NOVEMBRE 1918, 69622 VILLEURBANNE CEDEX, FRANCEURL:http://math.univ-lyon1.fr/∼begnac/
ARTEM CHERNIKOV
Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS, EDMOND J. SAFRA CAMPUS, GIVAT RAM, THE HEBREW UNIVERSITY OF JERUSALEM, JERUSALEM, 91904, ISRAELE-mail:[email protected], URL: http://chernikov.me
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish several results regarding dividing and forking in NTP2 theories. We show that dividing is the same as array-dividing. Combining it with existence of strictly invariant sequences we deduce that forking satisfies the chain condition over extension bases (namely, the forking ideal is S1, in Hrushovski’s terminology). Using it we prove an independence theorem over extension bases (which, in the case of simple theories, specializes to the ordinary independence theorem). As an application we show that Lascar strong type and compact strong type coincide over extension bases in an NTP2 theory.

We also define the dividing order of a theory—a generalization of Poizat’s fundamental order from stable theories—and give some equivalent characterizations under the assumption of NTP2. The last section is devoted to a refinement of the class of strong theories and its place in the classification hierarchy.

Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 1 , March 2014 , pp. 135 - 153
Copyright
Copyright © Association for Symbolic Logic 2014 

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

Adler, Hans, http://www.logic.univie.ac.at/ adler/docs/nip.pdf An introduction to theories without the independence property. Archive for Mathematical Logic, to appear.Google Scholar
Adler, Hans, Pre-independence relations, preprint.Google Scholar
Adler, Hans, http://www.logic.univie.ac.at/ adler/docs/strong.pdf Strong theories, burden, and weight, preprint.Google Scholar
Adler, Hans, Thorn-forking as local forking. Journal of Mathematical Logic, vol. 9 (2009), no. 1, pp. 21–38. http://dx.doi.org/10.1142/S0219061309000823doi:10.1142/S0219061309000823.CrossRefGoogle Scholar
Yaacov, ItaïBen, http://math.univ-lyon1.fr/ begnac/articles/catsim.pdf Simplicity in compact abstract theories. Journal of Mathematical Logic, vol. 3 (2003), no. 2, pp. 163–191. http://dx.doi.org/10.1142/S0219061303000297doi:10.1142/S0219061303000297.Google Scholar
Yaacov, ItaïBen, Pillay, Anand, and Vassiliev, Evgueni, http://math.univ-lyon1.fr/ begnac/articles/pairs.pdf Lovely pairs of models. Annals of Pure and Applied Logic, vol. 122 (2003), no. 1–3, pp. 235–261. http://dx.doi.org/10.1016/S0168-0072(03)00018-6doi:10.1016/S0168-0072(03)00018-6.Google Scholar
Casanovas, Enrique. Dividing and chain conditions, Archive for Mathematical Logic, vol. 42 (2003), no. 8, pp. 815–819. http://dx.doi.org/10.1007/s00153-003-0192-0doi:10.1007/s00153-003-0192-0.CrossRefGoogle Scholar
Chernikov, Artem, Theories without the tree property of the second kind. Annals of Pure and Applied Logic, vol. 165 (2004), no. 2, pp. 695–723.Google Scholar
Chernikov, Artem and Kaplan, Itay, Forking and dividing in NTP2 theories, this Journal, vol. 77 (2012), no. 1, pp. 1–20. http://arxiv.org/abs/0906.2806arXiv:0906.2806.Google Scholar
Chernikov, Artem, Kaplan, Itay, and Shelah, Saharon, On non-forking spectra, preprint, 2012, http://arxiv.org/abs/1205.3101arXiv:1205.3101.Google Scholar
Casanovas, Enrique, Lascar, Daniel, Pillay, Anand, and Ziegler, Martin, Galois groups of first order theories. Journal of Mathematical Logic, vol. 1 (2001), no. 2, pp. 305–319. doi:10.1142/S0219061301000119.Google Scholar
Dolich, Alfred. Weak dividing, chain conditions, and simplicity, Archive for Mathematical Logic, vol. 43 (2004), no. 2, pp. 265–283. http://dx.doi.org/10.1007/s00153-003-0176-0doi:10.1007/s00153-003-0176-0.Google Scholar
Grossberg, Rami, Iovino, José, and Lessmann, Olivier, A primer of simple theories. Archive for Mathematical Logic, vol. 41 (2002), no. 6, pp. 541–580. http://dx.doi.org/10.1007/s001530100126doi:10.1007/s001530100126.Google Scholar
Hrushovski, Ehud and Pillay, Anand, On NIP and invariant measures. Journal of the European Mathematical Society (JEMS), vol. 13 (2011), no. 4, pp. 1005–1061. http://dx.doi.org/10.4171/JEMS/274doi:10.4171/JEMS/274.Google Scholar
Hrushovski, Ehud, Stable group theory and approximate subgroups. Journal of the American Mathematical Society, vol. 25 (2012), no. 1, pp. 189–243. http://dx.doi.org/10.1090/S0894-0347-2011-00708-Xdoi:10.1090/S0894-0347-2011-00708-X.Google Scholar
Hrushovski, Ehud and Zilber, Boris, Zariski geometries. Journal of the American Mathematical Society, vol. 9 (1996), no. 1, pp. 1–56. http://dx.doi.org/10.1090/S0894-0347-96-00180-4doi:10.1090/S0894-0347-96-00180-4.Google Scholar
Kim, Bcyunghan, Simple first order theories, http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:9629800 Ph.D. thesis, University of Notre Dame, 1996, p. 96.Google Scholar
Kaplan, Itay, Onshuus, Alf, and Usvyatsov, Alexander, Additivity of the dp-rank. Transactions of the American Mathematical Society, vol. 365 (2013), no. 11, pp. 5783–5804. 03C45 (03C98 05Dxx 68R05).Google Scholar
Kaplan, Itay and Shelah, Saharon, Chain conditions in dependent groups. Annals of Pure and Applied Logic, vol. 164 (2013), no. 12, pp. 1322–1337. 03C45(03C98 05Dxx).Google Scholar
Kaplan, Itay and Usvyatsov, Alexander, Strict independence in dependent theories. Journal of Mathematical Logic, 2012, accepted.Google Scholar
Lessmann, Olivier, Counting partial types in simple theories. Colloquium Mathematicum, vol. 83 (2000), no. 2, pp. 201–208.Google Scholar
Onshuus, Alf and Usvyatsov, Alexander, On dp-minimality, strong dependence and weight, this Journal, vol. 76 (2011), no. 3, pp. 737–758. http://dx.doi.org/10.2178/jsl/1309952519doi:10.2178/jsl/1309952519.Google Scholar
Poizat, Bruno, Cours de théorie des modèles, Nur al-Mantiq wal-Ma’rifah, Lyon, 1985, Une introduction à la logique mathématique contemporaine.Google Scholar
Shelah, Saharon, Strongly dependent theories, preprint, http://arxiv.org/abs/math.LO/0504197arXiv:math.LO/0504197.Google Scholar
Shelah, Saharon, Simple unstable theories. Annals of Mathematical Logic, vol. 19 (1980), no. 3, pp. 177–203. http://dx.doi.org/10.1016/0003-4843(80)90009-1doi:10.1016/0003-4843(80)90009-1.CrossRefGoogle Scholar
Shelah, Saharon, Classification theory and the number of nonisomorphic models, second ed., Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, 1990.Google Scholar
Shelah, Saharon, Dependent first order theories, continued. Israel Journal of Mathematics, vol. 173 (2009), pp. 1–60. http://dx.doi.org/10.1007/s11856-009-0082-1doi:10.1007/s11856-009-0082-1.Google Scholar
Wagner, Frank O., Simple theories, Kluwer Academic Publishers, Dordrecht, 2000.Google Scholar