Skip to main content Accessibility help
×
Home
Hostname: page-component-5959bf8d4d-4p99k Total loading time: 0.356 Render date: 2022-12-10T03:42:35.834Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

AN INDEPENDENCE THEOREM FOR NTP2 THEORIES

Published online by Cambridge University Press:  17 April 2014

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:art.chernikov@gmail.com, URL: http://chernikov.me

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
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.
Adler, Hans, Pre-independence relations, preprint.
Adler, Hans, http://www.logic.univie.ac.at/ adler/docs/strong.pdf Strong theories, burden, and weight, preprint.
Adler, Hans, Thorn-forking as local forking. Journal of Mathematical Logic, vol. 9 (2009), no. 1, pp. 2138. http://dx.doi.org/10.1142/S0219061309000823doi:10.1142/S0219061309000823.CrossRef
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. 163191. http://dx.doi.org/10.1142/S0219061303000297doi:10.1142/S0219061303000297.
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. 235261. http://dx.doi.org/10.1016/S0168-0072(03)00018-6doi:10.1016/S0168-0072(03)00018-6.CrossRef
Casanovas, Enrique. Dividing and chain conditions, Archive for Mathematical Logic, vol. 42 (2003), no. 8, pp. 815819. http://dx.doi.org/10.1007/s00153-003-0192-0doi:10.1007/s00153-003-0192-0.CrossRef
Chernikov, Artem, Theories without the tree property of the second kind. Annals of Pure and Applied Logic, vol. 165 (2004), no. 2, pp. 695723.
Chernikov, Artem and Kaplan, Itay, Forking and dividing in NTP2 theories, this Journal, vol. 77 (2012), no. 1, pp. 120. http://arxiv.org/abs/0906.2806arXiv:0906.2806.
Chernikov, Artem, Kaplan, Itay, and Shelah, Saharon, On non-forking spectra, preprint, 2012, http://arxiv.org/abs/1205.3101arXiv:1205.3101.
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. 305319. doi:10.1142/S0219061301000119.
Dolich, Alfred. Weak dividing, chain conditions, and simplicity, Archive for Mathematical Logic, vol. 43 (2004), no. 2, pp. 265283. http://dx.doi.org/10.1007/s00153-003-0176-0doi:10.1007/s00153-003-0176-0.
Grossberg, Rami, Iovino, José, and Lessmann, Olivier, A primer of simple theories. Archive for Mathematical Logic, vol. 41 (2002), no. 6, pp. 541580. http://dx.doi.org/10.1007/s001530100126doi:10.1007/s001530100126.CrossRef
Hrushovski, Ehud and Pillay, Anand, On NIP and invariant measures. Journal of the European Mathematical Society (JEMS), vol. 13 (2011), no. 4, pp. 10051061. http://dx.doi.org/10.4171/JEMS/274doi:10.4171/JEMS/274.CrossRef
Hrushovski, Ehud, Stable group theory and approximate subgroups. Journal of the American Mathematical Society, vol. 25 (2012), no. 1, pp. 189243. http://dx.doi.org/10.1090/S0894-0347-2011-00708-Xdoi:10.1090/S0894-0347-2011-00708-X.
Hrushovski, Ehud and Zilber, Boris, Zariski geometries. Journal of the American Mathematical Society, vol. 9 (1996), no. 1, pp. 156. http://dx.doi.org/10.1090/S0894-0347-96-00180-4doi:10.1090/S0894-0347-96-00180-4.CrossRef
Kaplan, Itay, Onshuus, Alf, and Usvyatsov, Alexander, Additivity of the dp-rank. Transactions of the American Mathematical Society, vol. 365 (2013), no. 11, pp. 57835804. 03C45 (03C98 05Dxx 68R05).CrossRef
Kaplan, Itay and Shelah, Saharon, Chain conditions in dependent groups. Annals of Pure and Applied Logic, vol. 164 (2013), no. 12, pp. 13221337. 03C45(03C98 05Dxx).CrossRef
Kaplan, Itay and Usvyatsov, Alexander, Strict independence in dependent theories. Journal of Mathematical Logic, 2012, accepted.
Lessmann, Olivier, Counting partial types in simple theories. Colloquium Mathematicum, vol. 83 (2000), no. 2, pp. 201208.
Onshuus, Alf and Usvyatsov, Alexander, On dp-minimality, strong dependence and weight, this Journal, vol. 76 (2011), no. 3, pp. 737758. http://dx.doi.org/10.2178/jsl/1309952519doi:10.2178/jsl/1309952519.
Poizat, Bruno, Cours de théorie des modèles, Nur al-Mantiq wal-Ma’rifah, Lyon, 1985, Une introduction à la logique mathématique contemporaine.
Shelah, Saharon, Strongly dependent theories, preprint, http://arxiv.org/abs/math.LO/0504197arXiv:math.LO/0504197.
Shelah, Saharon, Simple unstable theories. Annals of Mathematical Logic, vol. 19 (1980), no. 3, pp. 177203. http://dx.doi.org/10.1016/0003-4843(80)90009-1doi:10.1016/0003-4843(80)90009-1.CrossRef
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. 160. http://dx.doi.org/10.1007/s11856-009-0082-1doi:10.1007/s11856-009-0082-1.
Wagner, Frank O., Simple theories, Kluwer Academic Publishers, Dordrecht, 2000.CrossRefGoogle Scholar
10
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

AN INDEPENDENCE THEOREM FOR NTP2 THEORIES
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

AN INDEPENDENCE THEOREM FOR NTP2 THEORIES
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

AN INDEPENDENCE THEOREM FOR NTP2 THEORIES
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *