Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-08T19:27:13.871Z Has data issue: false hasContentIssue false
  • English
  • Français

VARIATIONS SUR UN THÈME DE ALDAMA ET SHELAH

Published online by Cambridge University Press:  22 January 2016

CÉDRIC MILLIET
CÉDRIC MILLIET*
Affiliation:
UNIVERSITÉ GALATASARAY FACULTÉ DE SCIENCES ET DE LETTRES DÉPARTEMENT DE MATHÉMATIQUES ÇIRAĞAN CADDESI N°36 34357 ORTAKÖY, İSTANBUL,TÜRKIYE UNIVERSITÄT KONSTANZ FACHBEREICH MATEMATIK UND STATISTIK 78457 KONSTANZ, GERMANY PÔLE DE MATHÉMATIQUES DE L’INSA DE LYON BÂTIMENT LÉONARD DE VINCI 21, AVENUE JEAN CAPELLE 69621 VILLEURBANNE , FRANCEE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a group G that does not have the independence property and study the definability of certain subgroups of G, using parameters from a fixed elementary extension G of G. If X is a definable subset of G, its trace on G is called an externally definable subset. If H is a definable subgroup of G, we call its trace on G an external subgroup. We show the following. For any subset A of G and any external subgroup H of G, the centraliser of A, the A-core of H and the iterated centres of H are external subgroups. The normaliser of H and the iterated centralisers of A are externally definable. A soluble subgroup S of derived length ℓ is contained in an S-invariant externally definable soluble subgroup of G of derived length ℓ. The subgroup S is also contained in an externally definable subgroup X ∩ G of G such that X generates a soluble subgroup of G of derived length ℓ. Analogue results are discussed when G is merely a type definable group in a structure that does not have the independence property.

Keywords

03C4503C60Model theoryindependence propertyshattering typeVC-dimensionabeliannilpotentand soluble subgroupsnice subgroupdefinable and type definable envelope
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 1 , March 2016 , pp. 96 - 126
Copyright
Copyright © The Association for Symbolic Logic 2016 

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

Altınel, Tuna and Baginski, Paul, Definable envelopes of nilpotent subgroups of groups with chain conditions on centralizers, Proceedings of the American Mathematical Society, vol. 142 (2014), no. 5, pp. 1497–1506.CrossRefGoogle Scholar
Adler, Hans, An introduction to theories without the independence property, Archive for Mathematical Logic, to appear.Google Scholar
Bektur Sembiuly Baizhanov, Expansion of a model of a weakly o-minimal theory by a family of unary predicates, this Journal, vol. 66, (2001), no. 3, pp. 1382–1414.Google Scholar
Bélair, Luc, Panorama of p-adic model theory. Annales des sciences mathématiques du Québec, vol. 36 (2012), no. 1, pp. 43–75.Google Scholar
Baldwin, John T. and Saxl, Jan, Logical stability in group theory. Journal of the Australian Mathematical Society, Series A, vol. 21 (1976), no. 3, pp. 267–276.Google Scholar
Conversano, Annalisa and Pillay, Anand, Connected components of definable groups and o-minimality I. Advances in Mathematics, vol. 231 (2012), no. 2, pp. 605–623.CrossRefGoogle Scholar
de Aldama, Ricardo, Definable nilpotent and soluble envelopes in groups without the independence property. Mathematical Logic Quarterly, vol. 59 (2013), no. 3, pp. 201–205.CrossRefGoogle Scholar
Delon, Françoise, Types surC ((X)), Study Group on Stable Theories (Bruno Poizat), Second year: 1978/79 (French), Secrétariat Math., Paris, 1981, Exp. No. 5, 29 pages.Google Scholar
Delon, Françoise, Définissabilité avec paramètres extérieurs dans Qpet R. Proceedings of the American Mathematical Society, vol. 106 (1989), no. 1, pp. 193–198.Google Scholar
Fisher, Edward R., Abelian structures. I, Abelian group theory (Proc. Second New Mexico State Univ. Conf., Las Cruces, N.M., 1976), Lecture Notes in Mathematics, vol. 616, Springer, Berlin, 1977, pp. 270–322.Google Scholar
Gurevich, Yuri and Schmitt, Peter H., The theory of ordered abelian groups does not have the independence property. Transactions of the American Mathematical Society, vol. 284 (1984), no. 1, pp. 171–182.Google Scholar
Harnik, Victor and Harrington, Leo, Fundamentals of forking. Annals of Pure and Applied Logic, vol. 26 (1984), no. 3, pp. 245–286.CrossRefGoogle Scholar
Hrushovski, Ehud, Peterzil, Ya’acov, and Pillay, Anand, On central extensions and definably compact groups in o-minimal structures. Journal of Algebra, vol. 327 (2011), pp. 71–106.Google Scholar
Kaplan, Itay, Scanlon, Thomas, and Wagner, Frank O., Artin-Schreier extensions in NIP and simple fields. Israel Journal of Mathematics, vol. 185 (2011), pp. 141–153.Google Scholar
Laskowski, Michael C., Vapnik-Chervonenkis classes of definable sets. Journal of the London Mathematical Society (2), vol. 45 (1992), no. 2, pp. 377–384.Google Scholar
Lapidus, Michel L. and Lũ, Hùng, Nonarchimedean Cantor set and string. Journal of Fixed Point Theory and its Applications, vol. 3 (2008), no. 1, pp. 181–190.CrossRefGoogle Scholar
Matthews, Larry, The independence property in unstable algebraic structures I: p-adically closed fields, unpublished preprint.Google Scholar
Macpherson, Dugald and Steinhorn, Charles, One-dimensional asymptotic classes of finite structures. Transactions of the American Mathematical Society, vol. 360 (2008), no. 1, pp. 411–448 (electronic).Google Scholar
Macpherson, Dugald and Tent, Katrin, Pseudofinite groups with NIP theory and definability in finite simple groups, Groups and model theory, Contemporary Mathematics, vol. 576, American Mathematical Society, Providence, RI, 2012, pp. 255–267.Google Scholar
Neumann, B. H., Groups covered by permutable subsets. Journal of London Mathematical Society, vol. 29 (1954), pp. 236–248.CrossRefGoogle Scholar
Pillay, Anand, On groups and fields definable in o-minimal structures. Journal of Pure and Applied Algebra, vol. 53 (1988), no. 3, pp. 239–255.Google Scholar
Poizat, Bruno, Cours de théorie des modèles, Bruno Poizat, Lyon, 1985, Une introduction à la logique mathématique contemporaine. [An introduction to contemporary mathematical logic].Google Scholar
Poizat, Bruno, Groupes stables, Nur al-Mantiq wal-Ma’rifah [Light of Logic and Knowledge], 2, Bruno Poizat, Lyon, 1987, Une tentative de conciliation entre la géométrie algébrique et la logique mathématique. [An attempt at reconciling algebraic geometry and mathematical logic].Google 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. Isreal Journal of Mathematics, vol. 173 (2009), pp. 1–60.CrossRefGoogle Scholar
Szmielew, Wanda, Elementary properties of Abelian groups. Fundamenta Mathematicae, vol. 41 (1955), pp. 203–271.Google Scholar
Vapnik, Vladimir N. and Ya, Alexey. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities.Theory of Probability and Its Applications, vol. 16 (1971), pp. 264–280 (English).Google Scholar
van den Dries, Lou, Tarski’s problem and Pfaffian functions, Logic colloquium ’84 (Manchester, 1984), Studies in Logic and the Foundations of Mathematics, vol. 120, North-Holland, Amsterdam, 1986, pp. 59–90.Google Scholar
Wagner, Frank O., Stable groups, London Mathematical Society Lecture Note Series, vol. 240, Cambridge University Press, Cambridge, 1997.CrossRefGoogle Scholar
Weispfenning, Volker, Quantifier elimination for abelian structures, 1993, preprint.Google Scholar
Wood, Carol, Notes on the stability of separably closed fields, this Journal, vol. 44 (1979), no. 3, pp. 412–416.Google Scholar