Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T07:01:39.741Z Has data issue: false hasContentIssue false
  • English
  • Français

FORKING, IMAGINARIES, AND OTHER FEATURES OF $\text {ACFG}$

Part of: Model theory

Published online by Cambridge University Press:  07 June 2021

CHRISTIAN D’ELBÉE
CHRISTIAN D’ELBÉE*
Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS THE HEBREW UNIVERSITY OF JERUSALEM GIVAT RAM 9190401, JERUSALEM, ISRAELE-mail:[email protected]:http://choum.net/~chris/page_perso/
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the generic theory of algebraically closed fields of fixed positive characteristic with a predicate for an additive subgroup, called $\mathrm {ACFG}$ . This theory was introduced in [16] as a new example of $\mathrm {NSOP}_{1}$ nonsimple theory. In this paper we describe more features of $\mathrm {ACFG}$ , such as imaginaries. We also study various independence relations in $\mathrm {ACFG}$ , such as Kim-independence or forking independence, and describe interactions between them.

Keywords

fields of positive characteristic with generic subgroupNSOP1 theoriesforkingKim-forkingelimination of imaginaries

MSC classification

Primary: 03C10: Quantifier elimination, model completeness and related topics
Secondary: 03C30: Other model constructions 03C45: Classification theory, stability and related concepts 03C65: Models of other mathematical theories
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 2 , June 2021 , pp. 669 - 700
Check for updates
Copyright
© The Association for Symbolic Logic 2021

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, H., Around the strong order property and mock simplicity. Unpublished note, 2008.Google Scholar
Adler, H., A geometric introduction to forking and thorn-forking . Journal of Mathematical Logic , vol. 9 (2009), no. 1, pp. 1–20.10.1142/S0219061309000811CrossRefGoogle Scholar
Adler, H., Thorn-forking as local forking . Journal of Mathematical Logic, vol. 9 (2009), no. 1, pp. 21–38.10.1142/S0219061309000823CrossRefGoogle Scholar
Barbina, S. and Casanovas, E., Model theory of Steiner triple systems . Journal of Mathematical Logic, vol. 20 (2020), no. 2, p. 2050010.10.1142/S0219061320500105CrossRefGoogle Scholar
Blossier, T. and Martin-Pizarro, A., Un critère simple . Notre Dame Journal of Formal Logic , vol. 60 (2019), pp. 639–663.10.1215/00294527-2019-0023CrossRefGoogle Scholar
Bouscaren, E., Delon, F., Hindry, M., Hrushovski, E., Lascar, D., Marker, D., Pillay, A., Wood, C. and Ziegler, M., Model Theory and Algebraic Geometry: An Introduction to E. Hrushovski’s Proof of the Geometric Mordell-Lang Conjecture, Lecture Notes in Mathematics, vol. 1696, Springer-Verlag, Berlin, 1998.10.1007/978-3-540-68521-0CrossRefGoogle Scholar
Casanovas, E., Simple Theories and Hyperimaginaries , Lecture Notes in Logic, vol. 39, Association for Symbolic Logic, Chicago, 2011.CrossRefGoogle Scholar
Chatzidakis, Z., Model theory of finite fields and pseudo-finite fields . Annals of Pure and Applied Logic , vol. 88 (1997), nos. 2–3, pp. 95–108.10.1016/S0168-0072(97)00017-1CrossRefGoogle Scholar
Chatzidakis, Z., Properties of forking in omega-free pseudo-algebraically closed fields, this Journal, vol. 67 (2002), no. 3, pp. 957–996.Google Scholar
Chatzidakis, Z., Independence in (unbounded) pac fields, and imaginaries. Unpublished note, 2008.Google Scholar
Chatzidakis, Z. and Pillay, A., Generic structures and simple theories. Annals of Pure and Applied Logic, vol. 95 (1998), no. 1, pp. 71–92.10.1016/S0168-0072(98)00021-9CrossRefGoogle Scholar
Chernikov, A. and Kaplan, I., Forking and dividing in NTP2 theories, this Journal, vol. 77 (2012), no. 1, pp. 1–20.Google Scholar
Chernikov, A. and Ramsey, N., On model-theoretic tree properties . Journal of Mathematical Logic , vol. 16 (2016), no. 2, p. 1650009.10.1142/S0219061316500094CrossRefGoogle Scholar
Conant, G., Forking and dividing in Henson graphs . Notre Dame Journal of Formal Logic , vol. 58 (2017), no. 4, p. 2017.10.1215/00294527-2017-0016CrossRefGoogle Scholar
Conant, G. and Kruckman, A., Independence in generic incidence structures, 2017, 1709.09626v1 [math.LO].Google Scholar
d’Elbée, C., Generic expansions by a reduct. Journal of Mathematical Logic (2021), p. 2150016.10.1142/S0219061321500161CrossRefGoogle Scholar
de Cornulier, Y., On the Chabauty space of locally compact abelian groups. Algebric and Geometric Topology, vol. 11 (2011), no. 4, pp. 2007–2035.10.2140/agt.2011.11.2007CrossRefGoogle Scholar
Evans, D. and Hrushovski, E., On the automorphism groups of finite covers. Annals of Pure and Applied Logic, vol. 62 (1993), no. 2, pp. 83–112.10.1016/0168-0072(93)90168-DCrossRefGoogle Scholar
Fresnel, J., Anneaux , Hermann, Paris, 2001.Google Scholar
Granger, N., Stability, simplicity, and the model theory of bilinear forms. Ph.D. thesis, University of Manchester, 1999.Google Scholar
Humphreys, J. E., Linear Algebraic Groups, fourth ed., Graduate Texts in Mathematics, vol. 21, Springer, Berlin, 1998.Google Scholar
Kaplan, I. and Ramsey, N., On Kim-independence. Journal of the European Mathematical Society, vol. 22 (2020), no. 5, pp. 1423–1474.10.4171/JEMS/948CrossRefGoogle Scholar
Kim, B. and Kim, H.-J., Notions around tree property 1. Annals of Pure and Applied Logic, vol. 162 (2011), no. 9, pp. 698–709.10.1016/j.apal.2011.02.001CrossRefGoogle Scholar
Kim, B. and Pillay, A., Simple theories . Annals of Pure and Applied Logic , vol. 88 (1997), no. 2, pp. 149–164. Joint AILA-KGS Model Theory Meeting.10.1016/S0168-0072(97)00019-5CrossRefGoogle Scholar
Kruckman, A. and Ramsey, N., Generic expansion and skolemization in NSOP1 theories . Annals of Pure and Applied Logic , vol. 169 (2018), no. 8, pp. 755–774.10.1016/j.apal.2018.04.003CrossRefGoogle Scholar
Neumann, P. M., The structure of finitary permutation groups . Archiv der Mathematik (Basel) , vol. 27 (1976), no. 1, pp. 3–17.10.1007/BF01224634CrossRefGoogle Scholar