Skip to main content
Log in

Positive Jonsson Theories

  • Published:
Logica Universalis Aims and scope Submit manuscript

Abstract

This paper is a general introduction to Positive Logic, where only what we call h-inductive sentences are under consideration, allowing the extension to homomorphisms of model-theoric notions which are classically associated to embeddings; in particular, the existentially closed models, that were primitively defined by Abraham Robinson, become here positively closed models. It accounts for recent results in this domain, and is oriented towards the positivisation of Jonsson theories.

Résumé

Cet article est une introduction générale à la Logique Positive, où seuls sont considérés les énoncés dits h-inductifs, ce qui permet d’étendre aux homomorphismes les notions de Théorie des Modèles classiquement associées aux plongements; en particulier les modèles existentiellement clos, primitivement définis par Abraham Robinson, deviennent ici les modèles positivement clos. Il tient compte de résultats récents en ce domaine, et se focalise sur ce que deviennent les théories de Jonsson dans un contexte positif.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Barwise, J.: Handbook of Mathematical Logic, Model Theory, vol. 1. North Holland, Amsterdam (1982)

    MATH  Google Scholar 

  2. Ben Yaacov, I.: Positive model theory and compact abstract theories. J. Math. Log. 3, 85–118 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ben Yaacov, I.: Thickness, and a categoric point of view of type-space functors. Fundam. Math. 179, 199–224 (2003)

    Article  MATH  Google Scholar 

  4. Ben Yaacov, I.: Simplicity in compact abstract theories. J. Math. Log. 3, 163–191 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ben Yaacov, I.: Lovely pairs of models: the non first order case. J. Symb. Log. 69, 641–662 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ben Yaacov, I.: Uncountable dense categoricity in cats. J. Symb. Log. 70, 829–860 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ben Yaacov, I., Poizat, B.: Fondements de la Logique positive. J. Symb. Log. 72, 1141–1162 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Belkasmi, M.: Positive model theory and amalgamation. Notre Dame J. Form. Log. 55, 205–230 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cheng-Chung, C., Keisler, J.H.: Model Theory. North-Holland, Amsterdam (1973)

    Google Scholar 

  10. Daigneault, A.: Théorie des Modèles en Logique Mathématique, Université de Montréeal (1967)

  11. Fraïssé, R.: Sur l’extension aux relations de quelques propriétés connues des ordres. C.R. Acad. Sci. Paris 237, 508–510 (1953)

  12. Gödel, K.: Die Vollständigkeit der Axiome des logischen Funktionenkalkuls. Monat. Math. Phys. 37, 349–360 (1930)

    Article  MATH  Google Scholar 

  13. Jonsson, B.: Universal relational systems. Math. Scand. 4, 193–208 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jonsson, B.: Homogeneous universal relational systems. Math. Scand. 8, 137–142 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kaiser, K.: Über eine Verallgemeinerung der Robinsonschen Modell-vervollständigung. Z. Math. Logik Grundlagen Math. 15, 37–48 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kunghozhin, A.: Existentially closed and maximal models in positive logic (en russe). Algebra Log. 54, 496–506 (2013)

    Article  Google Scholar 

  17. Makkai, M., Reyes, G.: First Order Categorical Logic. Lectures Notes in Mathematics, vol. 611. Springer, Berlin (1977)

  18. Makowski, J.H.: On some conjectures connected with complete sentences. Fundam. Math. 81, 193–202 (1974)

    Article  MathSciNet  Google Scholar 

  19. Morley, M., Vaught, R.: Homogeneous universal models. Math. Scand. 11, 37–57 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mustafin, T.: Jonsson generalized conditions and generalized Jonsson theories of Boolean algebras (in Russian), Math. Trudy, Novosibirsk, 135–197; translated. Siberian Advances in Mathematics 10(2000), 1–58 (1998)

    MathSciNet  Google Scholar 

  21. Mustafin, Y.: Quelques propriétés des théories de Jonsson. J. Symb. Log. 67, 528–536 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mustafin, T., Nurkhaïdarov, E: Jonsson theories of polygons over a group (in Russian), Sbornik nauchnyh trudov, Qaraghandy, 67–73 (1995)

  23. Nurkhaïdarov, E.: Jonsson theories of abelian groups (in Russian), Sbornik nauchnyh trudov, Qaraghandy, 43–50 (1995)

  24. Nurtazin, A.T..: Countable existentially closed models of universally axiomatizable theories (in Russian), SORAN Trudy po Matematike, xx, 1–50 (2015)

  25. Pillay, A.: Forking in the category of existentially closed structures, Quaderni di Matematica, vol. 6 (2000)

  26. Poizat, B.: Univers positifs. J. Symb. Log. 71, 969–976 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Poizat, B.: Quelques effets pervers de la positivité. Ann. Pure Appl. Log. 6, 812–816 (2010)

    Article  MATH  Google Scholar 

  28. Poizat, B.: Parlons ladakhi. L’Harmattan (2018)

  29. Robinson, A.: Complete Theories. North Holland, Amsterdam (1956)

    MATH  Google Scholar 

  30. Yeshkeyev, A.: Description of Jonsson theories of unars (in Russian), Sbornik nauchnyh trudov, Qaraghandy, 51–57 (1995)

  31. Yeshkeyev, A.: Jonsson theories (in Russian), Qaraghandy (2009)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Bruno Poizat.

Additional information

This paper is dedicated to the memory of Tölendi Garifuly.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Poizat, B., Yeshkeyev, A. Positive Jonsson Theories. Log. Univers. 12, 101–127 (2018). https://doi.org/10.1007/s11787-018-0185-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11787-018-0185-8

Keywords

Mathematics Subject Classification

Navigation