Studia Logica

, Volume 65, Issue 1, pp 91–112 | Cite as

Algebraic Characterizations of Various Beth Definability Properties

  • Eva Hoogland
Article

Abstract

In this paper it will be shown that the Beth definability property corresponds to surjectiveness of epimorphisms in abstract algebraic logic. This generalizes a result by I. Németi (cf. [11, Theorem 5.6.10]). Moreover, an equally general characterization of the weak Beth property will be given. This gives a solution to Problem 14 in [20]. Finally, the characterization of the projective Beth property for varieties of modal algebras by L. Maksimova (see [15]) will be shown to hold for the larger class of semantically algebraizable logics.

Abstract algebraic logic Beth definability property epimorphisms 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    AndrÉka, H., Á. Kurucz, I. NÉmeti, and I. Sain, 'Methodolgy of applying algebraic logic to logic', in: M. Nivat, C. Rattray, and T. Rus (eds.), Algebraic Methodology and Software Technology (AMAST'93), Workshops in Computing, pages 7-28. Springer-Verlag, 1994. Extended version appeared in the Proceedings of the Summer School of Algebraic Logic 1994.Google Scholar
  2. [2]
    Beth, E. W., 'On Padoa's method in the theory of definition', Nederl. Akad. Welensch. Proc. Ser. A. 56 = Indagationes Math. 15: 330-339, 1953.Google Scholar
  3. [3]
    Barwise, J., and S. Feferman (eds.), Model-Theoretic Logics, New York, Springer-Verlag, 1985.Google Scholar
  4. [4]
    Blok, W. J., and D. Pigozzi, 'Algebraizable logics', Memoirs of the American Mathematical Society 77, 396: vi 78 pp., 1989.Google Scholar
  5. [5]
    Blok, W. J., and D. Pigozzi, 'Algebraic semantics for universal Horn logic without equality', in: J. D. H. Smith and A. Romanowska (eds.), Universal Algebra and Quasigroup Theory (Proc, Conf, Jadwisin, Poland, May 23–28, 1989), volume 19 of Research and Exposition in Mathematics, pages 1-56, Berlin, Heldermann Verlag, 1992.Google Scholar
  6. [6]
    Chang, C., and H. Keisler, Model Theory, volume 73 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, third edition, 1990.Google Scholar
  7. [7]
    Czelakowski, J., and D. Pigozzi, 'Amalgamation and interpolation in abstract algebraic logic', in: X. Caicedo and C. H. Montenegro (eds.), Models, Algebras and Proofs, volume 203 of Lecture Notes in Pure and Applied Matehmatics Series, pages 187-265, New York and Basel, Marcel Dekker, 1998.Google Scholar
  8. [8]
    Font, J. M., and R. Jansana, 'On the sentential logics associated with strongly nice and semi-nice general logics', Bulletin of the IGPL 2: 5-67, 1994.Google Scholar
  9. [9]
    Font, J. M., R. Jansana, and D. Pigozzi (eds.), Workshop on Abstract Algebraic Logic, vol. 10 of Quaderns, Bellaterra (Barcelona), 1998, Centre de Recerca Matemàtica.Google Scholar
  10. [10]
    Friedman, H. M., 'Beth's theorem in cardinality logics', Israel Journal of Mathematics 14: 205-212, 1973.Google Scholar
  11. [11]
    Henkin, L., J. D. Monk, and A. Tarski, Cylindric Algebras. Parls I & II, North-Holland, Amsterdam, 1971 & 1985.Google Scholar
  12. [12]
    Hoogland, E., 'Algebraic characterizations of two Beth definability properties', Master thesis, University of Amsterdam, 1996.Google Scholar
  13. [13]
    L. Maksimova, 'The Beth properties, interpolationa and amalgamability in varicties of modal algebras', Soviet Math. Dokl. 44(1): 327-331, 1992.Google Scholar
  14. [14]
    Maksimova, 'Definability and interpolation in classical modal logics', Contemporary Mathematics 131(3): 583-599, 1992.Google Scholar
  15. [15]
    Maksimova, L., 'On the Beth definability properties in varieties of modal algebras', in [9], pages 109-115.Google Scholar
  16. [16]
    NÉmeti, I., and H. AndrÉka, 'General algebraic logic: a perspective on “What is logic”', in D. M. Gabbay (ed.), What Is a Logical System, pages 393-444, Clarendron Press, Oxford, 1994.Google Scholar
  17. [17]
    NÉmeti, I., 'Beth definability is equivalent with surjectiveness of epis in general algebraic logic', Technical report, Math, Inst. Hungar, Acad. Sci., Budapest, 1984.Google Scholar
  18. [18]
    Padoa, A., 'Théorie algébrique des nombres entiers, précédé d'une introduction logique à une théorie déductive quelconque', in le Congrès International de Philosophie (1900 Paris), volume 3, pages 309-365, 1900.Google Scholar
  19. [19]
    Sain, I., 'Strong amalgamation and epimorphisms of cylindric algebras and Boolean algebras with operators', Preprint 17/1984, Math. Inst. Hungar. Acad. Sci., Budapest, 1984 (to appear in Studia Logica).Google Scholar
  20. [20]
    Sain, I., 'Beth's and Craig's properties via epimorphisms and amalgamation in algebraic logic', in: C. H. Bergman, R. D. Maddux, and D. Pigozzi (eds.), Algebraic Logic and Universal Algebra in Computer Science, volume 24 of Lecture Notes in Computer Science, pages 209-226, Springer-Verlag, Berlin, Heidelberg, New York, 1990.Google Scholar
  21. [21]
    Sain, I., 'On characterizations of definability properties in abstract algebraic logic', in [9], pages 162-175.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Eva Hoogland
    • 1
  1. 1.ILLCUniversiteit van AmsterdamAmsterdamThe Netherlands

Personalised recommendations