Advertisement

Epimorphisms, Definability and Cardinalities

  • T. Moraschini
  • J. G. RafteryEmail author
  • J. J. Wannenburg
Article
  • 7 Downloads

Abstract

We characterize, in syntactic terms, the ranges of epimorphisms in an arbitrary class of similar first-order structures (as opposed to an elementary class). This allows us to strengthen a result of Bacsich, as follows: in any prevariety having at most \(\mathfrak {s}\) non-logical symbols and an axiomatization requiring at most \(\mathfrak {m}\) variables, if the epimorphisms into structures with at most \(\mathfrak {m}+\mathfrak {s}+\aleph _0\) elements are surjective, then so are all of the epimorphisms. Using these facts, we formulate and prove manageable ‘bridge theorems’, matching the surjectivity of all epimorphisms in the algebraic counterpart of a logic \(\,\vdash \) with suitable infinitary definability properties of \(\,\vdash \), while not making the standard but awkward assumption that \(\,\vdash \) comes furnished with a proper class of variables.

Keywords

Epimorphism Prevariety Quasivariety Beth definability Algebraizable logic Equivalential logic 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

This work received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 689176 (project “Syntax Meets Semantics: Methods, Interactions, and Connections in Substructural logics”). The first author was also supported by the Project GA17-04630S of the Czech Science Foundation (GAČR). The second author was supported in part by the National Research Foundation of South Africa (UID 85407). The third author was supported by the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa. Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the CoE-MaSS.

References

  1. 1.
    Adámek, J., How many variables does a quasivariety need? Algebra Universalis 27:44–48, 1990.CrossRefGoogle Scholar
  2. 2.
    Bacsich, P. D., Model theory of epimorphisms, Canad. Math. Bull. 17:471–477, 1974.CrossRefGoogle Scholar
  3. 3.
    Banaschewski, B., and H. Herrlich, Subcategories defined by implications, Houston J. Math. 2:149–171, 1976.Google Scholar
  4. 4.
    Bezhanishvili, G., T. Moraschini, and J. G. Raftery, Epimorphisms in varieties of residuated structures, J. Algebra 492:185–211, 2017.CrossRefGoogle Scholar
  5. 5.
    Birkhoff, G., On the structure of abstract algebras, Proc. Cambridge Phil. Soc. 29:433–454, 1935.CrossRefGoogle Scholar
  6. 6.
    Blok, W. J., and E. Hoogland, The Beth property in algebraic logic, Studia Logica 83:49–90, 2006.CrossRefGoogle Scholar
  7. 7.
    Blok, W. J., and B. Jónsson, Equivalence of consequence operations, Studia Logica 83:91–110, 2006.CrossRefGoogle Scholar
  8. 8.
    Blok, W. J., and D. Pigozzi, Algebraizable Logics, Memoirs of the American Mathematical Society 396, Amer. Math. Soc., Providence, 1989.Google Scholar
  9. 9.
    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, Heldermann Verlag, Berlin, 1992, pp. 1–56.Google Scholar
  10. 10.
    Budkin, A., Dominions in quasivarieties of universal algebras, Studia Logica 78:107–127, 2004.CrossRefGoogle Scholar
  11. 11.
    Budkin, A., Dominions of universal algebras and projective properties, Algebra and Logic 47:304–313, 2008.CrossRefGoogle Scholar
  12. 12.
    Campercholi, M. A., Dominions and primitive positive functions, J. Symbolic Logic 83:40–54, 2018.CrossRefGoogle Scholar
  13. 13.
    Czelakowski, J., Equivalential logics (I), and (II), Studia Logica 40:227–236, and 355–372, 1981.Google Scholar
  14. 14.
    Czelakowski, J., Protoalgebraic Logics, Kluwer, Dordrecht, 2001.CrossRefGoogle Scholar
  15. 15.
    Czelakowski, J., and D. Pigozzi, Amalgamation and interpolation in abstract algebraic logic, in X. Caicedo, and C. H. Montenegro (eds.), Models, Algebras and Proofs, Lecture Notes in Pure and Applied Mathematics, No. 203, Marcel Dekker, New York, 1999, pp. 187–265.Google Scholar
  16. 16.
    Font, J. M., Abstract Algebraic Logic – An Introductory Textbook, Studies in Logic 60, College Publications, London, 2016.Google Scholar
  17. 17.
    Font, J. M., R. Jansana, and D. Pigozzi, A survey of abstract algebraic logic, and Update, Studia Logica 74:13–97, 2003, and 91:125–130, 2009.Google Scholar
  18. 18.
    Freyd, P., Abelian categories, Harper and Row, New York, 1964.Google Scholar
  19. 19.
    Gabbay, D. M., and L. Maksimova, Interpolation and Definability: Modal and Intuitionistic Logics, Oxford Logic Guides 46, Clarendon Press, Oxford, 2005.Google Scholar
  20. 20.
    Gorbunov, V. A., Algebraic Theory of Quasivarieties, Consultants Bureau, New York, 1998.Google Scholar
  21. 21.
    Grätzer, G., and H. Lakser, A note on the implicational class generated by a class of structures, Canad. Math. Bull. 16:603–605, 1973.CrossRefGoogle Scholar
  22. 22.
    Herrmann, B., Equivalential and algebraizable logics, Studia Logica 57:419–436, 1996.CrossRefGoogle Scholar
  23. 23.
    Herrmann, B., Characterizing equivalential and algebraizable logics by the Leibniz operator, Studia Logica 58:305–323, 1997.CrossRefGoogle Scholar
  24. 24.
    Henkin, L., J. D. Monk, and A. Tarski, Cylindric Algebras, Part II, North-Holland, Amsterdam, 1985.Google Scholar
  25. 25.
    Higgins, P., Epimorphisms and amalgams, Colloquium Mathematicum 56:1–17, 1988.CrossRefGoogle Scholar
  26. 26.
    Hoogland, E., Algebraic characterizations of various Beth definability properties, Studia Logica 65:91–112, 2000.CrossRefGoogle Scholar
  27. 27.
    Hoogland, E., Definability and interpolation: model-theoretic investigations, PhD. Thesis, Institute for Logic, Language and Computation, University of Amsterdam, 2001.Google Scholar
  28. 28.
    Isbell, J. R., Epimorphisms and dominions, in S. Eilenberg, et al. (eds.), Proceedings of the Conference on Categorical Algebra (La Jolla, California, 1965), Springer, New York, 1966, pp. 232–246.Google Scholar
  29. 29.
    Kreisel, G., Explicit definability in intuitionistic logic, J. Symbolic Logic 25:389–390, 1960.CrossRefGoogle Scholar
  30. 30.
    Łoś, J., and R. Suszko, Remarks on sentential logics, Proc. Kon. Nederl. Akad. van Wetenschappen, Series A 61:177–183, 1958.Google Scholar
  31. 31.
    Maksimova, L. L., Intuitionistic logic and implicit definability, Ann. Pure Appl. Logic 105:83–102, 2000.CrossRefGoogle Scholar
  32. 32.
    Maksimova, L. L., Implicit definability and positive logics, Algebra and Logic 42:37–53, 2003.CrossRefGoogle Scholar
  33. 33.
    Maltsev, A. I., Several remarks on quasivarieties of algebraic systems (Russian), Algebra i Logika 5:3–9, 1966.Google Scholar
  34. 34.
    Prucnal, T., and A. Wroński, An algebraic characterization of the notion of structural completeness, Bull. Sect. Logic 3:30–33, 1974.Google Scholar
  35. 35.
    Raftery, J. G., Correspondences between Gentzen and Hilbert systems, J. Symbolic Logic 71:903–957, 2006.CrossRefGoogle Scholar
  36. 36.
    Raftery, J. G., A non-finitary sentential logic that is elementarily algebraizable, J. Logic Comput. 20:969–975, 2010.CrossRefGoogle Scholar
  37. 37.
    Wasserman, D., Epimorphisms and Dominions in Varieties of Lattices, PhD thesis, University of California at Berkeley, 2001.Google Scholar
  38. 38.
    Wójcicki, R., Theory of Logical Calculi, Kluwer, Dordrecht, 1988.CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • T. Moraschini
    • 1
  • J. G. Raftery
    • 2
    Email author
  • J. J. Wannenburg
    • 2
    • 3
  1. 1.Institute of Computer ScienceAcademy of Sciences of the Czech RepublicPrague 8Czech Republic
  2. 2.Department of Mathematics and Applied MathematicsUniversity of PretoriaHatfield, PretoriaSouth Africa
  3. 3.DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS)JohannesburgSouth Africa

Personalised recommendations