Studia Logica

, Volume 107, Issue 2, pp 351–374 | Cite as

On Principal Congruences in Distributive Lattices with a Commutative Monoidal Operation and an Implication

  • Ramon Jansana
  • Hernán Javier San MartínEmail author


In this paper we introduce and study a variety of algebras that properly includes integral distributive commutative residuated lattices and weak Heyting algebras. Our main goal is to give a characterization of the principal congruences in this variety. We apply this description in order to study compatible functions.


Distributive lattices Operations Principal congruences Compatible functions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agliano, P., Ternary deductive terms in residuated structures, Acta Sci. Math. (Szeged) 68:397–429, 2002.Google Scholar
  2. 2.
    Balbes, R., Distributive Lattices, University of Missouri Press, 1974.Google Scholar
  3. 3.
    Bezhanishvili, N., and M. Gehrke, Finitely generated free Heyting algebras via Birkhoff duality and coalgebra, Logical Methods in Computer Science 7:1–24, 2011.CrossRefGoogle Scholar
  4. 4.
    Blok W. J., and D. Pigozzi, Abstract Algebraic Logics and the Deduction Theorem, Manuscript, 2001.
  5. 5.
    Blok, W., and D. Pigozzi, Local deduction theorems in algebraic logic, in H. Andreka, J.D. Monk, and I. Nemeti, (eds.), Algebraic logic, vol. 54 of Colloquia Math. Soc. Janos Bolyai. North-Holland, Amsterdam, 1991, pp. 75–109.Google Scholar
  6. 6.
    Burris, H., and H. P. Sankappanavar, A Course in Universal Algebra, Springer Verlag, New York, 1981.CrossRefGoogle Scholar
  7. 7.
    Caicedo, X., Implicit connectives of algebraizable logics, Studia Logica 78(3):155–170, 2004.CrossRefGoogle Scholar
  8. 8.
    Caicedo, X., Implicit operations in MV-algebras and the connectives of Lukasiewicz logic, Lecture Notes in Computer Science 4460(1):50–68, 2007.CrossRefGoogle Scholar
  9. 9.
    Caicedo, X., and R. Cignoli, An algebraic approach to intuitionistic connectives, Journal of Symbolic Logic 4:1620–1636, 2001.CrossRefGoogle Scholar
  10. 10.
    Castiglioni J. L., M. Menni, and M. Sagastume Compatible operations on commutative residuated lattices, JANCL 18:413–425, 2008.Google Scholar
  11. 11.
    Castiglioni, J. L., M. Sagastume, and H. J. San Martín, On frontal Heyting algebras, Reports on Mathematical Logic 45:201–224, 2010.Google Scholar
  12. 12.
    Castiglioni, J. L., and H. J. San Martín, Compatible operations on residuated lattices, Studia Logica 98(1-2):203–222, 2011.CrossRefGoogle Scholar
  13. 13.
    Celani, S. A., Distributive lattices with fusion and implication, Southeast Asian Bulletin of Mathematics 28:999–1010, 2004.Google Scholar
  14. 14.
    Celani, S. A., and R. Jansana, Bounded distributive lattices with strict implication, Mathematical Logic Quarterly 51:219–246, 2005.CrossRefGoogle Scholar
  15. 15.
    Celani, S. A., and H. J. San Martín, Frontal operators in weak Heyting algebras, Studia Logica 100:91–114, 2012.CrossRefGoogle Scholar
  16. 16.
    Cignoli R., I. D’Ottaviano, and D. Mundici, Algebraic Foundations of Many–Valued Reasoning, Trends in Logic, Studia Logica Library, vol. 7, Kluwer Academic Publishers, 2000.Google Scholar
  17. 17.
    Ertola, R., and H. J. San Martín, On some compatible operations on Heyting algebras, Studia Logica 98:331–345, 2011.CrossRefGoogle Scholar
  18. 18.
    Esakia, L., The modalized Heyting calculus: a conservative modal extension of the Intuitionistic Logic, Journal of Applied Non-Classical Logics 16:349–366, 2006.CrossRefGoogle Scholar
  19. 19.
    Font J. M., Abstract Algebraic logic. An Introductory Course, College Publications, London, 2016.Google Scholar
  20. 20.
    Fried E., G. Grätzer, and R. W. Quackenbush, The equational class generated by weakly associative lattices with the unique bound property, Ann. Univ. Sci. Budapest. Eštvšs Sect. Math. 205–211, 1979/80.Google Scholar
  21. 21.
    Fried, E., G. Grätzer, and R.W. Quackenbush, Uniform congruence schemes, Algebra Universalis 10:176–188, 1980.CrossRefGoogle Scholar
  22. 22.
    Gabbay, D. M., On some new intuitionistic propositional connectives, Studia Logica 36:127–139, 1977.CrossRefGoogle Scholar
  23. 23.
    Hart, J., L. Raftery, and C. Tsinakis, The structure of commutative residuated lattices, Internat. J. Algebra Comput. 12:509–524, 2002.CrossRefGoogle Scholar
  24. 24.
    Kaarli, K., and A. F. Pixley, Polynomial Completeness in Algebraic Systems, Chapman and Hall/CRC, 2001.Google Scholar
  25. 25.
    Kuznetsov, A. V., On the propositional calculus of intuitionistic provability, Soviet Math. Dokl. 32:18–21, 1985.Google Scholar
  26. 26.
    Muravitsky, A. Y., Logic KM: A biography, in Leo Esakia on Duality of Modal and Intuitionistic Logics, Series: Outstanding Contributions to Logic, vol. 4, Springer, 2014, pp. 147–177.Google Scholar
  27. 27.
    San Martín H. J., Compatible operations in some subvarieties of the variety of weak Heyting algebras, in 8th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2013), pp. 475–480.Google Scholar
  28. 28.
    San Martín H. J., Compatible operations on commutative weak residuated lattices, Algebra Universalis 73(2):143–155, 2015.CrossRefGoogle Scholar
  29. 29.
    San Martín, H. J., Principal congruences in weak Heyting algebras, Algebra Universalis 75(4):405–418, 2016.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Barcelona Graduate School of Mathematics, Philosophy DepartmentUniversitat de BarcelonaBarcelonaSpain
  2. 2.Departamento de MatemáticaFacultad de Ciencias Exactas (UNLP) and CONICETLa PlataArgentina

Personalised recommendations