Skip to main content
Log in

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

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

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.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agliano, P., Ternary deductive terms in residuated structures, Acta Sci. Math. (Szeged) 68:397–429, 2002.

    Google Scholar 

  2. Balbes, R., Distributive Lattices, University of Missouri Press, 1974.

  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.

    Article  Google Scholar 

  4. Blok W. J., and D. Pigozzi, Abstract Algebraic Logics and the Deduction Theorem, Manuscript, 2001. http://orion.math.iastate.edu/dpigozzi/.

  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. Burris, H., and H. P. Sankappanavar, A Course in Universal Algebra, Springer Verlag, New York, 1981.

    Book  Google Scholar 

  7. Caicedo, X., Implicit connectives of algebraizable logics, Studia Logica 78(3):155–170, 2004.

    Article  Google Scholar 

  8. Caicedo, X., Implicit operations in MV-algebras and the connectives of Lukasiewicz logic, Lecture Notes in Computer Science 4460(1):50–68, 2007.

    Article  Google Scholar 

  9. Caicedo, X., and R. Cignoli, An algebraic approach to intuitionistic connectives, Journal of Symbolic Logic 4:1620–1636, 2001.

    Article  Google Scholar 

  10. Castiglioni J. L., M. Menni, and M. Sagastume Compatible operations on commutative residuated lattices, JANCL 18:413–425, 2008.

    Google Scholar 

  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. Castiglioni, J. L., and H. J. San Martín, Compatible operations on residuated lattices, Studia Logica 98(1-2):203–222, 2011.

    Article  Google Scholar 

  13. Celani, S. A., Distributive lattices with fusion and implication, Southeast Asian Bulletin of Mathematics 28:999–1010, 2004.

    Google Scholar 

  14. Celani, S. A., and R. Jansana, Bounded distributive lattices with strict implication, Mathematical Logic Quarterly 51:219–246, 2005.

    Article  Google Scholar 

  15. Celani, S. A., and H. J. San Martín, Frontal operators in weak Heyting algebras, Studia Logica 100:91–114, 2012.

    Article  Google Scholar 

  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.

  17. Ertola, R., and H. J. San Martín, On some compatible operations on Heyting algebras, Studia Logica 98:331–345, 2011.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  19. Font J. M., Abstract Algebraic logic. An Introductory Course, College Publications, London, 2016.

    Google Scholar 

  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.

  21. Fried, E., G. Grätzer, and R.W. Quackenbush, Uniform congruence schemes, Algebra Universalis 10:176–188, 1980.

    Article  Google Scholar 

  22. Gabbay, D. M., On some new intuitionistic propositional connectives, Studia Logica 36:127–139, 1977.

    Article  Google Scholar 

  23. Hart, J., L. Raftery, and C. Tsinakis, The structure of commutative residuated lattices, Internat. J. Algebra Comput. 12:509–524, 2002.

    Article  Google Scholar 

  24. Kaarli, K., and A. F. Pixley, Polynomial Completeness in Algebraic Systems, Chapman and Hall/CRC, 2001.

  25. Kuznetsov, A. V., On the propositional calculus of intuitionistic provability, Soviet Math. Dokl. 32:18–21, 1985.

    Google Scholar 

  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.

  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.

  28. San Martín H. J., Compatible operations on commutative weak residuated lattices, Algebra Universalis 73(2):143–155, 2015.

    Article  Google Scholar 

  29. San Martín, H. J., Principal congruences in weak Heyting algebras, Algebra Universalis 75(4):405–418, 2016.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hernán Javier San Martín.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jansana, R., San Martín, H.J. On Principal Congruences in Distributive Lattices with a Commutative Monoidal Operation and an Implication. Stud Logica 107, 351–374 (2019). https://doi.org/10.1007/s11225-018-9796-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-018-9796-6

Keywords

Navigation