Skip to main content
SpringerLink
Log in

Expansions of Dually Pseudocomplemented Heyting Algebras

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

We investigate expansions of Heyting algebras (EHAs) in possession of a unary term describing the filters that correspond to congruences. Hasimoto proved that Heyting algebras equipped with finitely many (dual) normal operators have such a term, generalising a standard construction on finite-type boolean algebras with operators (BAOs). We utilise Hasimoto’s technique, extending the existence condition to a larger class of EHAs and some classes of double-Heyting algebras. Such a term allows us to characterise varieties with equationally definable principal congruences using a single equation. Moreover, in the presence of a dual pseudocomplement operation, discriminator varieties are characterised by a pair of equations. We also prove that a variety of dually pseudocomplemented EHAs with a normal filter term is semisimple if and only if it is a discriminator variety. This generalises two known results, one by Kowalski and Kracht for finite-type varieties of BAOs, and the other by the present author for dually pseudocomplemented Heyting algebras without additional operations.

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

Access this article

Log in via an institution

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. Balbes, R., and P. Dwinger, Distributive lattices, University of Missouri Press, Columbia, Mo., 1974.

    Google Scholar 

  2. Blok, W. J., P. Köhler, and D. Pigozzi, On the structure of varieties with equationally definable principal congruences. II, Algebra Universalis 18(3):334–379, 1984.

    Article  Google Scholar 

  3. Blok, W. J., and D. Pigozzi, On the structure of varieties with equationally definable principal congruences. I, Algebra Universalis 15(2):195–227, 1982.

    Article  Google Scholar 

  4. Blok, W. J., and D. Pigozzi, On the structure of varieties with equationally definable principal congruences. III, Algebra Universalis 32(4):545–608, 1994.

    Article  Google Scholar 

  5. Blok, W. J., and D. Pigozzi, On the structure of varieties with equationally definable principal congruences. IV, Algebra Universalis 31(1):1–35, 1994.

    Article  Google Scholar 

  6. Botur, M., J. Kühr, L. Liu, and C. Tsinakis, The Conrad program: from \(\ell \)-groups to algebras of logic, J. Algebra 450:173–203, 2016.

  7. Burris, S., and H. P. Sankappanavar, A course in universal algebra, vol. 78 of Graduate Texts in Mathematics, Springer, New York-Berlin, 1981.

  8. Goldblatt, R., Algebraic polymodal logic: a survey, Log. J. IGPL 8(4):393–450, 2000.

    Article  Google Scholar 

  9. Hasimoto, Y., Heyting algebras with operators, MLQ Math. Log. Q. 47(2):187–196, 2001.

    Article  Google Scholar 

  10. Jipsen, P., Discriminator varieties of Boolean algebras with residuated operators, in Algebraic methods in logic and in computer science (Warsaw, 1991), vol. 28 of Banach Center Publ., Polish Acad. Sci., Warsaw, pp. 239–252, 1993.

  11. Köhler, P., and D. Pigozzi, Varieties with equationally definable principal congruences, Algebra Universalis 11(2):213–219, 1980.

    Article  Google Scholar 

  12. Kowalski, T., Semisimplicity, EDPC and discriminator varieties of residuated lattices, Studia Logica 77(2):255–265, 2004.

    Article  Google Scholar 

  13. Kowalski, T., and I. Ferreirim, Semisimple varieties of residuated lattice and their reducts, Unpublished manuscript.

  14. Kowalski, T., and M. Kracht, Semisimple varieties of modal algebras, Studia Logica 83(1-3):351–363, 2006.

    Article  Google Scholar 

  15. Quackenbush, R. W., Structure theory for equational classes generated by quasi-primal algebras, Trans. Amer. Math. Soc. 187:127–145, 1974.

  16. Sankappanavar, H. P., Heyting algebras with dual pseudocomplementation, Pacific J. Math. 117:405–415, 1985.

    Article  Google Scholar 

  17. Takamura, H., Semisimplicity, EDPC and discriminator varieties of bounded weak-commutative residuated lattices with an S4-like modal operator, Studia Logica 100(6):1137–1148, 2012.

    Article  Google Scholar 

  18. Taylor, C., Discriminator varieties of double-Heyting algebras, Rep. Math. Logic. To appear, 2016.

  19. Werner, H., Discriminator-algebras, vol. 6 of Studien zur Algebra und ihre Anwendungen, Akademie-Verlag, Berlin. Algebraic representation and model theoretic properties, 1978.

Download references

Author information

Authors and Affiliations

  1. Mathematics and Statistics, La Trobe University, Melbourne, Australia

    Christopher J. Taylor

Authors
  1. Christopher J. Taylor
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christopher J. Taylor.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Taylor, C.J. Expansions of Dually Pseudocomplemented Heyting Algebras. Stud Logica 105, 817–841 (2017). https://doi.org/10.1007/s11225-017-9712-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-017-9712-5

Keywords

Search

Navigation