Skip to main content

Dualities for subresiduated lattices

Abstract

A subresiduated lattice is a pair (AD), where A is a bounded distributive lattice, D is a bounded sublattice of A and for every \(a,b\in A\) there is \(c\in D\) such that for all \(d\in D\), \(d\wedge a\le b\) if and only if \(d\le c\). This c is denoted by \(a\rightarrow b\). This pair can be regarded as an algebra \(\left<A,\wedge ,\vee ,\rightarrow ,0,1\right>\) of type (2, 2, 2, 0, 0) where \(D=\{a\in A\mid 1\rightarrow a=a\}\). The class of subresiduated lattices is a variety which properly contains to the variety of Heyting algebras. In this paper we present dual equivalences for the algebraic category of subresiduated lattices. More precisely, we develop a spectral style duality and a bitopological style duality for this algebraic category. Finally we study the connections of these results with a known Priestley style duality for the algebraic category of subresiduated lattices.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3

References

  1. 1.

    Balbes, R., Dwinger, P.: Distributive Lattices. University of Missouri Press, Columbia (1974)

    MATH  Google Scholar 

  2. 2.

    Bezhanishvili, G., Bezhanishvili, N., Gabelaia, D., Kurz, A.: Bitopological duality for distributive lattices and Heyting algebras. Math. Struct. Comput. Sci. 20(3), 359–393 (2010)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Celani, S., Jansana, R.: A closer look at some subintuitionistic logics. Notre Dame J. Formal Logic 42, 225–255 (2003)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Celani, S., Jansana, R.: Bounded distributive lattices with strict implication. Math. Logic Quart. 51, 219–246 (2005)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Cornish, W.H., On, H.: Priestley’s dual of the category of bounded distributive lattices. Matematički Vesnik 12 27(60), 329–332 (1975)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  7. 7.

    Epstein, G., Horn, A.: Logics which are characterized by subresiduated lattices. Math. Logic Quart. 22(1), 199–210 (1976)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Fleisher, I.: Priestley’s duality from Stone’s. Adv. Appl. Math. 25(3), 233–238 (2000)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Priestley, H.A.: Representation of bounded distributive lattice by means of orderer Stone spaces. Bull. Lond. Math. Soc. 2, 186–190 (1970)

    Article  Google Scholar 

  10. 10.

    San Martín H.J.: Compatible operations in some subvarieties of the variety of weak Heyting algebras. In: Proceedings of the 8th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2013). Advances in Intelligent Systems Research, pp. 475–480. Atlantis Press (2013)

  11. 11.

    Stone, M.: Topological representation of distributive lattices and Brouwerian logics. Časopis Pro Pěstování Matematiky a Fysiky 67(1), 1–25 (1938)

    Article  Google Scholar 

Download references

Acknowledgements

The authors acknowledge many helpful comments from the anonymous referee, which considerably improved the presentation of this paper. The second author was supported by a doctoral fellowship from Comisión de Investigaciones Científicas (CIC-Argentina).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Hernán J. San Martín.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Presented by N. Galatos.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Celani, S.A., Nagy, A.L. & Martín, H.J.S. Dualities for subresiduated lattices. Algebra Univers. 82, 59 (2021). https://doi.org/10.1007/s00012-021-00752-3

Download citation

Keywords

  • Distributive lattices
  • Subresiduated lattices
  • Spectral duality
  • Bitopological duality

Mathematics Subject Classification

  • 06D05
  • 06D50
  • 03G10