Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

A Topological Duality for Monotone Expansions of Semilattices

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

In this paper we provide a Stone style duality for monotone semilattices by using the topological duality developed in S. Celani, L.J. González (Appl Categ Struct 28:853–875, 2020) for semilattices together with a topological description of their canonical extension. As an application of this duality we obtain a characterization of the congruences of monotone semilattices by means of monotone lower-Vietoris-type topologies.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Adaricheva, K., Nation, J.: Lattices of quasi-equational theories as congruence lattices of semilattices with operators: part I. Int. J. Algebra Comput. 22(7), 27 (2012)

    MathSciNet  MATH  Google Scholar 

  2. Bezhanishvili, G., Jansana, R.: Priestley style duality for distributive meet-semilattices. Stud. Log. 98(1–2), 83–122 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bezhanishvili, G., Jansana, R.: Generalized Priestley quasi-orders. Order 28(2), 201–220 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bezhanishvili, G., Jansana, R.: Esakia style duality for implicative semilattices. Appl. Categ. Struct. 21(2), 181–208 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Birkhoff, G.: Rings of sets. Duke Math. J. 3, 443–454 (1937)

    Article  MathSciNet  MATH  Google Scholar 

  6. Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer, Berlin (1981)

    Book  MATH  Google Scholar 

  7. Celani, S.: Topological representation of distributive semilattices. Sci. Math. Jpn. 58(1), 55–66 (2003)

    MathSciNet  MATH  Google Scholar 

  8. Celani, S.: Topological Duality for Boolean Algebras with a Normal n-ary Monotonic Operator. Order 26(1), 49–67 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Celani, S., Calomino, I.: Some remarks on distributive semilattices. Commentat. Math. Univ. Carol. 54(3), 407–428 (2013)

    MathSciNet  MATH  Google Scholar 

  10. Celani, S., González, L.J.: A categorical duality for semilattices and lattices. Appl. Categ. Struct. 28(5), 853–875 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  11. Celani, S., Menchón, M.P.: Monotonic distributive semilattices. Order 36(3), 463–486 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chellas, B.: Modal Logic: An introduction. Cambridge University Press, Cambridge (1980)

    Book  MATH  Google Scholar 

  13. Dunn, J., Gehrke, M., Palmigiano, A.: Canonical extensions and relational completeness of some substructural logics. J. Symb. Log. 70(3), 713–740 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Düntsch, I., Orlowska, E., Rewitzky, I.: Structures with multirelations, their discrete dualities and applications. Fundam. Inform. 100(1–4), 77–98 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gehrke, M., Jónsson, B.: Monotone bounded distributive lattice expansions. Math. Jap. 52(2), 197–213 (2000)

    MathSciNet  MATH  Google Scholar 

  16. Gehrke, M., Jónsson, B.: Bounded distributive lattice expansions. Math. Scand. 94(1), 13–45 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gouveia, M.J., Priestley, H.A.: Canonical extensions and profinite completions of semilattices and lattices. Order 31(2), 189–216 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Grätzer, G.: General Lattice Theory. Birkhäuser, Basel (1998)

    MATH  Google Scholar 

  19. Hansen, H.H.: Monotonic Modal Logic. Master’s thesis, University of Amsterdam (2003)

  20. Hyndman, J., Nation, J.B., Nishida, J.: Congruence lattices of semilattices with operators. Stud. Log. 104(2), 305–316 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ivanova-Dimova, E.: Lower-Vietoris-type topologies on hyperspaces. Topology Appl. 220, 100–110 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jónsson, B., Tarski, A.: Boolean algebras with operators. I. Am. J. Math. 73, 891–939 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jónsson, B., Tarski, A.: Boolean algebras with operators. II. Am. J. Math. 74, 127–162 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  24. Moshier, M.A., Jipsen, P.: Topological duality and lattice expansions. I: A topological construction of canonical extensions. Algebra Univers. 71(2), 109–126 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Priestley, H.: Representation of distributive lattices by means of ordered Stone spaces. Bull. London Math. Soc. 2, 186–190 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rewitzky, I.: Binary multirelations. Theory and applications of relational structures as knowledge instruments. Lect. Notes Comput. Sci. 2929, 256–271 (2003)

    Article  MATH  Google Scholar 

  27. Stone, M.: Topological representations of distributive lattices and Brouwerian logics. Čas. Mat. Fys. 67, 1–25 (1937)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the anonymous referee for his/her careful reading of the first version of this paper. All his/her valuable suggestions considerably improved the presentation of the final version. In addition, we also would like to thank Sergio Celani for his clarifying comments on Boolean algebras with a monotone operator

Author information

Authors and Affiliations

  1. CIC and Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional del Centro, Tandil, Argentina

    Ismael Calomino

  2. Nicolaus Copernicus University, Toruń, Poland

    Paula Menchón

  3. CONICET and Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional del Centro, Tandil, Argentina

    Paula Menchón

  4. Laboratoire J. A. Dieudonné, Université Côte d’Azur, Nice, France

    William J. Zuluaga Botero

  5. Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional del Centro, Tandil, Argentina

    William J. Zuluaga Botero

Authors
  1. Ismael Calomino
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paula Menchón
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. William J. Zuluaga Botero
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ismael Calomino.

Additional information

Communicated by Jorge Picado.

Publisher's Note

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

This research was supported by ANPCyT under grant 2019-00882. The third author has been funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 670624).

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Calomino, I., Menchón, P. & Botero, W.J.Z. A Topological Duality for Monotone Expansions of Semilattices. Appl Categor Struct 30, 1257–1282 (2022). https://doi.org/10.1007/s10485-022-09690-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-022-09690-0

Keywords

Search

Navigation