Skip to main content

Discrete Duality for Nelson Algebras with Tense Operators


In this paper, we continue with the study of tense operators on Nelson algebras (Figallo et al. in Studia Logica 109(2):285–312, 2021, Studia Logica 110(1):241–263, 2022). We define the variety of algebras, which we call tense Nelson D-algebras, as a natural extension of tense De Morgan algebras (Figallo and Pelaitay in Logic J IGPL 22(2):255–267, 2014). In particular, we give a discrete duality for these algebras. To do this, we will extend the representation theorems for Nelson algebras given in Sendlewski (Studia Logica 43(3):257–280, 1984) to the tense case.

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


  1. Balbes, R., and P. Dwinger, Distributive Lattices, University of Missouri Press, 1974.

    Google Scholar 

  2. Bialynicki-Birula, A., and H. Rasiowa, On the representation of quasi-Boolean algebras, Bulletin L’Académie Polonaise des Science Cl. III 5:259–261, XXII, 1957.

  3. Chajda I., and J. Paseka, Algebraic Approach to Tense Operators, vol. 45 of Research and Exposition in Mathematics, Heldermann Verlag, 2015.

  4. Chajda, I., and J. Paseka, Set representation of partial dynamic De Morgan algebras, in 2016 IEEE 46th International Symposium on Multiple-Valued Logic, IEEE Computer Society, Los Alamitos, CA, 2016, pp. 119–124.

  5. Chajda I., and J. Paseka, De Morgan algebras with tense operators, Journal of Multiple-Valued Logic and Soft Computing 1:29–45, 2017.

    Google Scholar 

  6. Chajda I., and J. Paseka, The Poset-based logics for the De Morgan negation and set representation of partial dynamic De Morgan algebras, Journal of Multiple-Valued Logic and Soft Computing 31(3):213–237, 2018.

    Google Scholar 

  7. Figallo, A.V. and G. Pelaitay, Tense operators on De Morgan algebras, Logic Journal of IGPL 22(2):255–267, 2014.

    Article  Google Scholar 

  8. Figallo, A.V., G. Pelaitay, and J. Sarmiento, An algebraic study of tense operators on Nelson algebras, Studia Logica 109(2):285–312, 2021.

    Article  Google Scholar 

  9. Figallo, A.V., J. Sermento, and G. Pelaitay, A categorical equivalence for tense Nelson algebras, Studia Logica 110(1): 241–263, 2022.

    Article  Google Scholar 

  10. Järvinen, J., and S. Radeleczki, Monteiro spaces and rough sets determined by quasiorder relations: models for Nelson algebras, Funding Information 131(2):205–215, 2014.

    Google Scholar 

  11. Jónsson, B., and A. Tarski, Boolean algebras with operators: I, American Journal of Mathematics 73:891–939, 1951.

  12. Monteiro, A., Construction des algebres de Nelson finies (French), Bulletin L’Académie Polonaise des Science, Série des Sciences Mathématiques, Astronomiques et Physiques 11:359–362, 1963.

  13. Monteiro, A., and L. Monteiro, Axiomes indépendants pour les algébres de Nelson, de Łukasiewicz trivalentes, de De Morgan et de Kleene, in \(Unpublished\,papers,\,I,\,Notas\,de\,L\acute{o}gica\,Matem\acute{a}tica\), vol. 40, 13 pp. Univ. Nac. del Sur, Bahía Blanca, 1996.

  14. Orłowska, E., and I. Rewitzky, Duality via truth: semantic frameworks for lattice-based logics, Logic Journal of the IGPL 13:467–490, 2005.

  15. Orłowska, E., A.M. Radzikowska, and I. Rewitzky, Dualities for Structures of Applied Logics, College Publications, 2015.

  16. Priestley, H.A., Representation of distributive lattices by means of ordered Stone spaces, Bulletin of the London Mathematical Society 2:186–190, 1970.

    Article  Google Scholar 

  17. Rasiowa, H., \(\cal{N}\)-lattices and constructive logic with strong negation, Funding Mathematics 46:61–80, 1958.

  18. Rasiowa, H., An Algebraic Approach to Non-Classical Logics, North-Holland, Amsterdam, PWN, Warszawa, 1974.

    Google Scholar 

  19. Segura, C., Tense De Morgan \(S4\)-algebras, Asian-European Journal of Mathematics 15(1), paper no. 2250014, 2022.

  20. Sendlewski, A., Some investigations of varieties of \({N}\)-lattices, Studia Logica 43(3):257–280, 1984.

    Article  Google Scholar 

  21. Sendlewski, A., Nelson algebras through Heyting ones: I, Studia Logica 49(1):105–126, 1990.

  22. Sofronie–Stokkermans, V., Representation theorems and the semantics of non–classical logics, and applications to automated theorem proving, in: M. Fitting, and E. Orłowska, (eds.), Beyond Two: Theory and Applications of Multiple-Valued Logic, vol. 114 of Studies in Fuzziness and Soft Computing, Heidelberg, 2003, pp. 59–100.

  23. Vakarelov, D., Notes on \(\cal{N}\)-lattices and constructive logic with strong negation, Studia Logica 36(1-2):109–125, 1977.

Download references


Jonathan Sarmiento and Gustavo Pelaitay want to thank the institutional support of Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Jonathan Sarmiento.

Additional information

Publisher's Note

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

Presented by Jacek Malinowski

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

Verify currency and authenticity via CrossMark

Cite this article

Figallo, A.V., Pelaitay, G. & Sarmiento, J. Discrete Duality for Nelson Algebras with Tense Operators. Stud Logica 111, 1–19 (2023).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Nelson algebras
  • Tense Nelson algebras
  • Tense operators
  • Discrete duality