Skip to main content
SpringerLink
Log in

A Categorical Equivalence for Tense Nelson Algebras

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

In this paper we present a category equivalent to that of tense Nelson algebras. The objects in this new category are pairs consisting of an IKt-algebra and a Boolean IKt-congruence and the morphisms are a special kind of IKt-homomorphisms. This categorical equivalence permits understanding tense Nelson algebras in terms of the better–known IKt-algebras.

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. Ahmed, T. S., Space and time via Topological and Tense cylindric algebras, 2006.03421, 2020.

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

    Google Scholar 

  3. Bakhshi, M., Tense operators on non-commutative residuated lattices, Soft Computing 21(15):4257–4268, 2017.

    Article  Google Scholar 

  4. Botur, M., I. Chajda, R. Halaš, and M. Kolarik, Tense operators on Basic Algebras, International Journal of Theoretical Physics 50(12):3737–3749, 2011.

    Article  Google Scholar 

  5. Botur, M., and J. Paseka, Partial tense MV–algebras and related functions, Fuzzy Sets and Systems 326:24–33, 2017

    Article  Google Scholar 

  6. Burges, J., Basic tense logic, in D. M. Gabbay, and F. Gunter, (eds.), Handbook of Philosophical Logic, vol. II, Reidel, Dordrecht, 1984, pp. 89–139.

    Chapter  Google Scholar 

  7. Chajda, I., Algebraic axiomatization of tense intuitionistic logic, Central European Journal of Mathematics 9(5):1185–1191, 2011.

    Article  Google Scholar 

  8. Diaconescu, D., and G. Georgescu, Tense operators on MV-algebras and Lukasiewicz–Moisil algebras, Fundamenta Informaticae 81(4):379–408, 2007.

    Google Scholar 

  9. Dzik, W., J. Järvinen, and M. Kondo, Characterizing intermediate tense logics in terms of Galois connections, Logic Journal of the IGPL 22(6):992–1018, 2014.

    Article  Google Scholar 

  10. Ewald, W. B., Intuitionistic tense and modal logic, The Journal of Symbolic Logic 51(1):166–179, 1986.

    Article  Google Scholar 

  11. Figallo, A. V., and G. Pelaitay, Remarks on Heyting algebras with tense operators, Bulletin of the Section of Logic 41(1-2):71–74, 2012.

  12. Figallo, A. V., and G. Pelaitay, An algebraic axiomatization of the Ewald’s intuitionistic tense logic, Soft Computing 18(10):1873–1883, 2014.

    Article  Google Scholar 

  13. Figallo, A. V., I. Pascual, and G. Pelaitay, Subdirectly irreducible IKt-algebras, Studia Logica 105(4):673–701, 2017.

    Google Scholar 

  14. Figallo, A. V., I. Pascual, and G. Pelaitay, Principal and Boolean congruences on IKt-algebras, Studia Logica 106(4):857–882, 2018.

    Article  Google Scholar 

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

    Article  Google Scholar 

  16. Fitting, M. C., Intuitionistic logic, model theory and forcing, North-Holland Publishing Co., Amsterdam-London, 1969.

  17. Gabbay, D. M., Model theory for tense logics, Annals of Mathematical Logic 8:185–236, 1975.

    Article  Google Scholar 

  18. Georgescu, G., A representation theorem for tense polyadic algebras, Mathematica (Cluj) 21(44)(2):131–138, 1979.

  19. Kowalski, T., Varieties of tense algebras, Reports on Mathematical Logic 32:53–95, 1998.

    Google Scholar 

  20. Lemmon, E. J., Algebraic semantics for modal logics: I, The Journal of Symbolic Logic 31:46–65, 1966.

    Article  Google Scholar 

  21. Lemmon, E. J., Algebraic semantics for modal logics: II, The Journal of Symbolic Logic 31:191–218, 1966.

    Article  Google Scholar 

  22. Markov, A.A., A constructive logic, Uspehi Mathematiceskih Nauk (N.S.) 5:187–188, 1950.

  23. Monteiro, A., and L. Monteiro, Axiomes indépendants pour les algébres de Nelson, de Lukasiewicz trivalentes, de De Morgan et de Kleene, in A. Monteiro, Unpublished papers, I, vol. 40 of Notas de Lógica Matemática, Universidad Nacional del Sur, Bahía Blanca, 1996.

  24. Nelson, D., Constructible falsity, The Journal of Symbolic Logic 14:16–26, 1949.

    Article  Google Scholar 

  25. Pigozzi, D., and A. Salibra, Polyadic algebras over nonclassical logics, Banach Center Publications 28(1):51–66, 1993.

  26. Rasiowa, H., An algebraic aproach to non-classic logic, North–Holland, Amsterdam, 1974.

    Google Scholar 

  27. Rescher, N., and A. Urquhart, The Introduction of Tense Operators, in N. Rescher, and A. Urquhart, (eds.), Temporal Logic, vol. 3 of LEP Library of Exact Philosophy, Springer, Vienna 1971, pp. 50–54.

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

    Article  Google Scholar 

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

    Article  Google Scholar 

Download references

Acknowledgements

The authors acknowledge many helpful comments from the anonymous referee, which considerably improved the presentation of this paper. Jonathan Sarmiento wants to thanks the institutional support of Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET).

Author information

Authors and Affiliations

  1. Instituto de Ciencias Básicas, Universidad Nacional de San Juan, 5400, San Juan, Argentina

    Aldo V. Figallo, Jonathan Sermento & Gustavo Pelaitay

  2. Departamento de Matemática, Universidad Nacional del Sur, 8000, Bahía Blanca, Buenos Aires, Argentina

    Jonathan Sermento

Authors
  1. Aldo V. Figallo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jonathan Sermento
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gustavo Pelaitay
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gustavo Pelaitay.

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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Figallo, A.V., Sermento, J. & Pelaitay, G. A Categorical Equivalence for Tense Nelson Algebras. Stud Logica 110, 241–263 (2022). https://doi.org/10.1007/s11225-021-09960-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-021-09960-3

Keywords

Search

Navigation