Skip to main content
SpringerLink
Log in

Principal and Boolean Congruences on \(\varvec{IKt}\)-Algebras

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

The IKt-algebras were introduced in the paper An algebraic axiomatization of the Ewald’s intuitionistic tense logic by the first and third author. In this paper, our main interest is to investigate the principal and Boolean congruences on IKt-algebras. In order to do this we take into account a topological duality for these algebras obtained in Figallo et al. (Stud Log 105(4):673–701, 2017). Furthermore, we characterize Boolean and principal IKt-congruences and we show that Boolean IKt-congruence are principal IKt-congruences. Also, bearing in mind the above results, we obtain that Boolean IKt-congruences are commutative, regular and uniform. Finally, we characterize the principal IKt-congruences in the case that the IKt-algebra is linear and complete whose prime filters are complete and also the case that it is linear and finite. This allowed us to establish that the intersection of two principal IKt-congruences on these algebras is a principal one and also to determine necessary and sufficient conditions so that a principal IKt-congruence is a Boolean one on theses 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. Balbes, R., and P. Dwinger, Distributive lattices. University of Missouri Press Columbia, Mo., 1974.

    Google Scholar 

  2. Birkhoff, G., Lattice theory. Third edition. American Mathematical Society Colloquium Publications, Vol. XXV American Mathematical Society, Providence, R.I. 1967.

  3. Botur, M., I. Chajda, R. Halaš, and M. Kolařík, Tense operators on basic algebras. Internat. J. Theoret. Phys. 50(12):3737–3749, 2011.

    Article  Google Scholar 

  4. Botur, M., and J. Paseka, On tense \(MV\)-algebras. Fuzzy Sets and Systems 259:111–125, 2015.

    Article  Google Scholar 

  5. Burgess, J. P., Basic tense logic. Handbook of philosophical logic, Vol. II, 89–133, Synthese Lib., 165, Reidel, Dordrecht, 1984.

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

    Google Scholar 

  7. Chajda, I., R. Halaš, and J. Kühr, Semilattice structures. Research and Exposition in Mathematics, 30. Heldermann Verlag, Lemgo, 2007.

    Google Scholar 

  8. Chajda, I., and J. Paseka, Algebraic Approach to Tense Operators, Research and Exposition in Mathematics, 35, Heldermann Verlag, Lemgo, 2015.

    Google Scholar 

  9. Chajda, I., Algebraic axiomatization of tense intuitionistic logic. Cent. Eur. J. Math. 9(5):1185–1191, 2011.

    Article  Google Scholar 

  10. Chajda, I., and J. Paseka, Dynamic effect algebras and their representations, Soft Computing 16(10):1733–1741, 2012.

    Article  Google Scholar 

  11. Chajda, I., and J. Paseka, Tense Operators and Dynamic De Morgan Algebras, In: Proc. 2013 IEEE 43rd Internat. Symp. Multiple-Valued Logic, Springer, pp. 219–224, 2013.

  12. Chajda, I., and J. Paseka, Dynamic order algebras as an axiomatization of modal and tense logics. Internat. J. Theoret. Phys. 54(12):4327–4340, 2015.

    Article  Google Scholar 

  13. Chajda, I., and M. Kolařík, Dynamic effect algebras. Math. Slovaca 62(3):379–388, 2012.

    Article  Google Scholar 

  14. Chiriţă, C., Tense \(\theta \)-valued Łukasiewicz-Moisil algebras. J. Mult.-Valued Logic Soft Comput. 17(1):1–24, 2011.

    Google Scholar 

  15. Chiriţă, C., Polyadic tense \(\theta \)–valued Łukasiewicz-Moisil algebras, Soft Computing 16(6):979–987, 2012.

    Article  Google Scholar 

  16. Diaconescu, D., and G. Georgescu, Tense operators on MV-algebras and Łukasiewicz-Moisil algebras. Fund. Inform. 81(4):379–408, 2007.

    Google Scholar 

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

    Article  Google Scholar 

  18. Esakia L., Topological Kripke models. Soviet Math Dokl., 15:147–151, 1974.

    Google Scholar 

  19. Ewald, W. B., Intuitionistic tense and modal logic. J. Symbolic Logic 51(1):166–179, 1986.

    Article  Google Scholar 

  20. Figallo, A. V., and G. Pelaitay, A representation theorem for tense \(n\times m\)-valued Łukasiewicz-Moisil algebras. Math. Bohem. 140(3):345–360, 2015.

    Google Scholar 

  21. Figallo, A. V., and G. Pelaitay, Discrete duality for tense Łukasiewicz-Moisil algebras. Fund. Inform. 136(4):317–329, 2015.

    Google Scholar 

  22. Figallo, A. V., and G. Pelaitay, Note on tense \(SHn\)-algebras. An. Univ. Craiova Ser. Mat. Inform. 38(4):24–32, 2011.

    Google Scholar 

  23. Figallo, A. V., G. Pelaitay, and C. Sanza, Discrete duality for \(TSH\)-algebras. Commun. Korean Math. Soc. 27(1):47–56, 2012.

    Article  Google Scholar 

  24. Figallo, A. V., and G. Pelaitay, Remarks on Heyting algebras with tense operators. Bull. Sect. Logic Univ. Lódz 41(1-2):71–74, 2012.

    Google Scholar 

  25. Figallo, A. V., and G. Pelaitay, Tense polyadic \(n\times m\)-valued Łukasiewicz-Moisil algebras. Bull. Sect. Logic Univ. Lódz 44(3–4):155–181, 2015.

    Google Scholar 

  26. Figallo, A. V., and G. Pelaitay, Tense operators on De Morgan algebras. Log. J. IGPL 22(2):255–267, 2014.

    Article  Google Scholar 

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

    Article  Google Scholar 

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

  29. Johnstone, P. T., Stone Spaces. Cambridge Studies in Advanced Mathematics, 3. Cambridge University Press, Cambridge, 1982.

    Google Scholar 

  30. Kowalski, T., Varieties of tense algebras. Rep. Math. Logic 32:53–95, 1998.

    Google Scholar 

  31. Mac Lane, S., Categories for the working mathematician. Second edition. Graduate Texts in Mathematics, 5. Springer-Verlag, New York, 1998.

  32. Menni, M., and C. Smith, Modes of adjointness. J. Philos. Logic 43(2–3):365–391, 2014.

    Article  Google Scholar 

  33. Paseka, J., Operators on MV-algebras and their representations. Fuzzy Sets and Systems 232:62–73, 2013.

    Article  Google Scholar 

  34. Priestley, H. A., Representation of distributive lattices by means of ordered stone spaces. Bull. London Math. Soc. 2:186–190, 1970.

    Article  Google Scholar 

  35. Priestley, H. A., Ordered topological spaces and the representation of distributive lattices. Proc. London Math. Soc. 24(3):507–530, 1972.

    Article  Google Scholar 

  36. Priestley, H. A., Ordered sets and duality for distributive lattices. Orders: description and roles (L’Arbresle, 1982), 39–60, North-Holland Math. Stud., 99, North-Holland, Amsterdam, 1984.

Download references

Author information

Authors and Affiliations

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

    Aldo V. Figallo, Inés Inés Pascual & Gustavo Pelaitay

  2. Departamento de Matemática, Universidad Nacional de San Juan, San Juan, Argentina

    Inés Inés Pascual

Authors
  1. Aldo V. Figallo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Inés Inés Pascual
    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

Presented by Jacek Malinowski

Received July 12, 2017.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Figallo, A.V., Pascual, I.I. & Pelaitay, G. Principal and Boolean Congruences on \(\varvec{IKt}\)-Algebras. Stud Logica 106, 857–882 (2018). https://doi.org/10.1007/s11225-017-9770-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-017-9770-8

Keywords

Search

Navigation