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Subdirectly Irreducible IKt-Algebras

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Abstract

The IKt-algebras that we investigate in this paper were introduced in the paper An algebraic axiomatization of the Ewald’s intuitionistic tense logic by the first and third author. Now we characterize by topological methods the subdirectly irreducible IKt-algebras and particularly the simple IKt-algebras. Finally, we consider the particular cases of finite IKt-algebras and complete IKt-algebras.

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References

  1. Balbes, R., and P. Dwinger, Distributive lattices. University of Missouri Press, Columbia, Mo., 1974.

  2. Birkhoff, G., Lattice theory, 3rd edn, vol. XXV. American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, 1967.

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

    Article  Google Scholar 

  4. Botur M., Paseka J.: 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, pp. 89–133.

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

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

    Article  Google Scholar 

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

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

    Article  Google Scholar 

  10. Chajda I., Paseka J.: 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 Proceedings of 2013 IEEE 43rd International Symposium on Multiple-Valued Logic, Springer, 2013, pp. 219–224.

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

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

    Google Scholar 

  17. Dzik W., Järvinen J., Kondo M.: 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., Pelaitay G.: 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., Pelaitay G.: Discrete duality for tense Łukasiewicz-Moisil algebras. Fund. Inform. 136(4), 317–329 (2015)

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

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

    Google Scholar 

  30. Mac Lane, S., Categories for the working mathematician, in Graduate Texts in Mathematics, 2nd edn., Springer, New York, 1998, p. 5.

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

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Authors and Affiliations

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

    Aldo V. Figallo, Inés Pascual & Gustavo Pelaitay

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

    Inés Pascual & Gustavo Pelaitay

Authors
  1. Aldo V. Figallo
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  2. Inés Pascual
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  3. Gustavo Pelaitay
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Correspondence to Gustavo Pelaitay.

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Presented by Jacek Malinowski

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Figallo, A.V., Pascual, I. & Pelaitay, G. Subdirectly Irreducible IKt-Algebras. Stud Logica 105, 673–701 (2017). https://doi.org/10.1007/s11225-017-9707-2

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