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An Algebraic Study of Tense Operators on Nelson Algebras

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Abstract

Ewald considered tense operators G, H, F and P on intuitionistic propositional calculus and constructed an intuitionistic tense logic system called IKt. In 2014, Figallo and Pelaitay introduced the variety IKt of IKt-algebras and proved that the IKt system has IKt-algebras as algebraic counterpart. In this paper, we introduce and study the variety of tense Nelson algebras. First, we give some examples and we prove some properties. Next, we associate an IKt-algebra to each tense Nelson algebras. This result allowed us to determine the congruence of the tense Nelson algebras and also to characterize the subdirectly irreducible tense Nelson algebras and particularly the simple tense Nelson algebras. Finally, we prove that there exists an equivalence between the category of IKt-algebras and the category of tense centered Nelson algebras.

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References

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

    Article  Google Scholar 

  2. Botur, M., I. Chajda, R., Halaš, and M. Kolařik, Tense operators on Basic Algebras, Internat. J. Theoret. Phys. 50(12):3737–3749, 2011.

    Article  Google Scholar 

  3. Botur, M., and J. Paseka, Partial tense \(MV\)-algebras and related functions, Fuzzy Sets Syst. 326:24–33, 2017.

    Article  Google Scholar 

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

    Chapter  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  8. 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 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  11. Figallo, A. V., I. Pascual, and G. Pelaitay, A topological duality for tense \(LM_n\)-algebras and applications, Log. J. IGPL 26(4):339–380, 2018.

    Article  Google Scholar 

  12. 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 

  13. Figallo, A. V., I. Pascual, and G. Pelaitay, A topological duality for tense \(\theta \)-valued Łukasiewicz–Moisil algebras, Soft Comput. 23:3979, 2019.

    Article  Google Scholar 

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

    Google Scholar 

  15. 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 

  16. 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 

  17. Gabbay, D. M., Model theory for tense logics, Ann. Math. Logic 8:185–236, 1975.

    Article  Google Scholar 

  18. Jónsson, B., A survey of Boolean algebras with operators. Algebras and orders (Montreal, PQ, 1991), pp. 239–286, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 389, Kluwer Acad. Publ., Dordrecht, 1993.

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

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  22. Mac Lane, S., Categories for the Working Mathematician, 2nd edn. Springer-Verlag, NewYork, 1998.

    Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  25. Monteiro A., Algebras de Nelson semi-simples (Resumen), Rev. Unión Mat. Argentina 21:145–146, 1963.

    Google Scholar 

  26. Monteiro A., Constructions des algébres de Nelson finies, Bull. Acad. Pol. des Sc. 11:359–362, 1963.

    Google Scholar 

  27. Monteiro, A., Les elements reguliers d’ un \(N\)-lattice, Textos e Notas No 15, Univ. de Lisboa, 1978.

  28. Monteiro, A., Les \(N\)-lattice linéaires, Textos e Notas No 15, Univ. de Lisboa, 1978.

  29. Monteiro, A., and L. Monteiro, Axiomes indépendants pour les algébres de Nelson, de Lukasiewicz trivalentes, de De Morgan et de Kleene. In Unpublished papers, I, Notas de Lógica Matemática, vol. 40, 13 pp. Univ. Nac. del Sur, Bahía Blanca, 1996.

  30. Nelson, D., Constructible falsity, J. Symbolic Logic 14:16–26, 1949.

    Article  Google Scholar 

  31. Orłowska, E., A. M. Radzikowska, and I. Rewitzky, Dualities for structures of applied logics. Studies in Logic (London), 56. Mathematical Logic and Foundations. College Publications, London, 2015.

  32. Orłowska, E., and I. Rewitzky, Discrete dualities for Heyting algebras with operators, Fund. Inform. 81(1-3):275–295, 2007.

    Google Scholar 

  33. Rasiowa H., \(N\)-lattices and constructive logic with strong negation, Fund. Math. 46:61–80, 1958.

    Article  Google Scholar 

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

    Google Scholar 

  35. Rescher, N., and A. Urquhart, The Introduction of Tense Operators, in Temporal Logic. LEP Library of Exact Philosophy, vol 3. Springer, Vienna 1971.

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

    Article  Google Scholar 

  37. Sofronie–Stokkermans, V., Representation theorems and the semantics of non-classical logics, and applications to automated theorem proving. Beyond two: theory and applications of multiple-valued logic, Stud. Fuzziness Soft Comput., vol. 114, Physica, Heidelberg, 2003, pp. 59–100.

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

    Article  Google Scholar 

  39. von Karger, B., Temporal algebra. Mathematical Structures in Computer Science 8(3):277-320, 1998.

    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 want to thank the institutional support of Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET).

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Authors and Affiliations

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

    A. V. Figallo, G. Pelaitay & J. Sarmiento

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

    J. Sarmiento

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  1. A. V. Figallo
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  2. G. Pelaitay
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  3. J. Sarmiento
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Correspondence to G. Pelaitay.

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Figallo, A.V., Pelaitay, G. & Sarmiento, J. An Algebraic Study of Tense Operators on Nelson Algebras. Stud Logica 109, 285–312 (2021). https://doi.org/10.1007/s11225-020-09907-0

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