An Algebraic Study of S5-Modal Gödel Logic

Abstract

In this paper we continue the study of the variety \(\mathbb {MG}\) of monadic Gödel algebras. These algebras are the equivalent algebraic semantics of the S5-modal expansion of Gödel logic, which is equivalent to the one-variable monadic fragment of first-order Gödel logic. We show three families of locally finite subvarieties of \(\mathbb {MG}\) and give their equational bases. We also introduce a topological duality for monadic Gödel algebras and, as an application of this representation theorem, we characterize congruences and give characterizations of the locally finite subvarieties mentioned above by means of their dual spaces. Finally, we study some further properties of the subvariety generated by monadic Gödel chains: we present a characteristic chain for this variety, we prove that a Glivenko-type theorem holds for these algebras and we characterize free algebras over n generators.

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References

  1. 1.

    Abad, M., and L. Monteiro, On free \(L\)-algebras, Notas de Lógica Matemática Univ. Nac. del Sur 35:1–20, 1987.

    Google Scholar 

  2. 2.

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

    Google Scholar 

  3. 3.

    Bezhanishvili, G., Varieties of monadic Heyting Algebras. Part I, Studia Logica 61(3):367–402, 1998.

    Article  Google Scholar 

  4. 4.

    Bezhanishvili, G., Varieties of monadic Heyting Algebras. Part II: Duality Theory, Studia Logica 62(1):21–48, 1999.

    Article  Google Scholar 

  5. 5.

    Bezhanishvili, G., Glivenko Type Theorems for Intuitionistic Logics, Studia Logica 67:89–109, 2001.

    Article  Google Scholar 

  6. 6.

    Bezhanishvili, G., Locally finite varieties, Algebra Universalis 46(4):531–548, 2001.

    Article  Google Scholar 

  7. 7.

    Bezhanishvili, G., and R. Girgolia, Locally Tabular Extensions of MIPC, in M. Zakharyaschev, K. Segerberg, M. de Rijke, and H. Wansing, (eds.), Advances in Modal Logic, Vol. 2, CSLI Publications, 2000, pp. 119–138.

  8. 8.

    Blok, W. J., Varieties of interior algebras, Ph.D. Thesis, University of Amsterdam, 1976.

  9. 9.

    Caicedo, X., and R. O. Rodríguez, Bi-modal Gödel logic over \([0,1]\)-valued Kripke frames, J. Logic Comput. 25(1):37–55, 2015.

    Article  Google Scholar 

  10. 10.

    Castaño, D., C. Cimadamore, J. P. Díaz Varela, and L. Rueda, Monadic BL-algebras: The equivalent algebraic semantics of Hájek’s monadic fuzzy logic, Fuzzy Sets and Systems 320:40–59, 2017.

    Article  Google Scholar 

  11. 11.

    Castaño, D., C. Cimadamore, J. P. Díaz Varela, and L. Rueda, Completeness for monadic fuzzy logics via functional algebras, Fuzzy Sets and Systems, to appear, 2020, https://doi.org/10.1016/j.fss.2020.02.002.

  12. 12.

    Cignoli, R., Quantifiers on distributive lattices, Discrete Mathematics 96(3):183–197.

  13. 13.

    Davey, B. A., and H. A. Priestley, Introduction to lattices and order, second edition, Cambridge University Press, 2002.

    Google Scholar 

  14. 14.

    Di Nola, A., and R. Grigolia, On monadic MV-algebras, Annals of Pure and Applied Logic 128(1-3):125–139, 2004.

    Article  Google Scholar 

  15. 15.

    Esakia, L., Heyting algebras, translated from the Russian edition by Anton Evseev, Trends in Logic—Studia Logica Library 50, Springer, 2019.

  16. 16.

    Glivenko, V., Sur quelques points de la logique de M. Brouwer, Bulletin de la Classe des Sciences de l’Académie Royale de Belgique 15:183–188, 1929.

    Google Scholar 

  17. 17.

    Hájek, P., Metamathematics of fuzzy logic, Trends in Logic—Studia Logica Library 4, Kluwer Academic Publishers, 1998.

  18. 18.

    Halmos, P. R., Algebraic logic, I. Monadic boolean algebras, Compositio Mathematica 12:217–249, 1955.

    Google Scholar 

  19. 19.

    Hecht, T., and T. Katriňák, Equational classes of relative Stone algebras, Notre Dame J. Formal Logic 13(2):248–254, 1972.

    Article  Google Scholar 

  20. 20.

    Monteiro, A., Sur les Algèbres de Heyting Symétriques, Portugaliae Mathematica 39:1–237, 1980.

    Google Scholar 

  21. 21.

    Monteiro, A., and O. Varsavsky, Álgebras de Heyting monádicas, Actas de las X Jornadas de la Unión Matemática Argentina, 52–62, 1957.

  22. 22.

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

    Article  Google Scholar 

  23. 23.

    Priestley, H. A., Ordered sets and duality for distributive lattices, Ann. Discrete Math. 23:39–60, 1984.

    Google Scholar 

  24. 24.

    Rueda, L., The subvariety of \(Q\)-Heyting algebras generated by chains, Revista de la Unión Matemática Argentina 50(1):47–59, 2009.

    Google Scholar 

  25. 25.

    Werner, H., Discriminator-algebras: Algebraic Representation and Model Theoretic Properties, Studien zur Algebra und ihre Anwendungen [Studies in Algebra and its Applications] vol. 6, Akademie-Verlag, Berlin, 1978.

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Acknowledgements

We would like to thank the support of CONICET (PIP 11220170100195CO) and Departamento de Matemática (UNS) (PGI 24/L108). We would also like to thank the anonymous reviewer for his/her useful suggestions that helped us improve the article.

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Correspondence to Diego Castaño.

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Castaño, D., Cimadamore, C., Díaz Varela, J.P. et al. An Algebraic Study of S5-Modal Gödel Logic. Stud Logica (2021). https://doi.org/10.1007/s11225-020-09934-x

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Keywords

  • Gödel logic
  • S5-modal logic
  • Algebraic semantics
  • Priestley duality