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Completeness with respect to a chain and universal models in fuzzy logic

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Abstract

In this paper we investigate fuzzy propositional and first order logics which are complete or strongly complete with respect to a single chain, and we relate this properties with the existence of a universal chain for the logic.

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References

  1. Aglianò P., Ferreirim I.M.A., Montagna F.: Basic hoops: An algebraic study of continuous t-norms. Studia Logica 87(1), 73–98 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aglianò P., Montagna F.: Varieties of BL-algebras I: General properties. J. Pure Appl. Algebra 181, 105–129 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bianchi, M., Montagna, F.: n-contractive BL logics, Preprint 2010, submitted

  4. Blok W.J., Ferreirim I.M.A.: On the structure of hoops. Algebra Univers. 43, 233–257 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Blok, W., Pigozzi, D.: Algebraizable logics. Mem. Am. Math. Soc. 396, vol. 77, Am. Math. Soc., Providence (1989)

  6. Burris S., Sankappanavar H.P.: A Course in Universal Algebra, Graduate Texts in Mathematics. Springer, New York (1981)

    Google Scholar 

  7. Cignoli R., D’Ottaviano M.I., Mundici D.: Algebraic Foundations of Many-valued Reasoning. Kluwer, Dordrecht (2000)

    MATH  Google Scholar 

  8. Cignoli R., Torrens A.: An algebraic analysis of product logic. Mult. Valued Log. 5, 45–65 (2000)

    MATH  MathSciNet  Google Scholar 

  9. Cintula P.: Weakly implicative (fuzzy) logics I: Basic properties. Arch. Math. Log. 45(6), 673–704 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cintula P., Esteva F., Gispert J., Godo L., Montagna F., Noguera C.: Distinguished algebraic semantics for t-norm based fuzzy logics: Methods and algebraic equivalencies. Ann. Pure Appl. Log. 160(1), 53–81 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cintula P., Hájek P.: On theories and models in fuzzy predicate logics. J. Symb. Log. 71(3), 863–880 (2006)

    Article  MATH  Google Scholar 

  12. Di Nola A., Lettieri A.: Equational characterization of all varieties of MV-algebras. J. Algebra 221, 463–474 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Di Nola A., Lettieri A.: One chain generated varieties of MV-algebras. J. Algebra 25(2), 667–679 (2000)

    Article  MathSciNet  Google Scholar 

  14. Di Nola, A., Lenzi, G., Spada, L.: Representation of MV-algebras by regular ultrapowers, Technical Report 2009, to appear in Archive for Mathematical Logic

  15. Esteva F., Godo L.: Monoidal t-norm based logic: Towards a logic for left-continuous t-norms. Fuzzy Set Syst. 124, 271–288 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ferreirim, I. M. A.: On varieties and quasi varieties of hoops and their reducts. PhD thesis, University od Illinois at Chicago (1992)

  17. Galatos N., Jipsen P., Kowalski T., Ono H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Studies in Logic and the Foundations of Mathematics. Vol. 151, Elsevier, Amsterdam (2007)

    Google Scholar 

  18. Hájek P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)

    MATH  Google Scholar 

  19. Laskowski M., Malekpour S.: Provability in predicate product logic. Arch. Math. Log. 46, 365–378 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Montagna F.: Interpolation and Beth’s property in propositional many-valued logics: A semantic investigation. Ann. Pure Appl. Log. 141, 148–179 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Montagna F., Noguera C., Horčik R.: On weakly cancellative fuzzy logics. J. Log. Comput. 16, 423–450 (2006)

    Article  MATH  Google Scholar 

  22. Mundici D.: Free products in the category of abelian ℓ-groups with strong unit. J. Algebra 113, 89–109 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  23. Noguera C., Esteva F., Gispert J.: Triangular norm based axiomatic extensions of the weak Nilpotent minimum logic. Math. Log. Q. 54, 387–409 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Franco Montagna.

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Montagna, F. Completeness with respect to a chain and universal models in fuzzy logic. Arch. Math. Logic 50, 161–183 (2011). https://doi.org/10.1007/s00153-010-0207-6

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  • DOI: https://doi.org/10.1007/s00153-010-0207-6

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