, Volume 30, Issue 1, pp 195–210 | Cite as

Monadic Bounded Residuated Lattices

  • Jiří Rachůnek
  • Dana Šalounová


Bounded integral residuated lattices form a large class of algebras which contains algebraic counterparts of several propositional logics behind many-valued reasoning and intuitionistic logic. In the paper we introduce and investigate monadic bounded integral residuated lattices which can be taken as a generalization of algebraic models of the predicate calculi of those logics in which only a single variable occurs.


Bounded integral residuated lattice Monadic residuated lattice Algebras of logics Quantifier 

Mathematics Subject Classifications (2010)

03B50 06D20 06D35 06F05 


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  1. 1.
    Balbes, R., Dwinger, P.: Distributive Lattices. Univ. Missouri Press, Columbia (1974)MATHGoogle Scholar
  2. 2.
    Belluce, L.P., Grigolia, R., Lettieri, A.: Representations of monadic MV-algebras. Stud. Log. 81, 123–144 (2005)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bezhanisvili, G.: Varieties of monadic Heyting algebras I. Stud. Log. 61, 367–402 (1998)CrossRefGoogle Scholar
  4. 4.
    Bezhanisvili, G.: Varieties of monadic Heyting algebras II. Stud. Log. 62, 21–48 (1999)CrossRefGoogle Scholar
  5. 5.
    Bezhanisvili, G.: Varieties of monadic Heyting algebras III. Stud. Log. 63, 215–256 (2000)CrossRefGoogle Scholar
  6. 6.
    Bezhanisvili, G., Harding, J.: Functional monadic Heyting algebras. Alg. Univers. 48, 1–10 (2002)CrossRefGoogle Scholar
  7. 7.
    Chang, C.C.: Algebraic analysis of many valued logic. Trans. Am. Math. Soc. 88, 467–490 (1958)MATHCrossRefGoogle Scholar
  8. 8.
    Di Nola, A., Georgescu, G. and Iorgulescu, A.: Pseudo-BL algebras: Part I. Mult.-Valued Log. 8, 673–714 (2002)MathSciNetMATHGoogle Scholar
  9. 9.
    Di Nola, A., Georgescu, G. and Iorgulescu, A.: Pseudo-BL algebras: Part II. Mult.-Valued Log. 8, 717–750 (2002)MathSciNetMATHGoogle Scholar
  10. 10.
    Di Nola, A., Grigolia, R.: On monadic MV-algebras. Ann. Pure Appl. Logic 128, 125–139 (2004)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Dvurečenskij, A.: Every linear pseudo BL-algebra admits a state. Soft Comput. 11, 495–501 (2007)MATHCrossRefGoogle Scholar
  12. 12.
    Esteva, F., Godo, L.: Monoidal t-norm based logic: towards a logic for left continuous t norms. Fuzzy Sets Syst. 124, 271–288 (2001)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Flondor, P., Georgescu, G., Iorgulescu, A.: Pseudo t-norms and pseudo-BL-algebras. Soft Comput. 5, 355–371, (2001)MATHCrossRefGoogle Scholar
  14. 14.
    Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Elsevier Studies in Logic and Foundations, Amsterdam-Oxford (2007)MATHGoogle Scholar
  15. 15.
    Galatos, N., Tsinakis, C.: Generalized MV-algebras. J. Algebra 283, 254–291 (2005)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Georgescu, G., Popescu, A.: Non-commutative fuzzy structures and pair of weak negations. Fuzzy Sets Syst. 143, 129–155 (2004)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Georgescu, G., Iorgulescu, A.: Pseudo-MV algebras. Mult.-Valued Log. 6, 95–135 (2001)MathSciNetMATHGoogle Scholar
  18. 18.
    Georgescu, G., Iorgulescu, A., Leuştean, I.: Monadic and closure MV-algebras. Mult.-Valued Log. 3, 235–257 (1998)MATHGoogle Scholar
  19. 19.
    Grigolia, R.: Monadic BL-algebras. Georgian Math. J. 13, 267–276 (2006)MathSciNetMATHGoogle Scholar
  20. 20.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Amsterdam (1998)MATHCrossRefGoogle Scholar
  21. 21.
    Hájek, P.: Observations on non-commutative fuzzy logic. Soft Comput. 8, 39–43 (2003)CrossRefGoogle Scholar
  22. 22.
    Hájek, P.: Fuzzy logics with noncommutative conjuctions. J. Log. Comput. 13, 469–479 (2003)MATHCrossRefGoogle Scholar
  23. 23.
    Halmos, P.R.: Algebraic Logic. Chelsea Publ. Co., New York (1962)MATHGoogle Scholar
  24. 24.
    Henkin, L., Monk, J.D., Tarski, A.: Cylindric Algebras, Parts I and II. North-Holland, Amsterdam (1971, 1985)Google Scholar
  25. 25.
    Jipsen, P., Tsinakis, C.: A survey of residuated lattices. In: Martinez, J. (ed.) Ordered Algebraic Structures, pp. 19–56. Kluwer Academic, Dordrecht (2002)CrossRefGoogle Scholar
  26. 26.
    Neméti, I.: Algebraization of quantifier logics. Stud. Log. 50, 485–569 (1991)MATHCrossRefGoogle Scholar
  27. 27.
    Pigozzi, D., Salibra, A.: Polyadic algebras over nonclassical logics. In: Algebraic Methods in Logic and in Computer Science, vol. 28, pp. 51–66. Banach Center Publ. (1993)Google Scholar
  28. 28.
    Rachůnek, J.: A non-commutative generalization of MV-algebras. Czechoslov. Math. J. 52, 255–273 (2002)MATHCrossRefGoogle Scholar
  29. 29.
    Rachůnek, J., Šalounová, D.: Monadic GMV-algebras. Arch. Math. Log. 47, 277–297 (2008)MATHCrossRefGoogle Scholar
  30. 30.
    Rachůnek, J., Švrček, F.: Monadic bounded commutative residuated ℓ-monoids. Order 25, 157–175 (2008)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Rutledge, J.D.: A preliminary investigation of the infinitely many-valued predicate calculus. Ph.D. thesis, Cornell University (1959)Google Scholar
  32. 32.
    Varsavsky, O.: Quantifiers and equivalence realations. Rev. Mat. Cuyana 2, 29–51 (1956)MathSciNetGoogle Scholar

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© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Algebra and Geometry, Faculty of SciencesPalacký UniversityOlomoucCzech Republic
  2. 2.Department of Mathematical Methods in Economy, Faculty of EconomicsVŠB–Technical University OstravaOstravaCzech Republic

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