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Non-commutative Multiple-Valued Logic Algebras pp 107–134Cite as

Other Non-commutative Multiple-Valued Logic Algebras

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Abstract

In this chapter we present some specific properties of other non-commutative multiple-valued logic algebras: pseudo-MTL algebras, bounded Rℓ-monoids, pseudo-BL algebras and pseudo-MV algebras. As main results, we extend to the case of pseudo-MTL algebras some results regarding the prime filters proved for pseudo-BL algebras. The Glivenko property for multiple-valued logic algebras is defined and studied.

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References

  1. Birkhoff, G.: Lattice Theory. Am. Math. Soc., Providence (1967)

    MATH  Google Scholar 

  2. Buşneag, D., Piciu, D.: On the lattice of filters of a pseudo-BL algebra. J. Mult.-Valued Log. Soft Comput. 12, 217–248 (2006)

    MATH  Google Scholar 

  3. Di Nola, A., Georgescu, G., Iorgulescu, A.: Pseudo-BL algebras: part I. Mult. Valued Log. 8, 673–714 (2002)

    MathSciNet  MATH  Google Scholar 

  4. Di Nola, A., Georgescu, G., Iorgulescu, A.: Pseudo-BL algebras: part II. Mult. Valued Log. 8, 717–750 (2002)

    MathSciNet  MATH  Google Scholar 

  5. Di Nola, A., Dvurečenskij, A., Tsinakis, C.: Perfect GMV-algebras. Commun. Algebra 36, 1221–1249 (2008)

    Article  MATH  Google Scholar 

  6. Dvurečenskij, A.: On partial addition in pseudo-MV algebras. In: Proc. 4th International Symposium on Economic Informatics, pp. 952–960. INFOREC Printing House, Bucharest (1999)

    Google Scholar 

  7. Dvurečenskij, A.: States on pseudo-MV algebras. Stud. Log. 68, 301–327 (2001)

    Article  Google Scholar 

  8. Dvurečenskij, A.: Pseudo-MV algebras are intervals in ℓ-groups. J. Aust. Math. Soc. 70, 427–445 (2002)

    Article  Google Scholar 

  9. Dvurečenskij, A.: Every linear pseudo-BL algebra admits a state. Soft Comput. 6, 495–501 (2007)

    Article  Google Scholar 

  10. Dvurečenskij, A.: Aglianò-Montagna type decomposition of linear pseudo-hoops and its applications. J. Pure Appl. Algebra 211, 851–861 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dvurečenskij, A.: On n-perfect GMV-algebras. J. Algebra 319, 4921–4946 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dvurečenskij, A., Pulmannova, S.: New Trends in Quantum Structures. Kluwer Academic, Dordrecht (2000)

    Book  MATH  Google Scholar 

  13. Dvurečenskij, A., Rachůnek, J.: Probabilistic averaging in bounded non-commutative Rℓ-monoids. Semigroup Forum 72, 190–206 (2006)

    MathSciNet  Google Scholar 

  14. Dvurečenskij, A., Rachůnek, J.: On Riečan and Bosbach states for bounded non-commutative Rℓ-monoids. Math. Slovaca 56, 487–500 (2006)

    MathSciNet  MATH  Google Scholar 

  15. Dvurečenskij, A., Giuntini, R., Kowalski, T.: On the structure of pseudo-BL algebras and pseudo-hoops in quantum logics. Found. Phys. 49, 1519–1542 (2010)

    Article  Google Scholar 

  16. Dymek, G., Walendziak, A.: Semisimple, Archimedean and semilocal pseudo-MV algebras. Sci. Math. Jpn. 66, 217–226 (2007)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Flondor, P., Georgescu, G., Iorgulescu, A.: Pseudo-t-norms and pseudo-BL algebras. Soft Comput. 5, 355–371 (2001)

    Article  MATH  Google Scholar 

  19. Galatos, N., Ono, H.: Glivenko theorems for substructural logics over FL. J. Symb. Log. 71, 1353–1384 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Galatos, N., Tsinakis, C.: Generalized MV-algebras. J. Algebra 283, 254–291 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Elsevier, Amsterdam (2007)

    Google Scholar 

  22. Georgescu, G., Iorgulescu, A.: Pseudo-MV algebras: a non-commutative extension of MV-algebras. In: Proc. 4th International Symposium of Economic Informatics, pp. 961–968. INFOREC Printing House, Bucharest (1999)

    Google Scholar 

  23. Georgescu, G., Iorgulescu, A.: Pseudo-BL algebras: a non-commutative extension of BL-algebras. In: Abstracts of the 5th International Conference FSTA 2000, Slovakia, pp. 90–92 (2000)

    Google Scholar 

  24. Georgescu, G., Iorgulescu, A.: Pseudo-MV algebras. Mult. Valued Log. 6, 95–135 (2001)

    MathSciNet  MATH  Google Scholar 

  25. Georgescu, G., Leuştean, L.: Some classes of pseudo-BL algebras. J. Aust. Math. Soc. 73, 127–153 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. Georgescu, G., Leuştean, L., Preoteasa, V.: Pseudo-hoops. J. Mult.-Valued Log. Soft Comput. 11, 153–184 (2005)

    MATH  Google Scholar 

  27. Glivenko, V.: Sur quelques points de la logique de M. Brouwer. Bull. Cl. Sci., Acad. R. Belg. 15, 183–188 (1929)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  29. Hájek, P.: Basic fuzzy logic and BL-algebras. Soft Comput. 2, 124–128 (1998)

    Article  Google Scholar 

  30. Ioniţă, C.: Pseudo-MV algebras as semigroups. In: Proc. 4th International Symposium on Economic Informatics, pp. 983–987. INFOREC Printing House, Bucharest (1999)

    Google Scholar 

  31. Iorgulescu, A.: Classes of BCK-algebras—part III. Preprint 3, Institute of Mathematics of the Romanian Academy (2004)

    Google Scholar 

  32. Iorgulescu, A.: On pseudo-BCK algebras and porims. Sci. Math. Jpn. 16, 293–305 (2004)

    MathSciNet  Google Scholar 

  33. Iorgulescu, A.: Classes of pseudo-BCK algebras—part I. J. Mult.-Valued Log. Soft Comput. 12, 71–130 (2006)

    MathSciNet  MATH  Google Scholar 

  34. Iorgulescu, A.: Classes of pseudo-BCK algebras—part II. J. Mult.-Valued Log. Soft Comput. 12, 575–629 (2006)

    MathSciNet  MATH  Google Scholar 

  35. Iorgulescu, A.: Algebras of Logic as BCK-Algebras. ASE Ed., Bucharest (2008)

    MATH  Google Scholar 

  36. Iorgulescu, A.: Classes of examples of pseudo-MV algebras, pseudo-BL algebras and divisible bounded non-commutative residuated lattices. Soft Comput. 14, 313–327 (2010)

    Article  MATH  Google Scholar 

  37. Jenei, S., Montagna, F.: A proof of standard completeness for Esteva and Godo’s logic MTL. Stud. Log. 70, 183–192 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  38. Jipsen, P., Montagna, F.: On the structure of generalized BL-algebras. Algebra Univers. 55, 227–238 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  39. Kühr, J.: Pseudo-BL algebras and DRℓ-monoids. Math. Bohem. 128, 199–208 (2003)

    MathSciNet  MATH  Google Scholar 

  40. Kühr, J.: On a generalization of pseudo-MV algebras. J. Mult.-Valued Log. Soft Comput. 12, 373–389 (2006)

    MATH  Google Scholar 

  41. Leuştean, I.: Local pseudo-MV algebras. Soft Comput. 5, 386–395 (2001)

    Article  MATH  Google Scholar 

  42. Noguera, C., Esteva, F., Gispert, J.: On some varieties of MTL-algebras. Log. J. IGPL 13, 443–466 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  43. Noguera, C., Esteva, F., Gispert, J.: Perfect and bipartite IMTL-algebras and disconnected rotations of prelinear semihoops. Arch. Math. Log. 44, 869–886 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  44. Piciu, D.: Algebras of Fuzzy Logic. Universitaria Ed., Craiova (2007)

    Google Scholar 

  45. Rachůnek, J.: A duality between algebras of basic logic and bounded representable DRℓ-monoids. Math. Bohem. 126, 561–569 (2001)

    MathSciNet  MATH  Google Scholar 

  46. Rachůnek, J.: A non-commutative generalization of MV-algebras. Czechoslov. Math. J. 52, 255–273 (2002)

    Article  MATH  Google Scholar 

  47. Rachůnek, J., Slezák, V.: Bounded dually residuated lattice ordered monoids as a generalization of fuzzy structures. Math. Slovaca 56, 223–233 (2006)

    MathSciNet  MATH  Google Scholar 

  48. Torrens, A.: An approach to Glivenko’s theorem in algebraizable logics. Stud. Log. 88, 349–383 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Iowa, Iowa City, IA, USA

    Lavinia Corina Ciungu

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  1. Lavinia Corina Ciungu
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Ciungu, L.C. (2014). Other Non-commutative Multiple-Valued Logic Algebras. In: Non-commutative Multiple-Valued Logic Algebras. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-01589-7_4

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