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Characterization of Lattice-Valued Restricted Equivalence Functions

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Fuzzy Information Processing (NAFIPS 2018)

Abstract

In this paper we investigate about lattice-valued restricted equivalence functions and its characterization by means a particular class of lattice-valued implication operators.

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References

  1. Baczyński, M., Jayaram, B.: Fuzzy Implications. Studies in Fuzziness and Soft Computing. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-69082-5

    Book  MATH  Google Scholar 

  2. Bedregal, B.C., Santos, H.S., Callejas-Bedregal, R.: T-norms on bounded lattices: T-norm morphisms and operators. In: IEEE International Conference on Fuzzy Systems, pp. 22–28 (2006)

    Google Scholar 

  3. Bedregal, B.C.: On interval fuzzy negations. Fuzzy Sets Syst. 161, 2290–2313 (2010)

    Article  MathSciNet  Google Scholar 

  4. Bedregal, B.C.: Xor-Implications and E-Implications: classes of fuzzy implications based on fuzzy Xor. Electron. Notes Theor. Comput. Sci. 274, 5–18 (2009)

    Article  MathSciNet  Google Scholar 

  5. Bedregal, B., Beliakov, G., Bustince, H., Fernandez, J., Pradera, A., Reiser, R.: (S, N)-implications on bounded lattices. In: Baczyński, M., Beliakov, G., Bustince Sola, H., Pradera, A. (eds.) Advances in Fuzzy Implication Functions. Studies in Fuzziness and Soft Computing, vol. 300. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-35677-3_5

    Chapter  Google Scholar 

  6. Birkhoff, G.: Lattice Theory. American Mathematical Society, Providence (1973)

    MATH  Google Scholar 

  7. Bustince, H., Burillo, P., Soria, F.: Automorphisms, negations and implication operators. Fuzzy Sets Syst. 134(2), 209–229 (2003)

    Article  MathSciNet  Google Scholar 

  8. Bustince, H., Barrenechea, E., Pagola, M.: Restricted equivalence functions. Fuzzy Sets Syst. 157(17), 2333–2346 (2006)

    Article  MathSciNet  Google Scholar 

  9. Bustince, H., Barrenechea, E., Pagola, M.: Relationship between restricted dissimilarity functions, restricted equivalence functions and normal \({E}_N\)-functions: image thresholding invariant. Pattern Recogn. Lett. 29, 525–536 (2008)

    Article  Google Scholar 

  10. Calvo, T.: On mixed De Morgan triplets. Fuzzy Sets Syst. 50, 47–50 (1992)

    Article  MathSciNet  Google Scholar 

  11. Chen, G., Pham, T.T.: Fuzzy Sets, Fuzzy Logic and Fuzzy Control Systems. CRC Press, Boca Raton (2001)

    Google Scholar 

  12. De Cooman, G., Kerre, E.E.: Order norms on bounded partially ordered sets. J. Fuzzy Math. 2, 281–310 (1994)

    MATH  MathSciNet  Google Scholar 

  13. Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002)

    Book  Google Scholar 

  14. Fodor, J., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publisher, Dordrecht (1994)

    Book  Google Scholar 

  15. Hajek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht (1998)

    Book  Google Scholar 

  16. Hungerford, T.W.: Algebra. Graduate Texts in Mathematics. Springer, New York (2000)

    MATH  Google Scholar 

  17. Julio, A., Pagola, M., Paternain, D., Lopez-Molina, C., Melo-Pinto, P.: Interval-valued restricted equivalence functions applied on clustering techniques. In: IFSA-EUSFLAT, pp. 831–836 (2009)

    Google Scholar 

  18. Klement, E.P., Mesiar, R.: Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms. Elsevier B.V., Amsterdam (2005)

    MATH  Google Scholar 

  19. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)

    Book  Google Scholar 

  20. Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic, Theory and Applications. Prentice Hall PTR, Upper Saddle River (1995)

    MATH  Google Scholar 

  21. Mas, M., Monserrat, M., Torrens, J., Trillas, E.: A survey on fuzzy implications functions. IEEE Trans. Fuzzy Syst. 15(6), 1107–1121 (2007)

    Article  Google Scholar 

  22. Palmeira, E.S., Bedregal, B.C., Mesiar, R., Fernandez, J.: A new way to extend T-norms, T-conorms and negations. Fuzzy Sets Syst. 240, 1–21 (2014)

    Article  MathSciNet  Google Scholar 

  23. Palmeira, E.S., Bedregal, B.C.: Extension of fuzzy logic operators defined on bounded lattices via retractions. Comput. Math. Appl. 63, 1026–1038 (2012)

    Article  MathSciNet  Google Scholar 

  24. Palmeira, E.S., Bedregal, B.: Restricted equivalence function on L([0, 1]). In: Melin, P., Castillo, O., Kacprzyk, J., Reformat, M., Melek, W. (eds.) NAFIPS 2017. AISC, vol. 648, pp. 410–420. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-67137-6_45

    Chapter  Google Scholar 

  25. Palmeira, E.S., Bedregal, B.C., dos Santos, J.A.: Some results on extension of lattice-valued QL-implications. J. Braz. Comput. Soc. 22(1), 41–49 (2016)

    Article  MathSciNet  Google Scholar 

  26. Palmeira, E.S., Bedregal, B.C., Bustince, H., Paternain, D., De Miguel, L.: Application of two different methods for extending lattice-valued restricted equivalence functions used for constructing similarity measures on L-fuzzy sets. Inf. Sci. 441, 95–112 (2018)

    Article  MathSciNet  Google Scholar 

  27. Saminger-Platz, S., Klement, E.P., Mesiar, R.: On extensions of triangular norms on bounded lattices. Indag. Math. 19(1), 135–150 (2008)

    Article  MathSciNet  Google Scholar 

  28. Takano, M.: Strong completeness of lattice-valued logic. Arch. Math. Logic 41, 497–505 (2002)

    Article  MathSciNet  Google Scholar 

  29. Yager, R.R.: On the implication operator in fuzzy logic. Inf. Sci. 31(2), 141–164 (1983)

    Article  MathSciNet  Google Scholar 

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Correspondence to Eduardo Palmeira .

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Palmeira, E., Bedregal, B., Vargas, R.R. (2018). Characterization of Lattice-Valued Restricted Equivalence Functions. In: Barreto, G., Coelho, R. (eds) Fuzzy Information Processing. NAFIPS 2018. Communications in Computer and Information Science, vol 831. Springer, Cham. https://doi.org/10.1007/978-3-319-95312-0_15

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  • DOI: https://doi.org/10.1007/978-3-319-95312-0_15

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  • Online ISBN: 978-3-319-95312-0

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