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Implications Satisfying the Law of Importation with a Given T-norm

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Book cover Aggregation Functions in Theory and in Practise

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 228))

Abstract

The main goal of this paper is to characterize all fuzzy implications with continuous natural negation that satisfy the law of importation with a given continuous t-norm T. Particular cases when the fixed t-norm T is the minimum, the product and the Łukasiewicz t-norm are deduced from the general result and the corresponding characterizations are presented separately.

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References

  1. Aguiló, I., Suñer, J., Torrens, J.: A characterization of residual implications derived from left-continuous uninorms. Information Sciences 180(20), 3992–4005 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  2. Baczyński, M.: On the distributivity of fuzzy implications over continuous and archimedean triangular conorms. Fuzzy Sets and Systems 161(10), 1406–1419 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  3. Baczyński, M.: On the distributivity of fuzzy implications over representable uninorms. Fuzzy Sets and Systems 161(17), 2256–2275 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Baczyński, M., Jayaram, B.: Yager’s classes of fuzzy implications: some properties and intersections. Kybernetika 43, 157–182 (2007)

    MATH  MathSciNet  Google Scholar 

  5. Baczyński, M., Jayaram, B.: Fuzzy Implications. Springer, Heidelberg (2008)

    MATH  Google Scholar 

  6. Baczyński, M., Jayaram, B.: (S,N)- and R-implications: A state-of-the-art survey. Fuzzy Sets and Systems 159, 1836–1859 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Baczyński, M., Jayaram, B.: On the distributivity of fuzzy implications over nilpotent or strict triangular conorms. Transactions on Fuzzy Systems 17(3), 590–603 (2009)

    Article  Google Scholar 

  8. Baczyński, M., Jayaram, B.: (U,N)-implications and their characterizations. Fuzzy Sets and Systems 160, 2049–2062 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Baczyński, M., Jayaram, B.: Intersections between some families of (U,N)- and RU-implications. Fuzzy Sets and Systems 167(1), 30–44 (2011); Special Issue: Aggregation techniques and applications Selected papers from AGOP 2009 dedicated to Jaume Casasnovas

    Google Scholar 

  10. Balasubramaniam, J., Rao, C.: On the distributivity of implication operators over T and S norms. IEEE Transactions on Fuzzy Systems 12, 194–198 (2004)

    Article  Google Scholar 

  11. Bustince, H., Mohedano, V., Barrenechea, E., Pagola, M.: Definition and construction of fuzzy DI-subsethood measures. Information Sciences 176, 3190–3231 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bustince, H., Pagola, M., Barrenechea, E.: Construction of fuzzy indices from fuzzy DI-subsethood measures: application to the global comparison of images. Information Sciences 177, 906–929 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Combs, W., Andrews, J.: Combinatorial rule explosion eliminated by a fuzzy rule configuration. IEEE Transactions on Fuzzy Systems 6(1), 1–11 (1998)

    Article  Google Scholar 

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

    Book  MATH  Google Scholar 

  15. Gottwald, S.: Treatise on Many-Valued Logic. Research Studies Press, Baldock (2001)

    Google Scholar 

  16. Jayaram, B.: On the law of importation (x ∧ y) → z ≡ (x → (y → z)) in fuzzy logic. IEEE Transactions on Fuzzy Systems 16, 130–144 (2008)

    Article  Google Scholar 

  17. Jayaram, B.: Rule reduction for efficient inferencing in similarity based reasoning. International Journal of Approximate Reasoning 48(1), 156–173 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kerre, E., Huang, C., Ruan, D.: Fuzzy Set Theory and Approximate Reasoning. Wu Han University Press, Wu Chang (2004)

    Google Scholar 

  19. Kerre, E., Nachtegael, M.: Fuzzy techniques in image processing. STUDFUZZ, vol. 52. Springer, New York (2000)

    Book  MATH  Google Scholar 

  20. Klement, E., Mesiar, R., Pap, E.: Triangular norms. Kluwer Academic Publishers, Dordrecht (2000)

    Book  MATH  Google Scholar 

  21. Mas, M., Monserrat, M., Torrens, J.: Modus ponens and modus tollens in discrete implications. Int. J. Approx. Reasoning 49(2), 422–435 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Mas, M., Monserrat, M., Torrens, J.: The law of importation for discrete implications. Information Sciences 179, 4208–4218 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Mas, M., Monserrat, M., Torrens, J.: A characterization of (U,N), RU, QL and D-implications derived from uninorms satisfying the law of importation. Fuzzy Sets and Systems 161, 1369–1387 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  24. Mas, M., Monserrat, M., Torrens, J., Trillas, E.: A survey on fuzzy implication functions. IEEE Transactions on Fuzzy Systems 15(6), 1107–1121 (2007)

    Article  Google Scholar 

  25. Massanet, S., Torrens, J.: The law of importation versus the exchange principle on fuzzy implications. Fuzzy Sets and Systems 168(1), 47–69 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  26. Massanet, S., Torrens, J.: Intersection of Yager’s implications with QL and D-implications. International Journal of Approximate Reasoning 53, 467–479 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  27. Massanet, S., Torrens, J.: On a generalization of yager’s implications. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds.) IPMU 2012, Part II. CCIS, vol. 298, pp. 315–324. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  28. Massanet, S., Torrens, J.: On some properties of threshold generated implications. Fuzzy Sets and Systems 205, 30–49 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  29. Massanet, S., Torrens, J.: On the characterization of Yager’s implications. Information Sciences 201, 1–18 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  30. Massanet, S., Torrens, J.: Threshold generation method of construction of a new implication from two given ones. Fuzzy Sets and Systems 205, 50–75 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  31. Massanet, S., Torrens, J.: On the vertical threshold generation method of fuzzy implication and its properties. Fuzzy Sets and Systems (2013), doi:10.1016/j.fss.2013.03.003

    Google Scholar 

  32. Radzikowska, A.M., Kerre, E.E.: A comparative study of fuzzy rough sets. Fuzzy Sets and Systems 126(2), 137–155 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  33. Ruan, D., Kerre, E.E. (eds.): Fuzzy IF-THEN Rules in Computational Intelligence: Theory and Applications. Kluwer Academic Publishers, Norwell (2000)

    Google Scholar 

  34. Ruiz-Aguilera, D., Torrens, J.: Distributivity and conditional distributivity of a uninorm and a continuous t-conorm. IEEE Transactions on Fuzzy Systems 14(2), 180–190 (2006)

    Article  Google Scholar 

  35. Ruiz-Aguilera, D., Torrens, J.: Distributivity of residual implications over conjunctive and disjunctive uninorms. Fuzzy Sets and Systems 158, 23–37 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  36. Shi, Y., Van Gasse, B., Ruan, D., Kerre, E.: On a new class of implications in fuzzy logic. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds.) IPMU 2010. CCIS, vol. 80, pp. 525–534. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  37. Trillas, E., Alsina, C.: On the law [(p ∧ q) → r] = [(p → r) ∨ (q → r)] in fuzzy logic. IEEE Transactions on Fuzzy Systems 10(1), 84–88 (2002)

    Article  Google Scholar 

  38. Trillas, E., Alsina, C., Pradera, A.: On MPT-implication functions for fuzzy logic. Rev. R. Acad. Cien. Serie A. Mat. 98, 259–271 (2004)

    MATH  MathSciNet  Google Scholar 

  39. Trillas, E., Alsina, C., Renedo, E., Pradera, A.: On contra-symmetry and MPT condicionality in fuzzy logic. International Journal of Intelligent Systems 20, 313–326 (2005)

    Article  MATH  Google Scholar 

  40. Trillas, E., del Campo, C., Cubillo, S.: When QM-operators are implication functions and conditional fuzzy relations. International Journal of Intelligent Systems 15, 647–655 (2000)

    Article  MATH  Google Scholar 

  41. Yan, P., Chen, G.: Discovering a cover set of ARsi with hierarchy from quantitative databases. Information Sciences 173, 319–336 (2005)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. University of the Balearic Islands, Palma de Mallorca, 07122, Spain

    S. Massanet & J. Torrens

Authors
  1. S. Massanet
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  2. J. Torrens
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Corresponding author

Correspondence to S. Massanet .

Editor information

Editors and Affiliations

  1. Department of Automatica y Computacion, Universidad Publica de Navarra, Pamplona, Spain

    Humberto Bustince

  2. Department of Automatica y Computacion, Universidad Publica de Navarra, Pamplona, Spain

    Javier Fernandez

  3. Department of Mathematics and, Slovak University of Technology, Bratislava, Slovakia

    Radko Mesiar

  4. Department of Computer Science, Universidad de Alcalá de Henares, Madrid, Spain

    Tomasa Calvo

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Massanet, S., Torrens, J. (2013). Implications Satisfying the Law of Importation with a Given T-norm. In: Bustince, H., Fernandez, J., Mesiar, R., Calvo, T. (eds) Aggregation Functions in Theory and in Practise. Advances in Intelligent Systems and Computing, vol 228. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39165-1_40

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  • DOI: https://doi.org/10.1007/978-3-642-39165-1_40

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-39164-4

  • Online ISBN: 978-3-642-39165-1

  • eBook Packages: EngineeringEngineering (R0)

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