A New Look on Fuzzy Implication Functions: FNI-implications

  • Isabel Aguiló
  • Jaume Suñer
  • Joan TorrensEmail author
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 610)


Fuzzy implication functions are used to model fuzzy conditional and consequently they are essential in fuzzy logic and approximate reasoning. From the theoretical point of view, the study of how to construct new implication functions from old ones is one of the most important topics in this field. In this paper a construction method of implication functions from a t-conorm S (or any disjunctive aggregation function F), a fuzzy negation N and an implication function I is studied. Some general properties are analyzed and many illustrative examples are given. In particular, this method shows how to obtain new implications from old ones with additional properties not satisfied by the initial implication function.


Fuzzy implication function t-conorm Disjunctive aggregation function Construction methods Natural negation 



This paper has been partially supported by the Spanish grant TIN2013-42795-P.


  1. 1.
    Aguiló, I., Carbonell, M., Suñer, J., Torrens, J.: Dual representable aggregation functions and their derived S-implications. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds.) IPMU 2010. LNCS, vol. 6178, pp. 408–417. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  2. 2.
    Aguiló, I., Suñer, J., Torrens, J.: A characterization of residual implications derived from left-continuous uninorms. Inf. Sci. 180(20), 3992–4005 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aguiló, I., Suñer, J., Torrens, J.: New types of contrapositivisation of fuzzy implications with respect to fuzzy negations. Inf. Sci. 322, 223–226 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Baczyński, M., Beliakov, G., Bustince Sola, H., Pradera, A. (eds.): Advances in Fuzzy Implication Functions. STUDFUZZ, vol. 300. Springer, Heidelberg (2013)zbMATHGoogle Scholar
  5. 5.
    Baczyński, M., Jayaram, B.: Yager’s classes of fuzzy implications: Some properties and intersections. Kybernetika 43, 157–182 (2007)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Baczyński, M., Jayaram, B.: Fuzzy Implications. STUDFUZZ, vol. 231. Springer, Heidelberg (2008)zbMATHGoogle Scholar
  7. 7.
    Baczyński, M., Jayaram, B.: (U, N)-implications and their characterizations. Fuzzy Sets Syst. 160, 2049–2062 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Balasubramaniam, J.: Yager’s new class of implications \(J_f\) and some classical tautologies. Inf. Sci. 177, 930–946 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A Guide for Practitioners. STUDFUZZ, vol. 221. Springer, Heidelberg (2007)zbMATHGoogle Scholar
  10. 10.
    Jayaram, B., Mesiar, R.: On special fuzzy implications. Fuzzy Sets Syst. 160(14), 2063–2085 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    De Baets, B., Fodor, J.C.: Residual operators of uninorms. Soft. Comput. 3, 89–100 (1999)CrossRefzbMATHGoogle Scholar
  12. 12.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  13. 13.
    Hlinená, D., Kalina, M., Kral, P.: Implications functions generated using functions of one variable. In: [4], pp. 125–153 (2013)Google Scholar
  14. 14.
    Mas, M., Monserrat, M., Torrens, J.: Two types of implications derived from uninorms. Fuzzy Sets Syst. 158, 2612–2626 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mas, M., Monserrat, M., Torrens, J., Trillas, E.: A survey on fuzzy implication functions. IEEE Trans. Fuzzy Syst. 15(6), 1107–1121 (2007)CrossRefGoogle Scholar
  16. 16.
    Massanet, S., Riera, J.V., Ruiz-Aguilera, D.: On Fuzzy Polynomial Implications. In: Laurent, A., Strauss, O., Bouchon-Meunier, B., Yager, R.R. (eds.) IPMU 2014, Part I. CCIS, vol. 442, pp. 138–147. Springer, Heidelberg (2014)Google Scholar
  17. 17.
    Massanet, S., Torrens, J.: On a new class of fuzzy implications: h-Implications and generalizations. Inf. Sci. 181, 2111–2127 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Massanet, S., Torrens, J.: Threshold generation method of construction of a new implication from two given ones. Fuzzy Sets Syst. 205, 50–75 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Massanet, S., Torrens, J.: On the vertical threshold generation method of fuzzy implication and its properties. Fuzzy Sets Syst. 226, 232–252 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Massanet, S., Torrens, J.: An overview of construction methods of fuzzy implications. In: [4], pp. 1–30 (2013)Google Scholar
  21. 21.
    Ruiz, D., Torrens, J.: Residual implications and co-implications from idempotent uninorms. Kybernetika 40, 21–38 (2004)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Ruiz-Aguilera, D., Torrens, J.: S- and R-implications from uninorms continuous in ]0,1[2 and their distributivity over uninorms. Fuzzy Sets Syst. 160, 832–852 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    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)Google Scholar
  24. 24.
    Shi, Y., Van Hasse, B., Ruan, D., Kerre, E.: Fuzzy implications: Classification and a new class. In: [4], pp. 31–53 (2013)Google Scholar
  25. 25.
    Trillas, E., Mas, M., Monserrat, M., Torrens, J.: On the representation of fuzzy rules. Int. J. Approximate Reasoning 48, 583–597 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Vemuri, N.R., Jayaram, B.: Representations through a monoid on the set of fuzzy implications. Fuzzy Sets Syst. 247, 51–67 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Vemuri, N.R., Jayaram, B.: The \(\star \)-composition of fuzzy implications: Closures with respect to properties, powers and families. Fuzzy Sets Syst. 275, 58–87 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Yager, R.R.: On some new classes of implication operators and their role in approximate reasoning. Inf. Sci. 167, 193–216 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of the Balearic IslandsPalmaSpain

Personalised recommendations