A New Look on the Ordinal Sum of Fuzzy Implication Functions

  • Sebastia MassanetEmail author
  • Juan Vicente Riera
  • Joan Torrens
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 610)


Fuzzy implication functions are logical connectives commonly 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 new ordinal sum construction methods of implication functions based on fuzzy negations N are presented. Some general properties are analysed and particular cases when the considered fuzzy negation is the classical one or any strong negation are highlighted.


Ordinal sum Fuzzy implication function Fuzzy negation t-norm t-conorm \((S{, } N)\)-implication 



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


  1. 1.
    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
  2. 2.
    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)zbMATHGoogle Scholar
  3. 3.
    Baczyński, M., Jayaram, B.: Fuzzy Implications. Studies in Fuzziness and Soft Computing, vol. 231. Springer, Heidelberg (2008)zbMATHGoogle Scholar
  4. 4.
    Baczyński, M., Jayaram, B., Massanet, S., Torrens, J.: Fuzzy implications: past, present, and future. In: Kacprzyk, J., Pedrycz, W. (eds.) Springer Handbook of Computational Intelligence, pp. 183–202. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  5. 5.
    Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A Guide for Practitioners. Studies in Fuzziness and Soft Computing, vol. 221. Springer, Heidelberg (2007)zbMATHGoogle Scholar
  6. 6.
    Clifford, A.H.: Naturally totally ordered commutative semigroups. Am. J. Math. 76(3), 631–646 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    De Baets, B., Mesiar, R.: Residual implicators of continuous t-norms. In: Proceedings of EUFIT 1996, pp. 37–41 (1996)Google Scholar
  8. 8.
    Fodor, J.C., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht (1994)CrossRefzbMATHGoogle Scholar
  9. 9.
    Grabisch, M., Marichal, J.L., Mesiar, R., Pap, E.: Aggregation Functions. Encyclopedia of Mathematics and Its Applications, vol. 127. Cambridge University Press, New York (2009)CrossRefzbMATHGoogle Scholar
  10. 10.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  11. 11.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular norms as ordinal sums of semigroups in the sense of A.H. Clifford. Semigroup Forum 65, 71–82 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hlinená, D., Kalina, M., Kral, P.: Implications functions generated using functions of one variable. In: [2], pp. 125–153 (2013)Google Scholar
  13. 13.
    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
  14. 14.
    Massanet, S., Torrens, J.: On a new class of fuzzy implications: h-Implications and generalizations. Inf. Sci. 181, 2111–2127 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    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
  16. 16.
    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
  17. 17.
    Massanet, S., Torrens, J.: An overview of construction methods of fuzzy implications. In: [2], pp. 1–30 (2013)Google Scholar
  18. 18.
    Mesiar, R., Mesiarová, A.: Residual implications and left-continuous t-norms which are ordinal sums of semigroups. Fuzzy Sets Syst. 143, 47–57 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mesiarová, A.: Continuous triangular subnorms. Fuzzy Sets Syst. 142, 75–83 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Shi, Y., Van Hasse, B., Ruan, D., Kerre, E.: Fuzzy implications: classification and a new class. In: [2], pp. 31–53 (2013)Google Scholar
  21. 21.
    Su, Y., Xie, A., Liu, H.: On ordinal sum implications. Inf. Sci. 293, 251–262 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Trillas, E., Mas, M., Monserrat, M., Torrens, J.: On the representation of fuzzy rules. Int. J. Approximate Reason. 48, 583–597 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Vemuri, N.R., Jayaram, B.: Representations through a monoid on the set of fuzzy implications. Fuzzy Sets Syst. 247, 51–67 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Vemuri, N.R., Jayaram, B.: The composition of fuzzy implications: closures with respect to properties, powers and families. Fuzzy Sets Syst. 275, 58–87 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Sebastia Massanet
    • 1
    Email author
  • Juan Vicente Riera
    • 1
  • Joan Torrens
    • 1
  1. 1.Dept. Mathematics and Computer ScienceUniversity of the Balearic IslandsPalmaSpain

Personalised recommendations