Advertisement

About the Use of Admissible Order for Defining Implication Operators

  • Maria Jose Asiain
  • Humberto BustinceEmail author
  • Benjamin Bedregal
  • Zdenko Takáč
  • Michal Baczyński
  • Daniel Paternain
  • Graçaliz Dimuro
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 610)

Abstract

Implication functions are crucial operators for many fuzzy logic applications. In this work, we consider the definition of implication functions in the interval-valued setting using admissible orders and we use this interval-valued implications for building comparison measures.

Keywords

Interval-valued implication operator Admissible order Similarity measure 

Notes

Acknowledgements

H. Bustince was supported by Project TIN2013-40765-P of the Spanish Government. Z. Takáč was supported by Project VEGA 1/0420/15. B. Bedregal and G. Dimuro were supported by Brazilian funding agency CNPQ under Processes 481283/2013-7, 306970/2013-9, 232827/2014-1 and 307681/2012-2.

References

  1. 1.
    Baczyński, M., Beliakov, G., Bustince, H., Pradera, A.: Advances in Fuzzy Implication Functions, Advances in Fuzzy Implication Functions, Studies in Fuzziness and Soft Computing, 300. Springer, Berlin (2013)zbMATHGoogle Scholar
  2. 2.
    Baczyński, M., Jayaram, B.: Fuzzy Implications, Studies in Fuzziness and Soft Computing, vol. 231. Springer, Berlin (2008)zbMATHGoogle Scholar
  3. 3.
    Bedregal, B., Dimuro, G., Santiago, R., Reiser, R.: On interval fuzzy S-implications. Inf. Sci. 180(8), 1373–1389 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Burillo, P., Bustince, H.: Construction theorems for intuitionistic fuzzy sets. Fuzzy Sets Syst. 84, 271–281 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bustince, H., Barrenechea, E., Mohedano, V.: Intuitionistic fuzzy implication operators. an expression and main properties. Int. J. Uncertainty Fuzziness Knowl.-Based Syst. 12(3), 387–406 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    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(4), 525–536 (2008)CrossRefGoogle Scholar
  7. 7.
    Bustince, H., Barrenechea, E., Pagola, M.: Image thresholding using restricted equivalence functions and maximizing the measure of similarity. Fuzzy Sets Syst. 128(5), 496–516 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bustince, H., Barrenechea, E., Pagola, M.: Restricted equivalence functions. Fuzzy Sets Syst. 157(17), 2333–2346 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bustince, H., Barrenechea, E., Pagola, M., Fernandez, J., Xu, Z., Bedregal, B., Montero, J., Hagras, H., Herrera, F., De Baets, B.: A historical account of types of fuzzy sets and their relationship. IEEE Trans. Fuzzy Syst. 24(1), 179–194 (2016)CrossRefGoogle Scholar
  10. 10.
    Bustince, H., Burillo, P., Soria, F.: Automorphisms, negations and implication operators. Fuzzy Sets Syst. 134, 209–229 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bustince, H., Fernández, J., Kolesárová, A., Mesiar, R.: Generation of linear orders for intervals by means of aggregation functions. Fuzzy Sets Syst. 220, 69–77 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bustince, H., Galar, M., Bedregal, B., Kolesárová, A., Mesiar, R.: A new approach to interval-valued Choquet integrals and the problem of ordering in interval-valued fuzzy set applications. IEEE Trans. Fuzzy Syst. 21(6), 1150–1162 (2013)CrossRefGoogle Scholar
  13. 13.
    Cornelis, C., Deschrijver, G., Kerre, E.E.: Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application. Int. J. Approximate Reason. 35(1), 55–95 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Massanet, S., Mayor, G., Mesiar, R., Torrens, J.: On fuzzy implication: an axiomatic approach. Int. J. Approximate Reason. 54, 1471–1482 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pradera, A., Beliakov, G., Bustince, H., De Baets, B.: A review of the relationship between implication, negation and aggregation functions from the point of view of material implication. Inf. Sci. 329, 357–380 (2016)CrossRefGoogle Scholar
  16. 16.
    Riera, J.V., Torrens, J.: Residual implications on the set of discrete fuzzy numbers. Inf. Sci. 247, 131–143 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Xu, Z., Yager, R.R.: Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. Gen. Syst. 35, 417–433 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Maria Jose Asiain
    • 1
  • Humberto Bustince
    • 2
    • 3
    Email author
  • Benjamin Bedregal
    • 4
  • Zdenko Takáč
    • 5
  • Michal Baczyński
    • 6
  • Daniel Paternain
    • 2
  • Graçaliz Dimuro
    • 7
  1. 1.Dept. de MatemáticasUniversidad Pública de NavarraPamplonaSpain
  2. 2.Dept. de Automática y ComputaciónUniversidad Pública de NavarraPamplonaSpain
  3. 3.Institute of Smart CitiesUniversidad Pública de NavarraPamplonaSpain
  4. 4.Departamento de Informática e Matemática AplicadaUniversidade Federal do Rio Grande do NorteNatalBrazil
  5. 5.Institute of Information Engineering, Automation and MathematicsSlovak University of Technology in BratislavaBratislavaSlovakia
  6. 6.Institute of MathematicsUniversity of SilesiaKatowicePoland
  7. 7.Centro de Ciências ComputacionaisUniversidade Federal do Rio GrandeRio GrandeBrazil

Personalised recommendations