Skip to main content
SpringerLink
Log in

A New Look on the Ordinal Sum of Fuzzy Implication Functions

  • Conference paper
  • First Online:
Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2016)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 610))

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  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)

    Article  MathSciNet  Google Scholar 

  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)

    MATH  Google Scholar 

  3. Baczyński, M., Jayaram, B.: Fuzzy Implications. Studies in Fuzziness and Soft Computing, vol. 231. Springer, Heidelberg (2008)

    MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  5. Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A Guide for Practitioners. Studies in Fuzziness and Soft Computing, vol. 221. Springer, Heidelberg (2007)

    MATH  Google Scholar 

  6. Clifford, A.H.: Naturally totally ordered commutative semigroups. Am. J. Math. 76(3), 631–646 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  7. De Baets, B., Mesiar, R.: Residual implicators of continuous t-norms. In: Proceedings of EUFIT 1996, pp. 37–41 (1996)

    Google Scholar 

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

    Book  MATH  Google Scholar 

  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)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hlinená, D., Kalina, M., Kral, P.: Implications functions generated using functions of one variable. In: [2], pp. 125–153 (2013)

    Google Scholar 

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

    Article  Google Scholar 

  14. Massanet, S., Torrens, J.: On a new class of fuzzy implications: h-Implications and generalizations. Inf. Sci. 181, 2111–2127 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Massanet, S., Torrens, J.: On the vertical threshold generation method of fuzzy implication and its properties. Fuzzy Sets Syst. 226, 232–252 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Massanet, S., Torrens, J.: An overview of construction methods of fuzzy implications. In: [2], pp. 1–30 (2013)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mesiarová, A.: Continuous triangular subnorms. Fuzzy Sets Syst. 142, 75–83 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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. Su, Y., Xie, A., Liu, H.: On ordinal sum implications. Inf. Sci. 293, 251–262 (2015)

    Article  MathSciNet  Google Scholar 

  22. Trillas, E., Mas, M., Monserrat, M., Torrens, J.: On the representation of fuzzy rules. Int. J. Approximate Reason. 48, 583–597 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Vemuri, N.R., Jayaram, B.: Representations through a monoid on the set of fuzzy implications. Fuzzy Sets Syst. 247, 51–67 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgement

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

Author information

Authors and Affiliations

  1. Dept. Mathematics and Computer Science, University of the Balearic Islands, Crta. Valldemossa km 7,5, 07122, Palma, Spain

    Sebastia Massanet, Juan Vicente Riera & Joan Torrens

Authors
  1. Sebastia Massanet
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Juan Vicente Riera
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Joan Torrens
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sebastia Massanet .

Editor information

Editors and Affiliations

  1. INESC-ID,Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    Joao Paulo Carvalho

  2. LIP 6, Université Pierre et Marie Curie, Paris, France

    Marie-Jeanne Lesot

  3. School of Industrial Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

    Uzay Kaymak

  4. IDMEC,Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    Susana Vieira

  5. LIP6, Université Pierre et Marie Curie, CNRS, Paris, France

    Bernadette Bouchon-Meunier

  6. Machine Intelligence Institute, Iona College, New Rochelle, New York, USA

    Ronald R. Yager

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Massanet, S., Riera, J.V., Torrens, J. (2016). A New Look on the Ordinal Sum of Fuzzy Implication Functions. In: Carvalho, J., Lesot, MJ., Kaymak, U., Vieira, S., Bouchon-Meunier, B., Yager, R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2016. Communications in Computer and Information Science, vol 610. Springer, Cham. https://doi.org/10.1007/978-3-319-40596-4_34

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-40596-4_34

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-40595-7

  • Online ISBN: 978-3-319-40596-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation