Skip to main content
Springer Nature Link
Log in

Implications Satisfying the Law of Importation with a Given Uninorm

  • Conference paper
Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2014)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 442))

Abstract

In this paper a characterization of all fuzzy implications with continuous e-natural negation that satisfy the law of importation with a given uninorm U is provided. The cases when the considered uninorm U is representable or a uninorm in \({\cal U}_{\min}\) are studied separately and detailed descriptions of those implications with continuous natural negation with respect to e that satisfy the law of importation with a uninorm in these classes are done. In the process some important examples are included.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. BaczyÅ„ski, 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 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. BaczyÅ„ski, M., Jayaram, B.: Fuzzy Implications. STUDFUZZ, vol. 231. Springer, Heidelberg (2008)

    MATH  Google Scholar 

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

    Article  Google Scholar 

  5. BaczyÅ„ski, 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 

  6. 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 

  7. 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 

  8. 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 

  9. Cignoli, R., Esteva, F., Godo, L., Montagna, F.: On a class of left-continuous t-norms. Fuzzy Sets Syst. 131(3), 283–296 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. 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 

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

    Book  MATH  Google Scholar 

  12. Fodor, J.C., Yager, R.R., Rybalov, A.: Structure of uninorms. Int. J. of Uncertainty, Fuzziness and Knowledge-based Systems 5, 411–427 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  13. González-Hidalgo, M., Massanet, S.: A fuzzy mathematical morphology based on discrete t-norms: fundamentals and applications to image processing. Soft Computing, 1–15 (December 2013)

    Google Scholar 

  14. González-Hidalgo, M., Massanet, S., Mir, A., Ruiz-Aguilera, D.: On the choice of the pair conjunction-implication into the fuzzy morphological edge detector. IEEE Transactions on Fuzzy Systems (2013) (submitted)

    Google Scholar 

  15. Gottwald, S.: A 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. Jenei, S.: New family of triangular norms via contrapositive symmetrization of residuated implications. Fuzzy Sets Syst. 110(2), 157–174 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  19. Jenei, S.: How to construct left-continuous triangular norms–state of the art. Fuzzy Sets and Systems 143(1), 27–45 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  23. 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 

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

    Article  MATH  MathSciNet  Google Scholar 

  25. 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 

  26. 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 

  27. 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 

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

    Article  MATH  MathSciNet  Google Scholar 

  29. Massanet, S., Torrens, J.: 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. AISC, vol. 228, pp. 423–432. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

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

    Google Scholar 

  31. 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 

  32. 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 

  33. 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 

  34. 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 

  35. 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 

  36. 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 

  37. 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 

Download references

Author information

Authors and Affiliations

  1. University of the Balearic Islands, Crta. Valldemossa km 7.5, E-07122, Palma, Spain

    Sebastia Massanet & Joan Torrens

Authors
  1. Sebastia Massanet
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joan Torrens
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University Montpellier 2, LIRMM - CNRS UMR 5506, 161, Rue Ada, 34392, Montpellier Cedex 5, France

    Anne Laurent

  2. LIRMM, UMR CNRS/Universite Montpellier II, 161 rue Ada, 34392, Montpellier cedex 5, France

    Olivier Strauss

  3. LIP6, UPMC Univ. Paris 06, CNRS UMR 7606, F-75005, Paris, France

    Bernadette Bouchon-Meunier

  4. Dept. of Information Systems, Iona College, 710 North Ave, 10801, New Rochelle, NY, USA

    Ronald R. Yager

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Massanet, S., Torrens, J. (2014). Implications Satisfying the Law of Importation with a Given Uninorm. In: Laurent, A., Strauss, O., Bouchon-Meunier, B., Yager, R.R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2014. Communications in Computer and Information Science, vol 442. Springer, Cham. https://doi.org/10.1007/978-3-319-08795-5_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-08795-5_16

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-08794-8

  • Online ISBN: 978-3-319-08795-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation