Skip to main content
Springer Nature Link
Log in

Properties of Interval-Valued Fuzzy Relations, Atanassov’s Operators and Decomposable Operations

  • Conference paper
Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Methods (IPMU 2010)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 80))

Abstract

In this paper we study properties of interval-valued fuzzy relations which were introduced by L.A. Zadeh in 1975. Fuzzy set theory turned out to be a useful tool to describe situations in which the data are imprecise or vague. Interval-valued fuzzy set theory is a generalization of fuzzy set theory which was introduced also by Zadeh in 1965. We examine some properties of interval-valued fuzzy relations in the context of Atanassov’s operators and decomposable operations in interval-valued fuzzy set theory.

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. Agustench, E., Bustince, H., Mohedano, V.: Method of Least Squares Applied to the GMP with International-Valued Fuzzy Sets. Mathware Soft Comp. 6(2-3), 267–276 (1999)

    MathSciNet  MATH  Google Scholar 

  2. Atanassov, K.T., Gargov, G.: Interval Valued Intuitionistic Fuzzy Sets. Fuzzy Sets Syst. 31, 343–349 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Atanassov, K.: Intuitionistic Fuzzy Sets: Theory and Applications. Springer, Heidelberg (1999)

    Book  MATH  Google Scholar 

  4. Birkhoff, G.: Lattice Theory. AMS Coll. Publ. 25, Providence (1967)

    MATH  Google Scholar 

  5. Burillo, P., Bustince, H.: Intuitionistic Fuzzy Relations. Effect of Atanassov’s Operators on the Properties of the Intuitionistic Fuzzy Relations. Math. Soft Comp. 2, 117–148 (1995)

    MathSciNet  MATH  Google Scholar 

  6. Bustince, H., Burillo, P.: Mathematical Analysis of Interval-Valued Fuzzy Relations: Application to Approximate Reasoning. Fuzzy Sets Syst. 113, 205–219 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bustince, H., Barrenechea, E., Pagola, M.: Generation of Interval-Valued Fuzzy and Atannasov’s Intuitionistic Fuzzy Connectives from Fuzzy Connectives and from K α Operators: Laws for Conjunction and Disjunction. Amplitude. Inter. J. Intel. Syst. 23, 680–714 (2008)

    Article  MATH  Google Scholar 

  8. Bustince, H., Barrenechea, E., Pagola, M., Fernandez, J.: Interval-Valued Fuzzy Sets Constructed from Matrices: Application to Edge Detection. Fuzzy Sets Syst. 160, 1819–1840 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Deschrijver, G., Kerre, E.E.: On the Relationship between some Extensions of Fuzzy Set Theory. Fuzzy Sets Syst. 133(2), 227–235 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Deschrijver, G., Cornelis, C., Kerre, E.E.: On the Representation of Intuitonistic Fuzzy t-Norms and t-Conorms. IEEE Trans. Fuzzy Syst. 12, 45–61 (2004)

    Article  Google Scholar 

  11. Deschrijver, G., Kerre, E.E.: On the Position of Intuitionistic Fuzzy Set Theory in the Framework of Theories Modelling Imprecision. Inform. Sci. 177, 1860–1866 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Drygaś, P., Pȩkala, B.: Properties of Decomposable Operations on some Extension of the Fuzzy Set Theory. In: Atanassov, K.T., Hryniewicz, O., Kacprzyk, J., et al. (eds.) Advances in Fuzzy Sets, Intuitionistic Fuzzy Sets, Generalized Nets and Related Topics, pp. 105–118. EXIT, Warsaw (2008)

    Google Scholar 

  13. Dubois, D., Gottwald, S., Hajek, P., Kacprzyk, J., Prade, H.: Terminological Difficulties in Fuzzy Set Theory - The Case of ”Intuitionistic Fuzzy Sets”. Fuzzy Sets Syst. 156, 485–491 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dubois, D., Prade, H., Sessa, S.: Recent Literature. Fuzzy Sets Syst. 159, 857–865 (2008)

    Article  Google Scholar 

  15. Goguen, A.: L-Fuzzy Sets. J. Math. Anal. Appl. 18, 145–174 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gorzałczany, M.B.: A Method of Inference in Approximate Reasoning Based on Interval-Valued Fuzzy Sets. Fuzzy Sets Syst. 21(1), 1–17 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  17. Guh, Y.-Y., Yang, M.-S., Po, R.-W., Lee, E.S.: Interval-Valued Fuzzy Relation-Based Clustering with its Application to Performance Evaluetion. Comp. Math. Appl. 57, 841–849 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kaufmann, A.: Introduction to the Theory of Fuzzy Subsets. Acad. Press, New York (1975)

    MATH  Google Scholar 

  19. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Acad. Publ., Dordrecht (2000)

    Book  MATH  Google Scholar 

  20. Pȩkala, B.: Operations on Interval Matrices. In: Kryszkiewicz, M., Peters, J.F., Rybiński, H., Skowron, A. (eds.) RSEISP 2007. LNCS (LNAI), vol. 4585, pp. 613–620. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  21. Pȩkala, B.: Preservation of Properties of Interval-Valued Fuzzy Relations. In: Carvalho, J.P., et al. (eds.) IFSA/EUSFLAT 2009, pp. 1206–1210 (2009)

    Google Scholar 

  22. Sánchez, L., Couso, I.: Advocating the Use of Imprecisely Observed Data in Genetic Fuzzy Systems. IEEE Trans. Fuzzy Syst. 15(4), 551–562 (2007)

    Article  Google Scholar 

  23. Sambuc, R.: Fonctions φ-floues: Application á l’aide au diagnostic en pathologie thyroidienne. Ph.D. Thesis, Université de Marseille, France (1975) (in French)

    Google Scholar 

  24. Zadeh, L.A.: The Concept of a Linguistic Variable and its Application to Approximate Reasoning-I. Inform. Sci. 8, 199–249 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zadeh, L.A.: Fuzzy Sets. Inform. Control 8, 338–353 (1965)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Rzeszów, Rejtana 16a, 35-310, Rzeszów, Poland

    Barbara Pȩkala

Authors
  1. Barbara Pȩkala
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Marburg, Germany

    Eyke Hüllermeier

  2. Department of Knowledge Processing and Language Engineering, Otto-von-Guericke University of Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    Rudolf Kruse

  3. Fakultät für Elektrotechnik und Informationstechnik, Technische Universität Dortmund, 44221, Dortmund, Germany

    Frank Hoffmann

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Pȩkala, B. (2010). Properties of Interval-Valued Fuzzy Relations, Atanassov’s Operators and Decomposable Operations. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Methods. IPMU 2010. Communications in Computer and Information Science, vol 80. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14055-6_68

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-14055-6_68

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-14054-9

  • Online ISBN: 978-3-642-14055-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics