Skip to main content
SpringerLink
Log in

Computing a Transitive Opening of a Reflexive and Symmetric Fuzzy Relation

  • Conference paper
Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2005)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 3571))

Abstract

There are fast algorithms to compute the transitive closure of a fuzzy relation, but there are only a few different algorithms that compute transitive openings from a given fuzzy relation. In this paper a method to compute a transitive opening of a reflexive and symmetric fuzzy relation is given. Even though there is not a unique transitive opening of a fuzzy relation, it is proved that the computed transitive opening closure is maximal.

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 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alsina, C., Trillas, E., Valverde, L.: On some logical connectives for fuzzy set theory. J. Math. Ann. Appl. 93, 15–26 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  2. De Baets, B., De Meyer, H.: Transitive approximation of fuzzy relations by alternating closures and openings. Soft Computing 7, 210–219 (2003)

    Google Scholar 

  3. Esteva, F., Garcia, P., Godo, L., Rodriguez, R.O.: Fuzzy approximation relations, modal structures and possibilistic logic. Mathware and Soft Computing 5(2-3), 151–166 (1998)

    MATH  MathSciNet  Google Scholar 

  4. Dawyndt, P., De Meyer, H., De Baets, B.: The complete linkage clustering algorithm revisited. Soft Computing (available on-line) (in press)

    Google Scholar 

  5. Di Nola, A., Kolodziejczyk, W., Sessa, S.: Transitive solutions of relational equations on finite sets and linear latices. LNCS, vol. 521, pp. 173–182. Springer, Berlin (1991)

    Google Scholar 

  6. Dunn, J.C.: Some recent investigations of a new fuzzy partitioning algorithm and its application to pattern classification problems. J. Cybernet. 4, 5 (1974)

    Google Scholar 

  7. Garmendia, L., Campo, C., Cubillo, S., Salvador, A.: A Method to Make Some Fuzzy Relations T-Transitive. International Journal of Intelligence Systems 14(9), 873–882 (1999)

    Article  MATH  Google Scholar 

  8. Garmendia, L., Salvador, A.: On a new method to T-transitivize fuzzy relations. In: Information Processing and Management of Uncertainty in Knowledge - based Systems, IPMU 2000, pp. 864–869 (2000)

    Google Scholar 

  9. Garmendia, L., Salvador, A.: On a new method to T-transitivize fuzzy relations. In: Bouchon-Meunier, B., Gutierrez-Rios, J., Magdalena, L., Yager, R.R. (eds.) Technologies for Constructing Intelligent Systems, vol. 2, pp. 251–260. Springer, Heidelberg (2000)

    Google Scholar 

  10. Fu, G.: Guoyao Fu, An algorithm for computing the transitive closure of a fuzzy similarity matrix. Fuzzy Sets and Systems 5, 89–94 (1992)

    Google Scholar 

  11. Hashimoto, H.: Transitivity of generalised fuzzy matrices. Fuzzy Sets and Systems 17(1), 83–90 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  12. Jacas, J.: Similarity relations. The calculation of minimal generating families. Fuzzy Sets and Systems 35, 151–162 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kandel, Yelowitz, L.: Fuzzy chains. IEEE Trans. Systems Man Cybernet. 4, 472–475 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  14. Larsen, H., Yager, R.: Efficient computation of transitive closures. Fuzzy Sets Syst. 38, 81–90 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lee, H.-S.: An optimal algorithm for computing the max–min transitive closure of a fuzzy similarity matrix. Fuzzy Sets and Systems 123, 129–136 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  16. Naessens, H., De Meyer, H., De Baets, B.: Algorithms for the Computation of T-Transitive Closures. IEEE Trans Fuzzy Systems 10(4), 541–551 (2002)

    Article  Google Scholar 

  17. Ovchinnikov, S.: Representations of Transitive Fuzzy Relations. In: Skala, H.J., Termini, S., Trillas, E. (eds.) Aspects of Vagueness, pp. 105–118. Reidel Pubs. (1984)

    Google Scholar 

  18. Rodriguez, R.O., Esteva, F., Garcia, P., Godo, L.: On implicative closure operators in approximate reasoning. International Journal of Approximate Reasoning 33, 159–184 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Wagenknecht, M.: On pseudo-transitive approximations of fuzzy relations. Fuzzy Sets and Systems 44, 45–55 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  20. Wagenknecht, M.: On transitive solutions of fuzzy equations, inequalities and lower approximation of fuzzy relations. Fuzzy Sets and Systems 75, 229–240 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  21. Xiao, X.: An algorithm for calculating fuzzy transitive closure. Fuzzy Math 5(4), 71–73 (1985)

    Google Scholar 

  22. Zadeh, L.A.: Similarity relations and fuzzy orderings. Inform. Sci. 3, 177–200 (1971)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Facultad de Informática, Dpto. de Lenguajes y Sistemas Informáticos, Universidad Complutense of Madrid, 28040, Madrid, Spain

    Luis Garmendia

  2. E.T.S.I. Caminos Canales y Puertos, Dpto. de Matemática Aplicada, Technical University of Madrid, 28040, Madrid, Spain

    Adela Salvador

Authors
  1. Luis Garmendia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Adela Salvador
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. IIIA - CSIC, 08193, Bellaterra, Spain

    Lluís Godo

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Garmendia, L., Salvador, A. (2005). Computing a Transitive Opening of a Reflexive and Symmetric Fuzzy Relation. In: Godo, L. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2005. Lecture Notes in Computer Science(), vol 3571. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11518655_50

Download citation

  • DOI: https://doi.org/10.1007/11518655_50

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-27326-4

  • Online ISBN: 978-3-540-31888-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation