Skip to main content
SpringerLink
Log in

The Algebra of Truth Values of Type-2 Fuzzy Sets: A Survey

  • Chapter
Interval / Probabilistic Uncertainty and Non-Classical Logics

Part of the book series: Advances in Soft Computing ((AINSC,volume 46))

  • 653 Accesses

Summary

Type-2 fuzzy sets have come to play an increasingly important role in both applications and in the general theory of fuzzy sets. The basis of type-2 fuzzy sets is a certain algebra of truth values, as set forth by Zadeh. This paper is a survey of results about this algebra, along with some new material.

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 209.00
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.

References

  1. Dubois, D., Prade, H.: Operations in a fuzzy-valued logic. Inform. and Control 43, 224–240 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  2. Gehrke, M., Walker, C., Walker, E.: De Morgan Systems on the Unit Interval. International Journal of Intelligent Systems 11, 733–750 (1996)

    Article  MATH  Google Scholar 

  3. Gehrke, M., Walker, C., Walker, E.: Some Comments on Interval Valued Fuzzy Sets. International Journal of Intelligent Systems 11, 751–759 (1996)

    Article  MATH  Google Scholar 

  4. Gehrke, M., Walker, C., Walker, E.: A Mathematical Setting for Fuzzy Logics. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 5, 223–238 (1997)

    Article  MathSciNet  Google Scholar 

  5. Gehrke, M., Walker, C., Walker, E.: Algebraic aspects of fuzzy sets and fuzzy logic. In: Proceedings of the Workshop on Current Trends and Developments in Fuzzy Logic, Thessaloniki, Greece, October 16–20, 1998, pp. 101–170 (1999)

    Google Scholar 

  6. Greenfield, S., John, R.: Optimised Generalised Type-2 Join and Meet Operations. In: Proceedings of 2007 IEEE International Conference on Fuzzy systems, London, UK, July 23–26, 2007, pp. 141–146 (2007)

    Google Scholar 

  7. John, R.: Type-2 fuzzy sets: An appraisal of theory and applications. Int. J. Uncertainty, Fuzziness Knowledge-Based Systems 6(6), 563–576 (1998)

    Article  MATH  Google Scholar 

  8. Harding, J., Walker, C., Walker, E.: Lattices of Convex Normal Functions. Fuzzy Sets and Systems (to appear)

    Google Scholar 

  9. Karnik, N., Mendel, J.: Operations on type-2 fuzzy sets. Fuzzy Sets and Systems 122, 327–348 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  11. Mendel, J.M.: Uncertain Rule-Based Fuzzy Logic Systems. Prentice Hall PTR, Upper Saddle River (2001)

    MATH  Google Scholar 

  12. Mendel, J., John, R.: Type-2 Fuzzy Sets Made Simple. IEEE Transactions on Fuzzy Systems 10(2), 117–127 (2002)

    Article  Google Scholar 

  13. Mizumoto, M., Tanaka, K.: Some Properties of Fuzzy Sets of Type-2. Information and Control 31, 312–340 (1976)

    Article  MathSciNet  Google Scholar 

  14. Mizumoto, M., Tanaka, K.: Fuzzy sets of type-2 under algebraic product and algebraic sum. Fuzzy Sets and Systems 5, 277–290 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  15. Mukaidono, M.: Algebraic Structures of Truth Values in Fuzzy Logic. In: Driankov, D., L. Ralescu, A., Eklund, P.W. (eds.) IJCAI-WS 1991. LNCS, vol. 833, pp. 15–21. Springer, Heidelberg (1994)

    Google Scholar 

  16. Emoto, M., Mukaidono, M.: Necessary and Sufficient Conditions for Fuzzy Truth Values to Form a De Morgan Algebra. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 7(4), 309–316 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. Nieminen, J.: On the Algebraic Structure of Fuzzy Sets of Type 2. Kybernetika 13, 261–273 (1977)

    MathSciNet  Google Scholar 

  18. Nguyen, H., Walker, E.: A First Course in Fuzzy Logic. Chapman & Hall/CRC, Boca Raton (2000)

    MATH  Google Scholar 

  19. Walker, C., Walker, E.: The Algebra of Fuzzy Truth Values. Fuzzy Sets and Systems 149(2), 309–347 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Walker, C., Walker, E.: Automorphisms of the Algebra of Fuzzy Truth Values. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 11, 711–732 (2006)

    Article  Google Scholar 

  21. Walker, C., Walker, E.: Automorphisms of the Algebra of Fuzzy Truth Values II (preprint)

    Google Scholar 

  22. Walker, C., Walker, E.: Finite Algebras of Fuzzy Truth Values (in preparation)

    Google Scholar 

  23. Walker, C., Walker, E.: Sets With Type-2 Operations. International Journal of Approximate Reasoning (to appear)

    Google Scholar 

  24. Walker, C., Walker, E.: Points With Type-2 Operations. In: Melin, P., et al. (eds.) IFSA 2007. LNCS (LNAI), vol. 4529, Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  25. Walker, C., Walker, E.: Type-2 Intervals of Constant Height. In: Proceedings of the NAFIPS Conference, June 2007, San Diego (2007)

    Google Scholar 

  26. Zadeh, L.: The concept of a linguistic variable and its application to approximate reasoning. Inform Sci. 8, 199–249 (1975)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM, 88003

    Carol L. Walker & Elbert A. Walker

Authors
  1. Carol L. Walker
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elbert A. Walker
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Van-Nam Huynh Yoshiteru Nakamori Hiroakira Ono Jonathan Lawry Vkladik Kreinovich Hung T. Nguyen

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Walker, C.L., Walker, E.A. (2008). The Algebra of Truth Values of Type-2 Fuzzy Sets: A Survey. In: Huynh, VN., Nakamori, Y., Ono, H., Lawry, J., Kreinovich, V., Nguyen, H.T. (eds) Interval / Probabilistic Uncertainty and Non-Classical Logics. Advances in Soft Computing, vol 46. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77664-2_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-77664-2_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-77663-5

  • Online ISBN: 978-3-540-77664-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation