Skip to main content
SpringerLink
Log in

Compositions of Fuzzy Implications

  • Chapter
Advances in Fuzzy Implication Functions

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 300))

Abstract

This chapter considers fuzzy implications as a special case of fuzzy relations in [0,1]. We examine compositions of fuzzy implications based on a binary operation ∗ and discuss the dependencies between algebraic properties of the operation ∗ and the induced sup − ∗ composition. Under some simple assumptions the sup − ∗ composition of fuzzy implications gives also a fuzzy implication. This leads to an examination of ordered groupoids and ordered semigroups of fuzzy implications. Contrapositive and invariant fuzzy implications are also considered.

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
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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. Baczyński, M., Drewniak, J.: Conjugacy Classes of Fuzzy Implication. In: Reusch, B. (ed.) Fuzzy Days 1999. LNCS, vol. 1625, pp. 287–298. Springer, Heidelberg (1999)

    Google Scholar 

  2. Baczyński, M., Drewniak, J.: Monotonic Fuzzy Implication. In: Szczepaniak, P.S., Lisboa, P.J.G., Kacprzyk, J. (eds.) Fuzzy Systems in Medicine. STUDFUZZ, vol. 41, pp. 90–111. Physica Verlag, Heidelberg (2000)

    Chapter  Google Scholar 

  3. Baczyński, M., Drewniak, J., Sobera, J.: Semigroups of fuzzy implications. Tatra Mt. Math. Publ. 21, 61–71 (2001)

    MathSciNet  MATH  Google Scholar 

  4. Baczyński, M., Drewniak, J., Sobera, J.: On sup -∗ compositions of fuzzy implications. In: Rutkowski, L., Kacprzyk, J. (eds.) Neural Networks and Soft Somputing. Advances in Soft Computing, pp. 274–279. Physica Verlag, Heidelberg (2003)

    Google Scholar 

  5. Baczyński, M., Jayaram, B.: Fuzzy Implications. STUDFUZZ, vol. 231. Springer, Berlin (2008)

    MATH  Google Scholar 

  6. Baldwin, J.F., Pilsworth, B.W.: Axiomatic approach to implication for approximate reasoning with fuzzy logic. Fuzzy Sets Syst. 3, 193–219 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bandler, W., Kohout, L.J.: Fuzzy relational products as a tool for analysis and synthesis of the behaviour of complex natural and artificial systems. In: Wang, S.K., Chang, P.P. (eds.) Fuzzy Sets: Theory and Application to Policy Analysis and Information Systems, pp. 341–367. Plenum Press, New York (1980)

    Google Scholar 

  8. Bělohlávek, R.: Fuzzy Relational Systems. Kluwer, New York (2002)

    Book  MATH  Google Scholar 

  9. Czogała, E., Drewniak, J.: Associative monotonic operations in fuzzy set theory. Fuzzy Sets Syst. 12, 249–269 (1984)

    Article  MATH  Google Scholar 

  10. Drewniak, J., Kula, K.: Generalized compositions of fuzzy relations. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 10, 149–164 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Drewniak, J.: Invariant fuzzy implications. Soft Comput. 10, 506–513 (2006)

    Article  MATH  Google Scholar 

  12. Drewniak, J., Sobera, J.: Composition of invariant fuzzy implications. Soft Comput. 10, 514–520 (2006)

    Article  MATH  Google Scholar 

  13. Drewniak, J., Król, A.: A survey of weak connectives and the preservation of their properties by aggregations. Fuzzy Sets Syst. 161(2), 202–215 (2010)

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  15. Fodor, J.C., Yager, R.R., Rybalov, A.: Structure of uninorms. J. Uncertainty, Fuzzines Knowledge-Based System 5, 411–427 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Goguen, J.A.: L-fuzzy sets. J. Math. Anal. Appl. 18, 145–174 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gottwald, S.: A Treatise on Many Valued Logics. Research Studies Press, Baldock (2001)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  19. Łukasiewicz, J.: Interpretacja liczbowa teorii zdań. Ruch Filozoficzny 7, 92–93 (1923); english translation In: Borkowski, L. (ed.) Jan Łukasiewicz Selected Works. Studies in Logic and the Foundations of Mathematics, North Holland, Amsterdam, pp. 129–130. PWN Polish Sci. Publ., Warsaw (1970)

    Google Scholar 

  20. Mamdani, E.H., Assilian, S.: An experiment in linguistic synthesis with a fuzzy logic controller. Int. J. Man Machine Studies 7, 1–13 (1975)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Zadeh, L.A.: Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. Syst. Man Cyber. 3, 28–44 (1973)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Józef Drewniak

  2. Institute of Mathematics, University of Silesia, 40-007, Katowice, ul. Bankowa 14, Poland

    Jolanta Sobera

Authors
  1. Józef Drewniak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jolanta Sobera
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Józef Drewniak .

Editor information

Editors and Affiliations

  1. , Institute of Mathematics, University of Silesia, ul. Bankowa 14, Katowice, 40-007, Poland

    Michał Baczyński

  2. School of Information Technology, Deakin University, Burwood Highway 221, Burwood, 3125, Victoria, Australia

    Gleb Beliakov

  3. Departamento de Automática y, Computación, Universidad Pública de Navarra, Campus Arrosadia Campus s/n, Pamplona, 31006, Spain

    Humberto Bustince Sola

  4. , Departamento de Ciencias de la, Universidad Rey Juan Carlos, c/ Tulipan s/n, Mostoles, 28933, Spain

    Ana Pradera

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Drewniak, J., Sobera, J. (2013). Compositions of Fuzzy Implications. In: Baczyński, M., Beliakov, G., Bustince Sola, H., Pradera, A. (eds) Advances in Fuzzy Implication Functions. Studies in Fuzziness and Soft Computing, vol 300. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35677-3_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-35677-3_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-35676-6

  • Online ISBN: 978-3-642-35677-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation