Soft Computing

, Volume 14, Issue 10, pp 1037–1041 | Cite as

Properties of \(\text{max-}*\) fuzzy relation equations

Focus
First Online:

Abstract

We extend the result of Zhang et al. (J Fuzzy Math 14:53, 2006), who discussed the finite fuzzy relation equations with max–min and max–prod composition. In this article, the \(\text{max-}*\) composition is used for wide family of operations \(*\). In particular, families of solutions of two relation equations are compared.

Keywords

Fuzzy relation max-* composition max-* system of equations Solution family Solution invariant matrices Reduced matrix Matrix equivalence class 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

The support of the grant the University of Information Technology and Management in Rzeszów, Poland is kindly announced. The authors are grateful to the reviewers for their valuable comments and suggestions, which helped to improve the final version of the paper.

References

  1. Birkhoff G (1967) Lattice theory. AMS Coll Publ 25, Providence, RIGoogle Scholar
  2. Bour L, Lamotte M (1988) Equations de relations floues avec la composition conorme-norme triangulaires. BUSEFAL 34:86–94MATHGoogle Scholar
  3. Czogała E, Drewniak J (1984) Associative monotonic operations in fuzzy set theory. Fuzzy Sets Syst 12:249–269CrossRefMATHGoogle Scholar
  4. Czogała E, Drewniak J, Pedrycz W (1982) Fuzzy relation equations on a finite set. Fuzzy Sets Syst 7:98–101Google Scholar
  5. Di Nola A, Sessa S (1983) On the set of solutions of composite fuzzy relation equations. Fuzzy Sets Syst 9 (3):275–285CrossRefMathSciNetMATHGoogle Scholar
  6. Di Nola A, Sessa S, Pedrycz W (1983) Energy and entropy measures of fuzziness of solutions of fuzzy relation equations with continuous triangular norms. BUSEFAL 12:60–71MATHGoogle Scholar
  7. Drewniak J (1984) Fuzzy relation equations and inequalities. Fuzzy Sets Syst 14:237–247CrossRefMathSciNetMATHGoogle Scholar
  8. Drewniak J (1989) Fuzzy relation calculus. Uniwersytet Śląski, KatowiceMATHGoogle Scholar
  9. Drewniak J, Kula K (2002) Generalized compositions of fuzzy relations. Int J Uncertain Fuzziness Knowl Based Syst 10:149–163CrossRefMathSciNetMATHGoogle Scholar
  10. Fernández MJ, Gil P (2004) Some specific types of fuzzy relation equations. Inform Sci 164(1–4):189–195CrossRefMathSciNetMATHGoogle Scholar
  11. Goguen JA (1967) L-fuzzy sets. J Math Anal Appl 18:145–174CrossRefMathSciNetMATHGoogle Scholar
  12. Han SC, Li SC, Wang JY (2006) Resolution of finite fuzzy relation equations based on strong pseudo-t-norms. Appl Math Lett 19(8):752–757CrossRefMathSciNetMATHGoogle Scholar
  13. Higashi M, Klir GJ (1984) Resolution of finite fuzzy relation equations. Fuzzy Sets Syst 13:65–82CrossRefMathSciNetMATHGoogle Scholar
  14. Klement EP, Mesiar R, Pap E (2000) Triangular norms. Kluwer, DordrechtMATHGoogle Scholar
  15. Miyakoshi M, Shimbo M (1985) Solutions of composite fuzzy relational equations with triangular norms. Fuzzy Sets Syst 16:53–63CrossRefMathSciNetMATHGoogle Scholar
  16. Pedrycz W (1982) Fuzzy relational equations with triangular norms and their resolutions. BUSEFAL 11:24–32MATHGoogle Scholar
  17. Pedrycz W (1993) S-T fuzzy relational equations. Fuzzy Sets Syst 59:189–195CrossRefMathSciNetGoogle Scholar
  18. Sanchez E (1976) Resolution of composite fuzzy relation equations. Inform Control 30:38–48CrossRefMATHGoogle Scholar
  19. Shieh BS (2007) Solutions of fuzzy relation equations based on continuous t-norms. Inform Sci 177(19):4208–4215CrossRefMathSciNetMATHGoogle Scholar
  20. Stamou GB, Tzafestas SG (2001) Resolution of composite fuzzy relation equations based on Archimedean triangular norms. Fuzzy Sets Syst 120(3):395–407CrossRefMathSciNetMATHGoogle Scholar
  21. Wang XP, Xiong QQ (2005) The solution set of a fuzzy relational equation with sup-conjunctor composition in a complete lattice. Fuzzy Sets Syst 153(2):249–260CrossRefMathSciNetMATHGoogle Scholar
  22. Zhang Ch, Lu ChJ, Li DY (2006) On perturbation properties of fuzzy relation equations. J Fuzzy Math 14:53–63MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of RzeszówRzeszowPoland
  2. 2.University of Information Technology and ManagementRzeszowPoland

Personalised recommendations