Skip to main content
SpringerLink
Log in

A Note on Defuzzification of Fuzzy Pre-orders and Transitivity of Its Minimal Regular Strict Component

  • Conference paper
  • First Online:
Advances in Fuzzy Logic and Technology 2017 (EUSFLAT 2017, IWIFSGN 2017)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 642))

  • 923 Accesses

Abstract

Fono and Andjiga [Fuzzy strict preference and social choice, Fuzzy Sets and Systems, 115:372-389, 2005] established a factorization of a fuzzy binary relation into a unique indifference and a family of strict components including the minimal regular strict component. They studied some properties of strict components (pos-transitivity and negative transitivity).

In this note, based on these results, we study some structures of a fuzzy relation on a finite universe A. More precisely, we determine necessary and sufficient conditions under which a fuzzy T-pre-order (reflexive and max-T-transitive fuzzy binary relation where T is t-norm) on A generates a crisp complete pre-order on A. We deduce that the obtained defuzzification is true for a strongly complete fuzzy T-pre-order on A. We show that the minimal regular strict component of a fuzzy pre-order (reflexive and max-min transitive fuzzy relation) is max-min-transitive.

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 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
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

Notes

  1. 1.

    It is well known that the unique strict component of a crisp complete pre-order is transitive, pos-transitive and negative transitive.

  2. 2.

    For the interpretation of the two notions, see [10] page 383.

References

  1. Barrett, C.R., Pattanaik, P.K., Salles, M.: On choosing rationally when preferences are fuzzy. Fuzzy Sets Syst. 34, 197–212 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bodenhofer, U., De Baets, B., Fodor, J.: A compendium of fuzzy weak orders: representations and constructions. Fuzzy Sets Syst. 158, 811–829 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Diaz, S., De Baets, B., Montes, S.: Additive decomposition of fuzzy pre-orders. Fuzzy Sets Syst. 158, 830–842 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications. Academic Press, New york (1980)

    MATH  Google Scholar 

  5. Dubois, D., Prade, H.: A theorem on implications defined from triangular norms. Stochastica VIII(3), 267–279 (1980)

    MathSciNet  MATH  Google Scholar 

  6. Duru, G., Auray, J.P., Mougeot, M.: Mathé matiques appliquées aux sciences sociales, Tome 1 : Relations et Applications, (Document de travail no 8202) (1982)

    Google Scholar 

  7. Dutta, B.: Fuzzy preferences and social choice. Math. Soc. Sci. 13, 215–229 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fodor, J., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht (1994)

    Book  MATH  Google Scholar 

  9. Fono, L.A., Relations binaires floues et application au choix social, Thése de Doctorat/Ph.D en mathématiques, Faculté des sciences, Université de Yaoundé 1 (2004)

    Google Scholar 

  10. Fono, L.A., Andjiga, N.G.: Fuzzy strict preference and social choice. Fuzzy Sets Syst. 115, 372–389 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fotso, S., Fono, L.A.: On the consistency of some crisp choice functions based on a strongly complete fuzzy pre-order. New Math. Nat. Comput. 8, 257–272 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mordeson, J.N., Malik, D.S., Clark, T.D.: Application of Fuzzy Logic to Social Choice Theory, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2015)

    Book  Google Scholar 

  13. Ovchinnikov, S.V., Roubens, M.: On strict preference relations. Fuzzy Sets Syst. 43, 319–326 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Richardson, G.: The structure of fuzzy preferences: social choice implications. Soc. Choice Welf. 15, 359–369 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Salles, M.: Fuzzy utility. Cahier de recherches No. 96-6, Université de Caen (1996)

    Google Scholar 

  16. Sengupta, K.: Choice rules with fuzzy preferences: some characterizations. Soc. Choice Welf. 16, 259–272 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Van de Walle, B., De Baets, B., Kerre, E.: Characterizable fuzzy prefernce structures. Ann. Oper. Res. 80, 105–136 (1998)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, Ecole Normale Supérieure, Université de Yaoundé I, B.P. 47, Yaoundé, Cameroun

    Siméon Fotso

  2. Laboratoire de Mathématiques - Université de Douala, B.P. 24157, Douala, Cameroun

    Louis Aimé Fono

Authors
  1. Siméon Fotso
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Louis Aimé Fono
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Siméon Fotso .

Editor information

Editors and Affiliations

  1. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Janusz Kacprzyk

  2. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Eulalia Szmidt

  3. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Slawomir Zadrożny

  4. Department of Telecommunications and Information Processing, Bulgarian Academy of Sciences, Sofia, Bulgaria

    K. T. Atanassov

  5. WIT - Warsaw School of Information Technology, Warsaw, Poland

    Maciej Krawczak

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this paper

Cite this paper

Fotso, S., Fono, L.A. (2018). A Note on Defuzzification of Fuzzy Pre-orders and Transitivity of Its Minimal Regular Strict Component. In: Kacprzyk, J., Szmidt, E., Zadrożny, S., Atanassov, K., Krawczak, M. (eds) Advances in Fuzzy Logic and Technology 2017. EUSFLAT IWIFSGN 2017 2017. Advances in Intelligent Systems and Computing, vol 642. Springer, Cham. https://doi.org/10.1007/978-3-319-66824-6_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-66824-6_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66823-9

  • Online ISBN: 978-3-319-66824-6

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation