Skip to main content
Erschienen in: Soft Computing 1/2015

01.01.2015 | Foundations

Fuzzy relation and fuzzy function over fuzzy sets: a retrospective

verfasst von: Soma Dutta, Mihir K. Chakraborty

Erschienen in: Soft Computing | Ausgabe 1/2015

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

In this paper, some earlier results dealing with fuzzy relations on fuzzy sets by one of the present authors are recapitulated to clarify the motivational background of the present study. Then concepts like fuzzy homorelation, correlation, cohomorelation, fuzzy function and fuzzy morphism are introduced and discussed. The presentation is uniform and integrated in the sense that every concept is developed on fuzzy set as base and is organized from the angle of foundational aspects of fuzzy set theory.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literatur
Zurück zum Zitat Bandler W, Kohout LJ (1986a) On new types of homomorphisms and congruences for partial algebraic structures and n-ary relations. Intern J Gener Syst 12(2):149–157CrossRefMATHMathSciNet Bandler W, Kohout LJ (1986a) On new types of homomorphisms and congruences for partial algebraic structures and n-ary relations. Intern J Gener Syst 12(2):149–157CrossRefMATHMathSciNet
Zurück zum Zitat Bandler W, Kohout LJ (1986b) On the general theory of relational morphisms. Intern J Gener Syst 13(1):47–68CrossRefMathSciNet Bandler W, Kohout LJ (1986b) On the general theory of relational morphisms. Intern J Gener Syst 13(1):47–68CrossRefMathSciNet
Zurück zum Zitat Banerjee M, Chakraborty MK (2003) Foundations on vagueness: a category theoretic approach. In: Proceedings of electronic notes in theoretical computer science 82(4) (2003) Banerjee M, Chakraborty MK (2003) Foundations on vagueness: a category theoretic approach. In: Proceedings of electronic notes in theoretical computer science 82(4) (2003)
Zurück zum Zitat Bodenhofer U (2002) A general framework for ordering fuzzy sets. In: Bouchon-Meunier B, Guitérrez-Ríoz J, Magdalena L, Yager RR (eds) Technologies of constructing intelligent systems 1: tasks, studies in fuzziness and soft computing, vol 89. Physica, pp 213–224 Bodenhofer U (2002) A general framework for ordering fuzzy sets. In: Bouchon-Meunier B, Guitérrez-Ríoz J, Magdalena L, Yager RR (eds) Technologies of constructing intelligent systems 1: tasks, studies in fuzziness and soft computing, vol 89. Physica, pp 213–224
Zurück zum Zitat Bodenhofer U (1999) A new approach to fuzzy orderings, vol 16. Tatra Mountains Mathematical Publications, Nottingham Bodenhofer U (1999) A new approach to fuzzy orderings, vol 16. Tatra Mountains Mathematical Publications, Nottingham
Zurück zum Zitat Bodenhofer U (2008) Orderings of fuzzy sets based on fuzzy orderings part I: the basic approach. Mathw Soft Comput 15:201–218MATHMathSciNet Bodenhofer U (2008) Orderings of fuzzy sets based on fuzzy orderings part I: the basic approach. Mathw Soft Comput 15:201–218MATHMathSciNet
Zurück zum Zitat Chakraborty MK, Ahsanullah TMG (1992) Fuzzy topology on fuzzy sets and tolerance topology. Fuzzy Sets Syst 45:103–108 Chakraborty MK, Ahsanullah TMG (1992) Fuzzy topology on fuzzy sets and tolerance topology. Fuzzy Sets Syst 45:103–108
Zurück zum Zitat Chakraborty MK, Banerjee M (1991) A categorical approach to fuzzy set theory, Recent advances in fuzzy mathematics. In: Proceedings of the national seminar on fuzzy mathematics and its applications, Tripura University, Tripura Chakraborty MK, Banerjee M (1991) A categorical approach to fuzzy set theory, Recent advances in fuzzy mathematics. In: Proceedings of the national seminar on fuzzy mathematics and its applications, Tripura University, Tripura
Zurück zum Zitat Chakraborty MK, Das M (1981) On the product of fuzzy subsets. J Pure Math I:55–60 Chakraborty MK, Das M (1981) On the product of fuzzy subsets. J Pure Math I:55–60
Zurück zum Zitat Chakraborty MK, Sarkar S, Das M (1985) Some aspects of [0,1]-fuzzy relation and a few suggestions towards its use. In: Gupta MM, Kandel A, Bandler W, Kiszka JB (eds) Approximate resoning in expert systems, Elsevier Science Publishers, North Holland, pp 139–156 Chakraborty MK, Sarkar S, Das M (1985) Some aspects of [0,1]-fuzzy relation and a few suggestions towards its use. In: Gupta MM, Kandel A, Bandler W, Kiszka JB (eds) Approximate resoning in expert systems, Elsevier Science Publishers, North Holland, pp 139–156
Zurück zum Zitat Das M, Chakraborty MK, Ghoshal TK (1998) Fuzzy tolerance relation, fuzzy tolerance space and basis. Fuzzy Sets Syst 97:361–369CrossRefMATHMathSciNet Das M, Chakraborty MK, Ghoshal TK (1998) Fuzzy tolerance relation, fuzzy tolerance space and basis. Fuzzy Sets Syst 97:361–369CrossRefMATHMathSciNet
Zurück zum Zitat Di Nola A, Pedrycz W, Sessa S (1989) An aspect of discrepancy in the implementation of modus ponens in the presence of fuzzy quantities. Intern J Approx Reason 3(3):259–265CrossRefMATH Di Nola A, Pedrycz W, Sessa S (1989) An aspect of discrepancy in the implementation of modus ponens in the presence of fuzzy quantities. Intern J Approx Reason 3(3):259–265CrossRefMATH
Zurück zum Zitat Etyan M (1981) Fuzzysets: a topos-logical view. Fuzzy Sets Syst 5:47–67CrossRef Etyan M (1981) Fuzzysets: a topos-logical view. Fuzzy Sets Syst 5:47–67CrossRef
Zurück zum Zitat Gilles M (1984) PhD thesis, University of Claude Bernard of Lyon, France Gilles M (1984) PhD thesis, University of Claude Bernard of Lyon, France
Zurück zum Zitat Ha\(\acute{j}\)ek P (1998) Metamathematics of fuzzy logic. Kluwer Academic Publishers, Dordrecht Ha\(\acute{j}\)ek P (1998) Metamathematics of fuzzy logic. Kluwer Academic Publishers, Dordrecht
Zurück zum Zitat Higgs D (1973) A categorical approach to Boolean-valued set theory (pre-print). University of Waterloo, Waterloo Higgs D (1973) A categorical approach to Boolean-valued set theory (pre-print). University of Waterloo, Waterloo
Zurück zum Zitat Kaufmann A (1975) Introduction to the theory of fuzzy subsets. Academic Press, New YorkMATH Kaufmann A (1975) Introduction to the theory of fuzzy subsets. Academic Press, New YorkMATH
Zurück zum Zitat Klawonn F (2000) Fuzzy points, fuzzy relations, and fuzzy functions, In: Novak V, Perfilieva I (eds) Discovering the world with fuzzy logic, Springer, Berlin pp 431–453 Klawonn F (2000) Fuzzy points, fuzzy relations, and fuzzy functions, In: Novak V, Perfilieva I (eds) Discovering the world with fuzzy logic, Springer, Berlin pp 431–453
Zurück zum Zitat Klir GJ, Yuan B (1995) Fuzzy sets and fuzzy logic: theory and applications. Prentice hall of India, DelhiMATH Klir GJ, Yuan B (1995) Fuzzy sets and fuzzy logic: theory and applications. Prentice hall of India, DelhiMATH
Zurück zum Zitat Ovchinnikov SV, Riera T (1982) Fuzzy sets and possibility theory. In: Yager (ed) On fuzzy classifications. Pergamon Press Inc., Elmsford, New York, pp 119–132 Ovchinnikov SV, Riera T (1982) Fuzzy sets and possibility theory. In: Yager (ed) On fuzzy classifications. Pergamon Press Inc., Elmsford, New York, pp 119–132
Zurück zum Zitat Perfilieva I (2011) Fuzzy function: theoretical and practical point of view. In: Proceedings of advances in intelligent system research, vol 1, issue 1, EUSFLAT, Atlantic Press, Aix-les-Bains, France, pp 480–486 Perfilieva I (2011) Fuzzy function: theoretical and practical point of view. In: Proceedings of advances in intelligent system research, vol 1, issue 1, EUSFLAT, Atlantic Press, Aix-les-Bains, France, pp 480–486
Zurück zum Zitat Rosenfold A (1975) Fuzzy graphs. In: Zadeh LA, Fu KS, Tanaka K, Shimura M (eds) Fuzzy sets and their applications to cognitive and decision processes. Academic Press, New York, pp 77–96 Rosenfold A (1975) Fuzzy graphs. In: Zadeh LA, Fu KS, Tanaka K, Shimura M (eds) Fuzzy sets and their applications to cognitive and decision processes. Academic Press, New York, pp 77–96
Zurück zum Zitat Sanchez E (1976) Resolution in composite fuzzy relation equations. Inf Control 30(1):38–48CrossRefMATH Sanchez E (1976) Resolution in composite fuzzy relation equations. Inf Control 30(1):38–48CrossRefMATH
Zurück zum Zitat Yeh RT, Bang SY (1975) Fuzzy relations, fuzzy graphs and their applications. In: Zadeh LA, Fu KS, Tanaka K, Shimura M (eds) Fuzzy sets and their applications to cognitive and decision processes. Academic Press, New York, pp 125–149CrossRef Yeh RT, Bang SY (1975) Fuzzy relations, fuzzy graphs and their applications. In: Zadeh LA, Fu KS, Tanaka K, Shimura M (eds) Fuzzy sets and their applications to cognitive and decision processes. Academic Press, New York, pp 125–149CrossRef
Zurück zum Zitat Zadeh LA (1971a) Similarity relations and fuzzy orderings. Inf Sci 3:117–200 Zadeh LA (1971a) Similarity relations and fuzzy orderings. Inf Sci 3:117–200
Zurück zum Zitat Zadeh LA (1971b) Towards a theory of fuzzy systems, In: Kalman RE, De Clairis RN (eds) Aspects of networks and systems theory, Holt, Rinehart & Winston, New York, pp 469–490 Zadeh LA (1971b) Towards a theory of fuzzy systems, In: Kalman RE, De Clairis RN (eds) Aspects of networks and systems theory, Holt, Rinehart & Winston, New York, pp 469–490
Metadaten
Titel
Fuzzy relation and fuzzy function over fuzzy sets: a retrospective
verfasst von
Soma Dutta
Mihir K. Chakraborty
Publikationsdatum
01.01.2015
Verlag
Springer Berlin Heidelberg
Erschienen in
Soft Computing / Ausgabe 1/2015
Print ISSN: 1432-7643
Elektronische ISSN: 1433-7479
DOI
https://doi.org/10.1007/s00500-014-1356-z

Weitere Artikel der Ausgabe 1/2015

Soft Computing 1/2015 Zur Ausgabe

Premium Partner