Skip to main content
Top
Published in: Soft Computing 1/2015

01-01-2015 | Foundations

Fuzzy relation and fuzzy function over fuzzy sets: a retrospective

Authors: Soma Dutta, Mihir K. Chakraborty

Published in: Soft Computing | Issue 1/2015

Log in

Activate our intelligent search to find suitable subject content or patents.

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.

Dont have a licence yet? Then find out more about our products and how to get one now:

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!

Literature
go back to reference 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
go back to reference 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)
go back to reference 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
go back to reference 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
go back to reference 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
go back to reference 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
go back to reference 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
go back to reference 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
go back to reference 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
go back to reference 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
go back to reference Gilles M (1984) PhD thesis, University of Claude Bernard of Lyon, France Gilles M (1984) PhD thesis, University of Claude Bernard of Lyon, France
go back to reference 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
go back to reference 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
go back to reference 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
go back to reference 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
go back to reference 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
go back to reference 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
go back to reference 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
go back to reference 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
go back to reference 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
go back to reference 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
go back to reference 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
go back to reference 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
Metadata
Title
Fuzzy relation and fuzzy function over fuzzy sets: a retrospective
Authors
Soma Dutta
Mihir K. Chakraborty
Publication date
01-01-2015
Publisher
Springer Berlin Heidelberg
Published in
Soft Computing / Issue 1/2015
Print ISSN: 1432-7643
Electronic ISSN: 1433-7479
DOI
https://doi.org/10.1007/s00500-014-1356-z

Other articles of this Issue 1/2015

Soft Computing 1/2015 Go to the issue

Premium Partner