Fundamentals of Fuzzy Relation Calculus

  • Siegfried Gottwald
Part of the International Series in Intelligent Technologies book series (ISIT, volume 7)


This chapter introduces the basic notions of the fuzzy relation calculus, also in their t-norm based form. Typical properties of fuzzy relations are discussed in connection with particular types of relations like fuzzy equivalence or fuzzy ordering relations. Finally there is a discussion of fuzzy relational equations, their relationship to fuzzy control, and their solvability resp. approximate solvability.


Membership Function Relational Equation Fuzzy Controller Relational Product Membership Degree 
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  1. [1]
    Dinola, A.: An algorithm of calculation of lower solutions of fuzzy relation equation. Stochastica 3, 33–40 (1984).MathSciNetGoogle Scholar
  2. [2]
    Dinola, A.: Relational equations in totally ordered lattices and their complete resolution. J. Math. Anal. Appl. 107, 148–155 (1985).MathSciNetCrossRefGoogle Scholar
  3. [3]
    Dinola, A.; Sessa, S.; Pedrycz, W.; Sanchez, E.: Fuzzy Relation Equations and Their Applications to Knowledge Engineering. Theory and Decision Libr., ser. D, Kluwer Academic Publ., Dordrecht 1989.Google Scholar
  4. [4]
    Gottwald, S.: On the existence of solutions of systems of fuzzy equations. Fuzzy Sets and Systems 12, 301–302 (1984).MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Gottwald, S.: Criteria for non-interactivity of fuzzy logic controller rules. In: Large Scale Systems: Theory and Applications 1983 (A. Straczak, ed.), Pergamon Press, Oxford 1984 229–233.Google Scholar
  6. [6]
    Gottwald, S.: Characterizations of the solvability of fuzzy equations. Elektron. Informationsverarb. Kybernetik 22, 67–91 (1986).MathSciNetzbMATHGoogle Scholar
  7. [7]
    Gottwald, S.: Fuzzy Sets and Fuzzy Logic. Foundation of Application — from a Mathematical Point of View. Vieweg, Braunschweig/Wiesbaden and Teknea, Toulouse 1993.zbMATHGoogle Scholar
  8. [8]
    Gottwald, S.; Pedrycz, W.: Analysis and synthesis of fuzzy controller. Problems Control Inform. Theory 14, 33–45 (1985).MathSciNetzbMATHGoogle Scholar
  9. [9]
    Gottwald, S.; Pedrycz, W.: On the suitability of fuzzy models: an evaluation through fuzzy integrals. Internat. J. Man-Machine Studies 24, 141–151 (1986).zbMATHCrossRefGoogle Scholar
  10. [10]
    Gottwald, S.; Pedrycz, W.: On the methodology of solving fuzzy relational equations and its impact on fuzzy modelling. In: Fuzzy Logic in Knowledge-Based Systems, Decision and Control (M. M. Gupta, T. Yamakawa, eds.), North-Holland Publ. Comp., Amsterdam 1988, 197–210.Google Scholar
  11. [11]
    Klaua, D.: Über einen zweiten Ansatz zur mehrwertigen Mengenlehre. Monatsber. Deut. Akad. Wiss. Berlin 8, 161–177 (1966).MathSciNetzbMATHGoogle Scholar
  12. [12]
    Klaua, D.: Grundbegriffe einer mehrwertigen Mengenlehre. Monatsber. Deut. Akad. Wiss. Berlin 8, 781–802 (1966).Google Scholar
  13. [13]
    Klir, G.; Yuan, B.: Approximate solutions of systems of fuzzy relation equations. In: Proc. 3rd IEEE Internat. Conf. Fuzzy Systems, FUZZ-IEEE ‘94, Orlando/FL, IEEE Soc. 1994, 1452–1457.CrossRefGoogle Scholar
  14. [14]
    Nauck, D.; Klawonn, F.; Kruse, R.: Neuronale Netze und Fuzzy-Systeme. Vieweg, Braunschweig/Wiesbaden 1994.zbMATHGoogle Scholar
  15. [15]
    Sanchez, E.: Solutions in composite fuzzy relation equations: application to medical diagnosis in Brouwerian logic. In: Fuzzy Automata and Decision Processes (M. M. Gupta, G. N. Saridis, B. N. Gaines, eds.), North-Holland Publ. Comp., Amsterdam 1977, 221–234.Google Scholar
  16. [16]
    Sanchez, E.: Solution of fuzzy equations with extended operations. Fuzzy Sets and Systems 12, 237–248 (1984).MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    Sessa, S.: Some results in the setting of fuzzy relation equations theory. Fuzzy Sets and Systems 14, 281–297 (1984).MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Wagenknecht, M.; Hartmann, K.: On the solution of direct and inverse problems for fuzzy equation systems with tolerances. In: Fuzzy Sets Applications, Methodological Approaches, and Results (St. Bocklisch et al., eds.); Mathematical Research, 30, Akademie-Verlag, Berlin 1986, 37–44.Google Scholar
  19. [19]
    Wagenknecht, M.; Hartmann, K.: Fuzzy modelling with tolerances. Fuzzy Sets and Systems 20, 325–332 (1986).MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Wu, W.-M.: Fuzzy reasoning and fuzzy relational equations. Fuzzy Sets and Systems 20, 67–78 (1986).MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    Zadeh, L.A.: Similarity relations and fuzzy orderings. Information Sci. 3, 159–176 (1971).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Siegfried Gottwald
    • 1
  1. 1.Institute of Logic and Philosophy of ScienceLeipzig UniversityLeipzigGermany

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