Fuzzy Relation Equations in Residuated Lattices

  • Antonio di Nola
  • Salvatore Sessa
  • Witold Pedrycz
  • Elie Sanchez
Part of the Theory and Decision Library book series (TDLD, volume 3)


We would like to underline the following statement of Goguen [5]: “The importance of relations is almost self-evident. Science is, in a sense, the discovery of relations between observables... Difficulties arise in the so-called “soft” sciences because the relations involved do not appear to be “hard”, as they are, say, in classical physics. A thoroughgoing application of probability theory has relieved many difficulties, but it is clear that others remain. We suggest that further difficulties might be cleared up through a systematic exploitation of fuzziness”.


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  1. [1]
    G. Birkhoff, Lattice Theory, 3rd Ed., Vol.XXV, AMS Colloquium Publications, Providence, Rhode Islands, 1967.zbMATHGoogle Scholar
  2. [2]
    T.S. Blyth, Matrices over ordered algebraic structures, J. London Math. Soc. 39 (1964), 427 - 432.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    A. Di Nola, On solving relational equations in Brouwerian lattices, Fuzzy Sets and Systems, to appear.Google Scholar
  4. [4]
    A. Di Nola and A. Lettieri, Relation equations in residuated lattices, BUSEFAL 34 (1988), 95-106; final version in Rend. Circ. Mat. Palermo, to appear.Google Scholar
  5. [5]
    J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967), 145 - 174.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    R.D. Luce, A note on Boolean matrix theory, Proc. Amer. Math. Soc. 3 (1952), 382 - 388.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    S. Rudeanu, Boolean Functions and Equations, North-Holland, Amsterdam, 1974.zbMATHGoogle Scholar
  8. [8]
    E. Sanchez, Resolution of composite fuzzy relation equations, Inform. and Control 30 (1976), 38 - 48.zbMATHCrossRefGoogle Scholar
  9. [9]
    E. Sanchez, 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 and B.R. Gaines, Eds.), North-Holland, New York (1977), pp. 221 - 234.Google Scholar
  10. [10]
    E. Sanchez, Compositions of fuzzy relations, in: Advances in Fuzzy Set Theory and Applications ( M.M. Gupta, R.K. Ragade and R.R. Yager, Eds.), North-Holland, Amsterdam (1979), pp. 421 - 433.Google Scholar
  11. [11]
    E. Sanchez, Eigen fuzzy sets and fuzzy relations, J. Math. Anal. Appl. 81 (1981), 399 - 421.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    G. Szàsz, Introduction to Lattice Theory, 3rd Ed., Academic Press, New York, 1963.Google Scholar
  13. [13]
    E. Turunen, On generalized fuzzy relation equations: necessary and sufficient conditions for the existence of solutions, Acta Univ. Carolinae — Mathematica et Physica 28 (1987), 33 - 37.MathSciNetzbMATHGoogle Scholar
  14. [14]
    L.A. Zadeh and C.A. Desoer, Linear System Theory, Mc.Graw-Hill, New York, 1963.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1989

Authors and Affiliations

  • Antonio di Nola
    • 1
  • Salvatore Sessa
    • 1
  • Witold Pedrycz
    • 2
  • Elie Sanchez
    • 3
  1. 1.Facoltà di ArchitetturaUniversità di NapoliNapoliItaly
  2. 2.Department of Electrical EngineeringWinnipegCanada
  3. 3.Faculté de MédecineUniversité Aix-Marseille IIMarseilleFrance

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