Abstract
Many general problems on the resolution of composite fuzzy relation equations have been investigated for nearly thirty years. This paper now shows how to solve a class of SUP-min fuzzy relational equations, such as T = RoQ, where Q is a functional relation. Such equations are related to systems of truth-qualified propositions and the concept of the decomposition of a fuzzy relation by a fuzzy set is introduced Finally, the problem of ‘decomposability’ of a fuzzy relation by a given fuzzy set is solved by combining these methodologies.
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References
Birkhoff G (1967) Lattice Theory, American Mathematical Society Colloquium Publications, Vol. XXV, Providence, R.I.
De Baets B (2000) Analytical solution methods for fuzzy relational equations, In: Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Sets Series (Dubois D and Prade H, Eds.), Vol.1, Kluwer Academic Publishers, 291–340.
Di Nola A, Pedrycz W, Sanchez E, Sessa S (1989) Fuzzy Relation Equations and their Applications to Knowledge Engineering, Kluwer Academic Publishers.
Dubois D, Prade H, (1992) Possibility theory as a basis for preference propagation in automated reasoning, Proceedings of FUZZ-IEEE, San Diego, CA, 821–832.
Kaufmann A (1977) Introduction à la Théorie des Sous-Ensembles Flous, Vol. 4, Compléments et Nouvelles Applications, Masson, Paris.
Peeva KG, Kyosev YK (to appear in 2003 ) Fuzzy Relational Calculus - Theory, Applications and Software, World Scientific Pub Co.
Prade H (automne 1980) Compatibilité, qualification, modification, niveau de précision, Busefal N°4, 71–78.
Sanchez E (1972) Matrices et Fonctions en Logique Symbolique, Ph.D. Thesis in Mathematics, Faculty of Science of Marseille, France.
Sanchez E (1974) Equations de Relations Floues, Ph.D. Thesis in Human Biology, Faculty of Medicine of Marseille, France.
Sanchez E (1976) Resolution of Composite Fuzzy Relation Equations, Information and Control, 30, 1, 38–48.
Sanchez E (1977) Solutions in Composite Fuzzy Relation Equations: Application to Medical Diagnosis in Brouwerian Logic, In: Fuzzy Automata and Decision Processes (Gupta MM, Ed.), North-Holland, 221–234. Also reproduced in (1993) Readings in Fuzzy Sets for Intelligent Systems (Dubois D, Prade H, Yager RR, Eds.), Morgan Kaufmann Pub., 159–165.
Sanchez E (1978) On Truth-Qualification in Natural Languages, Proceedings of the International Conference on Cybernetics and Society ( Tokyo-Kyoto, Japan ) 1233–1236.
Sanchez E (1996) Truth-qualification and fuzzy relations in natural languages, Fuzzy Sets and Systems 84, 155–167.
Schweizer B, Sklar A (1963) Associative functions and abstract semigroups, Publ. Math. Debrecen, 10, 69–81.
Zadeh LA (1965) Fuzzy Sets, Information and Control, 8, 338–353.
Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning, Part I, Information Sciences, 8, 199–249;
Zadeh LA (1975) Part II, Information Sciences, 8, 301–357;
Zadeh LA (1975) Part III, Information Sciences, 9, 43–80.
Zadeh LA (1978a) PRUF–A Meaning Representation Language for Natural Languages, Int. J. of Man-Machine Studies, 395–460.
Zadeh LA (1978b) Fuzzy Sets as a Basis for a Theory of Possibility, Fuzzy Sets and Systems, 1, 3–28.
Zadeh LA (1979) A Theory o f A pproximate Reasoning (AR), I n: Machine Intelligence, Vol. 9 ( Hayes JE, Mitchie, Mikulich LI, Eds), J. Wiley, New York, 149–194.
Zadeh LA (1999) From Computing with Numbers to Computing with Words — From Manipulation of Measurements to Manipulation of Perceptions, IEEE Transactions on Circuits and Systems, 45, 1, 105–119.
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Sanchez, E. (2004). Decomposition of Fuzzy Relations and Functional Relations. In: Nikravesh, M., Zadeh, L.A., Korotkikh, V. (eds) Fuzzy Partial Differential Equations and Relational Equations. Studies in Fuzziness and Soft Computing, vol 142. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39675-8_10
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DOI: https://doi.org/10.1007/978-3-540-39675-8_10
Publisher Name: Springer, Berlin, Heidelberg
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