Abstract
Fuzzy relational equations are without doubt the most important inverse problems arising from fuzzy set theory, and in particular from fuzzy relational calculus. Indeed, the calculus of fuzzy relations is a powerful one, with applications in fuzzy control and fuzzy systems modelling in general, approximate reasoning, relational databases, clustering, etc. In this paper, fuzzy relational equations are approached from an order-theoretical point of view. It is shown how all inverse problems can be reduced to systems of polynomial lattice equations. The exposition is limited to the description of exact solutions of systems of sup-T equations, and analytical ways are presented for obtaining the complete solution set when working in a broad and interesting class of distributive lattices.
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References
Bandler W. and Kohout L. (1980a). Fuzzy relational products as a tool for analysis and synthesis of the behaviour of complex natural and artificial systems, Fuzzy Sets: Theory and Application to Policy Analysis and Information Systems (Wang P. and Chang S., eds. ), Plenum Press, 341–367.
Bandler W. and Kohout L. (1980b). Semantics of implication operators and fuzzy relational products, Int. J. Man-Machine Studies, 12, 89–116.
Birkhoff G. (1967). Lattice Theory, AMS Colloquium Publications, Volume XXV, American Mathematical Society.
Davey B. and Priestley H. (1990). Introduction to Lattices and Order,Cambridge University Press.
De Baets B. (1995a). An order-theoretic approach to solving sup-T equations, Fuzzy Set Theory and Advanced Mathematical Applications ( Ruan D., ed. ), Kluwer Academic Publishers, 67–87.
De Baets B. (1995b). Oplossen van vaagrelationele vergelijkingen: een ordetheoretische benadering, Ph.D. dissertation, University of Gent, 389 p.
De Baets B. (1996). Disjunctive and conjunctive fuzzy modelling, Proc Symposium on Qualitative System Modelling, Qualitative Fault Diagnosis and Fuzzy Logic and Control ( Budapest and Balatonfüred, Hungary ), 63–70.
De Bacts B. (1998). Sup-T equations: state of the art, Computational Intelligence: Soft Computing and Fuzzy-Neural Integration with Applications (Kaynak O., Zadeh L., Turksen B. and Rudas I., eds.), NATO ASI Series F: Computer and Systems Sciences, Vol. 162, Springer Verlag, 80–93.
De Baets B. and Kerre E. (1993). A revision of Bandler—Kohout compositions of relations, Math. Pannon., 4, 59–78.
De Baets B. and Kerre E. (1995). Fuzzy relations and applications, Advances in Electronics and Electron Physics (Hawkes P., ed.), 89, Academic Press, 255–324.
De Baets B. and Mesiar R. (1999). Triangular norms on product lattices, Fuzzy Sets and Systems, 104, 61–75.
De Cooman G. and Kerre E. (1994). Order norms on bounded partially ordered sets, J. Fuzzy Math., 2, 281–310.
Di Nola A. (1987). Fuzzy equations in infinitely distributive lattices, Proc Second IFSA World Congress ( Tokyo, Japan ), 533–534.
Di Nola A. (1990). On solving relational equations in Brouwerian lattices, Fuzzy Sets and Systems, 34, 365–376.
Di Nola A., Pedrycz W. and Sessa S. (1987). Fuzzy relation equations under lsc and use t-norms and their Boolean solutions, Stochastica, 11, 151–183.
Di Nola A., Sessa S., Pedrycz W. and Sanchez E. (1989). Fuzzy Relation Equations and their Applications to Knowledge Engineering, Theory and Decision Library. Series D. System Theory, Knowledge Engineering and Problem Solving, Kluwer Academic Publishers.
Dubois D. and Prade H. (1992). Upper and lower images of a fuzzy set induced by a fuzzy relation: applications to fuzzy inference and diagnosis, Inform. Sc., 64, 203–232.
Goguen J. (1967). L-fuzzy sets, J. Math. Anal. Appl., 18, 145–174.
Gottwald S. (1994). Approximately solving fuzzy relational equations: some mathematical results and some heuristic proposals, Fuzzy Sets and Systems, 66, 175–193.
Pedrycz W. (1985). Applications of fuzzy relational equations for methods of reasoning in presence of fuzzy data, Fuzzy Sets and Systems, 16, 163–175.
Pedrycz W. (1988). Approximate solutions of fuzzy relational equations, Fuzzy Sets and Systems, 28, 183–202.
Rudeanu S. (1974). Boolean Functions and Equations,North-Holland.
Sanchez E. (1977). Solutions in composite fuzzy relation equations: application to medical diagnosis in Brouwerian logic, Fuzzy Automata and Decision Processes (Gupta M., Saridis G. and Gaines B., eds.), North-Holland, New York, 221–234.
Schweizer B. and Sklar A. (1983). Probabilistic Metric Spaces,Elsevier.
Zhao C. (1987). On matrix equations in a class of complete and completely distributive lattices, Fuzzy Sets and Systems, 22, 303–320.
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De Baets, B. (2004). FREs: the ODEs and PDEs of the Fuzzy Modelling Paradigm. 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_8
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DOI: https://doi.org/10.1007/978-3-540-39675-8_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-05789-2
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