×

Openings and closures of fuzzy preorderings: theoretical basics and applications to fuzzy rule-based systems. (English) Zbl 1110.03048

Fuzzy relations, the full images \(R''A\) of a fuzzy set \(A\) under a fuzzy relation \(R\), as well as extensional fuzzy sets with respect to a fuzzy equivalence relation play crucial rôles in fuzzy modelling, particularly in fuzzy control.
In the context of some t-norm based logic the authors generalize the case of extensionality to fuzzy preorderings, and they consider a kind of dual to the full image \(R''A\) determined as \(\overline{R''{\overline{A}}}\) with the complement \(\overline{A}\) defined from the standard negation understood as “implies falsum”.
First they prove some general theoretical results, and then they apply their formal machinery to the discussion of the use of order related fuzzy modifiers in fuzzy control problems.

MSC:

03E72 Theory of fuzzy sets, etc.
93C42 Fuzzy control/observation systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bauer P. Klement E.P. Moser B. Leikermoser A. 1995 Modeling of control functions by fuzzy controllers Nguyen H.T. Sugeno M. Tong R.M. Yager R.R.Theoretical Aspects of Fuzzy ControlWiley New York Chapter 5 91 116 · Zbl 0851.93038
[2] Bellman R.E. Zadeh L.A. 1970 Decision making in a fuzzy environment Man. Sci. 17 4 141 164
[3] Bělohlávek R. 2002Fuzzy Relational Systems. Foundations and PrinciplesIFSR Int. Series on Systems Science and Engineering, Kluwer Academic New York
[4] Bodenhofer U. 2000 A similarity-based generalization of fuzzy orderings preserving the classical axioms Int. J. Uncertainty Fuzziness Knowledge-based Syst. 8 5 593 610 · Zbl 1113.03333
[5] Bodenhofer, U. (2002) ”Binary ordering-based modifiers,”in Proc. 9th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-based Systems · Zbl 1009.68150
[6] Bodenhofer U. 2003 Representations and constructions of similarity-based fuzzy orderings Fuzzy Sets Syst. 137 1 113 116 · Zbl 1052.91032
[7] Boixader D. Jacas J. Recasens J. 2000 Fuzzy equivalence relations: advanced material Dubois D. Prade H.Fundamentals of Fuzzy SetsThe Handbooks of Fuzzy Sets, Kluwer Academic Publishers Boston 261 290 · Zbl 0987.03047
[8] De Baets B. 2000 Analytical solution methods for fuzzy relational equations Dubois D. Prade H.Fundamentals of Fuzzy SetsThe Handbooks of Fuzzy Sets, Kluwer Academic Publishers Boston 291 340
[9] di Nola A. Pedrycz W. Sessa S. Sanchez E. 1991 Fuzzy relational equations theory as a basis for fuzzy modelling: An overview Fuzzy Sets Syst. 40 415 429 · Zbl 0727.04005
[10] Fodor J. Roubens M. 1994Fuzzy Preference Modelling and Multicriteria Decision SupportKluwer Academic Publishers Dordrecht
[11] Gottwald S. 1993Fuzzy Sets and Fuzzy LogicVieweg Braunschweig
[12] Gottwald S. 2001A Treatise on Many-Valued Logics, Studies in Logic and ComputationResearch Studies Press Baldock · Zbl 1048.03002
[13] DOI: 10.1007/978-94-011-5300-3 · doi:10.1007/978-94-011-5300-3
[14] Höhle U. Blanchard N. 1985 Partial ordering inL-underdeterminate sets Inf. Sci. 35 133 144 · Zbl 0576.06004
[15] Jacas J. 1988 On the generators ofT-indistinguishability operators Stochastica 12 49 63 · Zbl 0682.04004
[16] Jacas J. Recasens J. 1995 FuzzyT-transitive relations: eigenvectors and generators Fuzzy Sets Syst. 72 147 154 · Zbl 0844.04006
[17] Kerre, E.E., ed. (1993)Introduction to the Basic Principles of Fuzzy Set Theory and Some of its Applications
[18] Klawonn F. 1993 Mamdani’s model in the view of equality relations Proc. EUFIT 93 1 364 369
[19] Klawonn F. Castro J.L. 1995 Similarity in fuzzy reasoning Mathware Soft Comput. 3 2 197 228 · Zbl 0859.04006
[20] Klawonn F. Kruse R. 1993 Equality relations as a basis for fuzzy control Fuzzy Sets Syst. 54 2 147 156 · Zbl 0785.93059
[21] Klawonn F. Gebhardt J. Kruse R. 1995 Fuzzy control on the basis of equality relations–with an example from idle speed control IEEE Trans. Fuzzy Syst. 3 336 356
[22] Klement E.P. Mesiar R. Pap E. 2000Triangular NormsTrends in Logic, Kluwer Academic Publishers Dordrecht
[23] Kóczy L.T. Hirota K. 1993 Ordering, distance and closeness of fuzzy sets Fuzzy Sets Syst. 59 3 281 293 · Zbl 1002.03530
[24] Kóczy L.T. Hirota K. 1997 Size reduction by interpolation in fuzzy rule bases IEEE Trans. Syst. Man Cybern. 27 1 14 25
[25] Kruse R. Gebhardt J. Klawonn F. 1994Foundations of Fuzzy SystemsWiley New York
[26] Lowen R. 1980 Convex fuzzy sets Fuzzy Sets Syst. 3 291 310 · Zbl 0439.52001
[27] Miyakoshi M. Shimbo M. 1985 Solutions of composite fuzzy relational equations with triangular norms Fuzzy Sets Syst. 16 53 63 · Zbl 0582.94031
[28] Sanchez E. 1984 Solution of fuzzy equations with extended operations Fuzzy Sets Syst. 12 237 248 · Zbl 0556.04001
[29] Valverde L. 1985 On the structure ofF-indistinguishability operators Fuzzy Sets Syst. 17 3 313 328 · Zbl 0609.04002
[30] Zadeh L.A. 1965 Fuzzy sets Inf. Control 8 338 353 · Zbl 0139.24606
[31] Zadeh L.A. 1971 Similarity relations and fuzzy orderings Inf. Sci. 3 177 200 · Zbl 0218.02058
[32] Zadeh L.A. 1973 Outline of a new approach to the analysis of complex systems and decision processes IEEE Trans. Syst. Man Cybern. 3 1 28 44 · Zbl 0273.93002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.