Fuzzy Equivalence Relations: Advanced Material

This Chapter presents an overview of the different aspects of the concept of fuzzy equivalence relation (FER) as the extension to the fuzzy framework of the classical idea of equivalence. In this setting, new concepts like generator, dimension and base arise naturally. On the other hand, these relations can be dualy related with some kind of generalized metrics that allows a metric-like study of their properties. This chapter starts introducing some general ideas extending, for any triangular continuous norm, the concept of similarity relation already presented in Chapter 4. Then, we explain different methods for its effective construction. The relationship between fuzzy equivalence relations and generalized metrics is also studied. Next, based on the representation theorem, the concepts of generator, dimension and base are introduced. The structure of the generators set is studied and some procedures for calculating bases are presented.

[1]  K Menger.  Probabilistic Theories of Relations. , 1951, Proceedings of the National Academy of Sciences of the United States of America.

[2]  C. Chang,et al.  Algebraic analysis of many valued logics , 1958 .

[3]  H. Poincaré.  La science et l'hypothèse , 1968 .

[4]  Lotfi A. Zadeh,et al.  Similarity relations and fuzzy orderings , 1971, Inf. Sci..

[5]  R. Lowen.  Fuzzy topological spaces and fuzzy compactness , 1976 .

[6]  J. Bezdek,et al.  Fuzzy partitions and relations; an axiomatic basis for clustering , 1978 .

[7]  L. Valverde,et al.  On Some Logical Connectives for Fuzzy Sets Theory , 1983 .

[8]  S. Ovchinnikov.  Representations of Transitive Fuzzy Relations , 1984 .

[9]  H. J. Skala,et al.  Aspects of vagueness , 1984 .

[10]  L. Valverde.  On the structure of F-indistinguishability operators , 1985 .

[11]  Janusz Kacprzyk,et al.  Management decision support systems using fuzzy sets and possibility theory , 1985 .

[12]  J. Jacas.  On the generators of T-indistinguishability operator. , 1988 .

[13]  U. Höhle.  Quotients with respect to similarity relations , 1988 .

[14]  P. García,et al.  Isomorphisms between de Morgan triplets , 1989 .

[15]  J. Jacas.  Similarity relations: the calculation of minimal generating families , 1990 .

[16]  Ulrich Höhle,et al.  The Poincare Paradox and the Cluster Problem , 1990 .

[17]  J. Jacas,et al.  Fuzzy values in fuzzy logic , 1991 .

[18]  U. Höhle.  M-valued Sets and Sheaves over Integral Commutative CL-Monoids , 1992 .

[19]  U. Höhle,et al.  Applications of category theory to fuzzy subsets , 1992 .

[20]  Frank Klawonn,et al.  Equality Relations as a Basis for Fuzzy Control , 1993 .

[21]  Joan Jacas,et al.  Fuzzy numbers and equality relations , 1993, [Proceedings 1993] Second IEEE International Conference on Fuzzy Systems.

[22]  E. Trillas,et al.  ON IMPLICATION AND INDISTINGUISHABILITY IN THE SETTING OF FUZZY LOGIC , 1993 .

[23]  Frank Klawonn,et al.  Foundations of fuzzy systems , 1994 .

[24]  Jacek M. Zurada,et al.  Computational Intelligence: Imitating Life , 1994 .

[25]  Joan Jacas,et al.  Fixed Points and Generators of Fuzzy Relations , 1994 .

[26]  Michael Wagenknecht.  On transitive solutions of fuzzy equations, inequalities and lower approximations of fuzzy relations , 1995, Fuzzy Sets Syst..

[27]  Marc Roubens,et al.  Structure of transitive valued binary relations , 1995 .

[28]  J. Recasens,et al.  Fuzzy T-transitive relations: eigenvectors and generators , 1995 .

[29]  Frank Klawonn,et al.  Similarity in fuzzy reasoning , 1995 .

[30]  Dionís Boixader,et al.  Similarity-based approach to defuzzification , 1997, Proceedings of 6th International Fuzzy Systems Conference.

[31]  J. Recasens,et al.  UPPER AND LOWER APPROXIMATIONS OF FUZZY SETS , 2000 .