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.