Dionís Boixader

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.

Extensionality based approximate reasoning

The aim of this paper is to analyze Approximate Reasoning (AR) through extensionality with respect to the natural T-indistinguishability operator, by considering the indistinguishability level between fuzzy sets as a formal measure of its degree of similarity, resemblance or closeness, having in all these terms an intuitive meaning.

Transitive closure and betweenness relations

Indistinguishability operators fuzzify the concept of equivalence relation and have been proved a useful tool in theoretical studies as well as in different applications such as fuzzy control or approximate reasoning. One interesting problem is their construction. There are different ways depending on how the data are given and on their future use. In this paper, the length of an indistinguishability operator is defined and it is used to relate its generation via max-T product and via the representation theorem when T is an Archimedean t-norm. The link is obtained taking into account that indistinguishability operators generate betweenness relations. The study is also extended to decomposable operators.

On the relationship between T-transitivity and approximate equality

The paper discusses the suitability of T-transitivity for modelling approximate equality. It explains how T-transitivity, a property suitable for modelling vagueness, can still be used to handle the imprecision inherent in approximate equality. Further remarks on resemblance relations are made to clarify how far they improve traditional fuzzy equivalence relations.

Approximate fuzzy preorders and equivalences

Although Fuzzy Preorders and Fuzzy Equivalences are well established types of fuzzy relations, they are defined in terms of properties to be fulfilled in a crisp way. In this paper we relax those requirements by making them gradual in some way. As a result, a number of relations which are not strictly reflexive, symmetric or transitive with respect to any t-norm T, would be regarded as (Approximate) Fuzzy Preorders or Equivalences.

T-indistinguishability Operators and Approximate Reasoning via CRI

This section is devoted to review some concepts concerning T-indistinguishability operators which are used in different contexts like control processes [Klawonn and Kruse, 1993], CADG environments [Jacas et al., 1995] and cluster analysis [Kandel, 1992]. An overview on this topic can be found in [Dubois and Prade, 1994].

Searching for meaning on defuzzification

Defuzzification is an essential problem in fuzzy systems that it is always solved in a heuristic way. The aim of this work is to give a semantic interpretation to this process with the help of indistinguishability operators

Similarity-based approach to defuzzification

Defuzzification is an essential problem in fuzzy systems that it is always solved in a heuristic way. The aim of this work is to give a semantic interpretation to this process with the help of indistinguishability operators.

Powers with Respect to t-Norms and t-Conorms and Aggregation Functions

Aggregation functions A stable with respect to powers of t-norms and t-conorms (i.e.: satisfying \(A(x^{(r)},y^{(r)})=(A(x,y))^{(r)}\)) where \(x^{(r)}\) is the r-th power of \(x \in [0,1]\) with respect to a t-norm or t-conorm) are characterized. This result generalizes the characterization of power stable aggregation functions in [5].

A characterization of the columns of a T-indistinguishability operator

Two different ways to characterize the columns of a operators are studied. The first one by giving explicit conditions, and second one by defining a new operator (/spl Lambda//sub E/) that has columns as fixed points. Some relevant properties of two known operators (/spl phi//sub E/ and /spl psi//sub E/) are also reviewed, in order to compare them with /spl Lambda//sub E/.