Joan Jacas

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.

On contradiction in fuzzy logic

Abstract We clarify which space of functions in [0, 1]E would be reasonable in fuzzy logic in order to avoid self-contradiction.

Similarity relations: the calculation of minimal generating families

Abstract The structure of similarity relations is studied from the point of view of the representation theorem of T-indistinguishabilities. The generators are characterized, the concept of dimension is introduced and algoriths in oder to calculate a minimal generating family in the finite case are developed.

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.

Aggregation of T‐transitive relations

This article studies the aggregation of transitive fuzzy relations. We first find operators that preserve transitivity and then extend the results to aggregating operators. As special cases, means and some kind of suitable ordered weighted averaging (OWAs) are used to aggregate transitive fuzzy relations with respect to an Archimedean t‐norm. Three families of transitive relations that allow us to modify the entries of a given relation R continuously towards the smallest and the greatest ones in our universe are given. Aggregation of nonfinite families of transitive relations also is studied and applied to calculate the degree of inclusion or similarity of fuzzy quantities (fuzzy subsets of an interval of the real line).

Fuzzy numbers and equality relations

A general approach to the concept of a fuzzy number associated to a generalized equality on the real line is given. As a result, the use of triangular and trapezoidal fuzzy numbers, among other types, is justified and a representation theorem that allows the construction of indistinguishability operators on the real line based on t-norms is presented. The families of fuzzy numbers invariant under translations are characterized.<<ETX>>

Maps and isometries between indistinguishability operators

Abstract In this paper, some geometric aspects of indistinguishability operators are studied by using the concept of morphism between them. Among all possible types of morphisms, the paper is focused on the following cases: Maps that transform a T-indistinguishability operator into another of such operators with respect to the same t-norm T and maps that transform a T-indistinguishability operator into another one of such operators with respect to a different t-norm T′. The group of isometries of a given T-indistinguishability operator is also studied and it is determined for the case of one-dimensional operators, in particular for the natural indistinguishability operators ET on [0, 1]. Finally, the indistinguishability operators invariant under translations on the real line are characterized.

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 logical connectives for a fuzzy set theory with or without nonempty self-contradictions

We show how noncontradiction and excluded‐middle laws can hold in a fuzzy logic where the concepts of contradiction and incompatibility are clearly distinguished. However, if we want to avoid self‐contradictions in fuzzy set theory then one needs to consider only fuzzy sets with supremum 1 and infimum 0. We clarify in this new context (on which the applications usually take place) which logical connectives make sense. ©2000 John Wiley & Sons, Inc.

One-dimensional indistinguishability operators

The main result of the paper is an algorithm that allows us to decide when a given fuzzy relation is a one-dimensional T-indistinguishability operator for some archimedean t-norm T (in the sense of the Representation Theorem of Valverde (Fuzzy Sets and Systems 17 (1985) 313–328)). The algorithm also finds all t-norms with this property.