Ana Pradera

Some properties of overlap and grouping functions and their application to image thresholding

In this paper we study under which conditions overlap and grouping functions satisfy some commonly demanded properties such as migrativity or homogeneity. We also recall that the convex combination of overlap (grouping) functions is a new overlap (grouping) function. This property allows us to achieve a consensus between different methods that solve certain problems by means of these functions. We also show one application of this property in image processing.

A general class of triangular norm-based aggregation operators: quasi-linear T–S operators

Abstract This paper generalizes the well-known exponential and linear convex T–S aggregation operators into a wider class of compensatory aggregation operators, built as the composition of an arbitrary quasi-linear mean with a t-norm and a t-conorm, which are called quasi-linear T–S operators. These new operators are compared with other existing ones, and their main properties, such as the existence of neutral or annihilator elements, are studied. In particular, the self-duality property is investigated, and a characterization of an important family of self-dual quasi-linear T–S operators is provided.

Advances in Fuzzy Implication Functions

Fuzzy implication functions are one of the main operations in fuzzy logic. They generalize the classical implication, which takes values in the set {0,1}, to fuzzy logic, where the truth values belong to the unit interval [0,1]. These functions are not only fundamental for fuzzy logic systems, fuzzy control, approximate reasoning and expert systems, but they also play a significant role in mathematical fuzzy logic, in fuzzy mathematical morphology and image processing, in defining fuzzy subsethood measures and in solving fuzzy relational equations. This volume collects 8 research papers on fuzzy implication functions. Three articles focus on the construction methods, on different ways of generating new classes and on the common properties of implications and their dependencies. Two articles discuss implications defined on lattices, in particular implication functions in interval-valued fuzzy set theories. One paper summarizes the sufficient and necessary conditions of solutions for one distributivity equation of implication. The following paper analyzes compositions based on a binary operation * and discusses the dependencies between the algebraic properties of this operation and the induced sup-* composition. The last article discusses some open problems related to fuzzy implications, which have either been completely solved or those for which partial answers are known. These papers aim to present todays state-of-the-art in this area.

Generalization of the weighted voting method using penalty functions constructed via faithful restricted dissimilarity functions

In this paper we present a generalization of the weighted voting method used in the exploitation phase of decision making problems represented by preference relations. For each row of the preference relation we take the aggregation function (from a given set) that provides the value which is the least dissimilar with all the elements in that row. Such a value is obtained by means of the selected penalty function. The relation between the concepts of penalty function and dissimilarity has prompted us to study a construction method for penalty functions from the well-known restricted dissimilarity functions. The development of this method has led us to consider under which conditions restricted dissimilarity functions are faithful. We present a characterization theorem of such functions using automorphisms. Finally, we also consider under which conditions we can build penalty functions from Kolmogoroff and Nagumo aggregation functions. In this setting, we propose a new generalization of the weighted voting method in terms of one single variable functions. We conclude with a real, illustrative medical case, conclusions and future research lines.

Searching for the roots of non-contradiction and excluded-middle

This paper tries to obtain frameworks in which one can prove as theorems, and with few assumptions, the laws of non-contradiction (NC) and excluded-middle (EM), for a large class of very general systems including orthocomplemented and De Morgan lattices, as well as the theories of L -fuzzy sets. Its goal is nothing else than to begin with a search for the algebraic roots of these laws. For such a goal, and similarly to ancient Aristotelian logic, the laws are referred to self-contradiction and not to falsity as it is done in modern logics. It is additionally shown that there are some structures where it is possible to find special operations that separate those operations that verify NC from those verifying EM.

AN OVERVIEW ON THE CONSTRUCTION OF FUZZY SET THEORIES

This paper analyzes the main issues involved in the construction of Fuzzy Set Theories. It reviews both standard solutions (based on the well-known triangular norms and conorms) as well as less the conventional proposals that provide alternative views on, for example, the definition of fuzzy connectives or the study of their properties.

On the aggregation of some classes of fuzzy relations

This paper deals with the aggregation of fuzzy relations under a closure constraint, that is, it studies how to combine a collection of fuzzy relations that present some common Properties in order to obtain a single one with the same properties. This aggregation problem is addressed for two important classes of closely related fuzzy relations: generalized distances and indistinguishability operators.

On a class of Fuzzy Set Theories

This paper studies the main properties of a broad class of Fuzzy Set Theories that we call Minimal Fuzzy Set Theories. We investigate the fulfillment of well-known properties, such as idempotency or the existence of absorbent elements. The paper shows, in particular, that these theories do always verify the Kleenes law, as well as the Non-Contradiction and Excluded-Middle principles when these are understood as in ancient logic.

Double aggregation operators

This paper introduces the so-called double aggregation operators, which allow to combine two input lists of information, coming from different sources, into a single output. Several basic properties of these operators, such as symmetry, neutral and annihilator elements, idempotency, etc., are studied and illustrated with some examples. In addition, the classes of additive and comonotone additive double aggregation operators are characterized.

Consequences and conjectures in preordered sets

In a preordered set, or preset, consequence operators in the sense of Tarski, defined on families of subsets, are introduced. From them, the corresponding sets of conjectures, hypotheses, speculations and refutations are considered, studying the relationships between these sets and those previously defined on ortholattices. All the concepts introduced are illustrated with three particular consequence operators, whose behavior is studied in detail. The results obtained are applied to the case of fuzzy sets endowed with the usual pointwise ordering.