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## Publications

Publications (53)

When a decision maker is asked to compare a set of alternatives, it may happen that the information provided is incomplete because she has no time to compare all the options or is unable to compare some alternatives against others. This contribution departs from an incomplete fuzzy weak preference relation by completing it on a consistent way with...

Characterizing the degree of similarity or difference between two sets is a very important topic, since it has many applications in different areas, including image processing or decision making. Several studies have been done about the comparison of fuzzy sets and its extensions, in particular for interval-valued fuzzy sets. However, in most of th...

We investigate two related problems of categorization of alternatives. We are particularly concerned about liberalism and dictatorship, two noteworthy and opposing principles, in the context of fuzzy expressions of the opinions. First we study the problem of self-distributing the individuals in a society among fixed classes or categories. We identi...

We focus on the relationships among some consistency axioms in the framework of fuzzy choice functions. In order to help disclose the role of the t-norm in existing analyses, we start to study the situation that arises when we replace the standard t-norm with other t-norms. Our results allow us to conclude that unless we impose further structure on...

Probabilistic and fuzzy choice functions are used to describe decision situations in which some degree of uncertainty or imprecision is involved. We propose a way to equate these two formalisms by means of residual implication operations. Furthermore, a set of new rationality conditions for probabilistic choice functions is proposed and proved to b...

The link between information theory and fuzzy logic has been proven in several previous papers. From this starting point, we propose here a review about the concept of divergence measures, which was proposed as a tool for comparing two fuzzy sets. The initial definition comes from the ideas behind the classical concept of divergence between two pro...

The link between information theory and fuzzy logic has been proven in several previous papers. From this starting point, we propose here a review about the concept of divergence measures, which was proposed as a tool for comparing two fuzzy sets. The initial definition comes from the ideas behind the classical concept of divergence between two pro...

We analyze the existence of fuzzy sets of a universe that are convex with respect to certain particular classes of fusion operators that merge two fuzzy sets. In addition, we study aggregation operators that preserve various classes of generalized convexity on fuzzy sets. We focus our study on fuzzy subsets of the real line, so that given a mapping...

Fuzzy relations and reciprocal relations are two popular tools for representing degrees of preference. It is important to note that they carry a different semantics and cannot be equated directly. We propose a simple transformation based on implication operators that allows to establish a one-to-one correspondence between both formalisms. It sets t...

In this paper MN-convex and MN-concave functions are examined and compared for particular cases of M and N. Characterization of pairs for the family of MN-convex (MN-concave) functions consisting of all functions and partial characterization of pairs for the family of MN-convex functions consisting of all constant functions will be presented. The c...

Partitions are at the basis of many processes as classification. They are classically defined in the context of crisp sets. However, partitions based on fuzzy sets have been proven to be more useful in real-life problems. This has motivated several different proposals to extend the definition of partition to the fuzzy sets context. Nevertheless, fu...

Genetic algorithms can be used to construct knowledge bases. They are based on the idea of “survival of the fittest” in the same way as natural evolution. Nature chooses the fittest ones in real life. In artificial intelligence we need a method that carries out the comparison and choice. Traditionally, this choice is based on fitness functions. Eac...

Probabilistic and fuzzy choice theory are used to describe decision situations in which a certain degree of imprecision is involved. In this work we propose a correspondence between probabilistic and fuzzy choice functions, based on implication operators. Given a probabilistic choice function a fuzzy choice function can be constructed and, furtherm...

Probabilistic and fuzzy choice theory are used to describe decision situations in which a certain degree of imprecision is involved. In this work we propose a correspondence between probabilistic and fuzzy choice functions, based on implication operators. Given a probabilistic choice function a fuzzy choice function can be constructed and, furtherm...

Stochastic orders are methods that allow the comparison of random quantities. One of the most used stochastic orders is stochastic dominance. This method is based on the direct comparison of the cumulative distribution functions of the random variables, and it is characterized by comparing the expectations of the adequate transformation of the vari...

The Arrow-Sen Theorem establishes the equivalence between different rationality conditions for a choice function. In this contribution we deal with fuzzy versions of these rationality conditions and we study the connection between them. We recall results found in the literature and we prove that they are valid under weaker conditions. We also prese...

Choices among alternatives in a set can be expressed in three different ways: by means of choice functions, by means of preference relations or using choice probabilities. The connection between the two first formalizations has been widely studied in the literature, both in the crisp or classical context and in the setting of fuzzy relations. Howev...

For crisp relations the transitivity of a relation and the negative transitivity of its dual are equivalent conditions. Particularly, a crisp complete large preference relation is transitive if and only if its associated strict preference relation is negatively transitive. In this contribution we focus on one of those implications for fuzzy relatio...

The Arrow-Sen theorem is one of the most important results concerning rationality of choice functions. It states that under suitable hypothesis, several definitions of rationality given by different authors can be considered equivalent. Following the same spirit, other authors have proved that further definitions can also be considered equivalent t...

Different definitions of the concept of a fuzzy semiorder are compared. It is proved that their α-cuts are crisp binary relations that may fail to be Ferrers and semitransitive, in general. Consequently, we analyze the preservation of semiorders when coming back from the fuzzy to the crisp setting using α-cuts. In the final sections, a discussion i...

For crisp relations the concept of a semi-order can be stated in a number of equivalent ways. When trying to extend this concept to the fuzzy setting, we observe that the (generalized) definitions fail to be equivalent. In this contribution, we discuss which is the most natural definition of a fuzzy semi-order, and study the hierarchy among the alt...

Given a set of alternatives we consider a fuzzy relation and a probabilistic relation defined on such a set. We investigate the relation between the T-transitivity of the fuzzy relation and the cycle-transitivity of the associated probabilistic relation. We provide a general result, valid for any t-norm and we later provide explicit expressions for...

We study the relationship between the Ferrers property and the notion of interval order in the context of valued relations.
Given a crisp preference structure without incomparability, the strict preference relation satisfies the Ferrers property
if and only if the associated weak preference relation does. These conditions characterize a total inter...

Fuzzy choice functions obtained from preference relations have been recently used to develop automated negotiation systems. This contribution contains a theoretical study of coherence conditions in the process of creating a fuzzy choice function from a preference relation. In particular, the role of the acyclicity property of fuzzy preference relat...

In the context of crisp or classical relations, one may find several alternative characterizations of the concept of a total
preorder. In this contribution, we first discuss the way of translating those characterizations to the framework of fuzzy
relations. Those new properties depend on t-norms. We focus on two important families of t-norms, namel...

A (crisp) binary relation is transitive if and only if its dual relation is negatively transitive. In preference modelling,
if a weak preference relation is complete, the associated strict preference relation is its dual relation. It follows from
here this well-known result: given a complete weak preference relation, it is transitive if and only if...

This is a first approach to the study of the connection between fuzzy preference relations and fuzzy choice functions. In
particular we depart from a fuzzy preference relation and we study the conditions it must satisfy in order to get a fuzzy
choice function from it. We are particulary interested in one function: G-rationalization. We discuss the...

In some cases the fitness value of a knowledge base is not completely determined, but just bounded in an interval. In this case the fitness value is modelled by a random variable. Thus the comparison of random variables allows to compare the fitness values when they are not completely determined. In this contribution we consider a quite new proposa...

Classically, the comparison of random variables have been done by means of a crisp order, which is known as stochastic dominance. In the last years, the classical stochastic dominance have been extended to a graded version by means of a probabilistic relation. In this work we propose different ways of measuring the gradual order among random variab...

Classical complete preorders can be characterized in several ways. However, when we work with complete fuzzy preorders this
equivalences do not hold in general. In previous works we have proven some connections among them when using the minimum and
the Łukasiewicz t-norms. In this contribution we generalize the study and we work with two important...

We study the transitivity of fuzzy preference relations, often considered as a fundamental property providing coherence to
a decision process. We consider the transitivity of fuzzy relations w.r.t. conjunctors, a general class of binary operations
on the unit interval encompassing the class of triangular norms usually considered for this purpose. H...

In this paper, we isolate an interesting transformation of binary aggregation functions from a result on the propagation of transitivity in additive preference structures. This transformation is based on a projection technique and results in a smaller (or equal) binary aggregation function. The transformation is idempotent and acts internally on th...

The concept of (classical) complete preorder can be characterized in several ways. In previous works we have studied whether complete fuzzy preorders can be characterized by the same properties as in the crisp case. We have proven that this is not usually the case. We have studied five possible characterizations and we have proven that only one sti...

Transitivity is a very important property in order to provide coherence to a preference relation. Usually, t-norms are con- sidered to define the transitivity of fuzzy relations. In this paper we deal with conjunctors, a wider family than t-norm, to define the tran- sitivity. This more general definition allows to impove the results found in the li...

Complete pre-orders can be characterized in terms of the transitivity of the corresponding strict preference and indifference relations. In this paper, we investigate this characterization in a fuzzy setting. We consider two types of completeness (weak completeness and strong completeness) and decompose a fuzzy pre-order by means of an indifference...

In this paper we consider a decision maker who shows his/her preferences for different alternatives through a finite set of ordinal values. We analyze the problem of consistency taking into account some transitivity properties within this framework. These properties are based on the very general class of conjunctors on the set of ordinal values. Ea...

In this paper we study the behaviour of a kind of partitions formed by fuzzy sets, the ϵ-partitions, with respect to three important operations: refinement, union and product of partitions. In the crisp set theory, the previous operations lead to new partitions: every refinement of a partition is also a partition; the union of partitions of disjoin...

Fuzzy pre-orders (reflexive and min-transitive fuzzy relations) constitute an important class of fuzzy relations. By means of an indifference generator, a fuzzy pre-order can be decomposed additively into two parts: an indifference relation and a strict preference relation. When using a Frank t-norm as indifference generator, we fully characterize...

We study different characterizations for the transitivity of reflexive fuzzy rela-tions under completeness. We depart from some properties equivalent to the definition of a crisp preorder. We study if the same conditions characterize the transitivity of reflexive relations in the fuzzy set context. Instead of restrict-ing ourselves to t-norms, we w...

Transitivity is a fundamental notion in preference modelling. In this work we study this property in the framework of additive fuzzy preference structures. In particular, we depart from a large preference relation that is transitive w.r.t. the nilpotent minimum t-norm and decompose it into an indifference and strict preference relation by means of...

In this paper, we study the Ferrers property of relations in the context of fuzzy preference modelling. A logical approach leads us to the notion of T-Ferrers relations, while a relational approach brings us to T-biorders. We characterize the t-norms for which both notions coincide. We also describe the kind of completeness exhibited by re∞exive T-...

Transitivity is an essential property in preference modelling. In this work we study this property in the framework of fuzzy prefe- rence structures. In particular, we discuss the relationship between the transitivity of a fuzzy large preference relation R and the transitivity of the fuzzy indifierence relation I obtained from R by some of the most...

The aim of this work is to study an essential property in the framework of fuzzy preference structures, which is the transitivity. In particular, we discuss the relationship between the transitivity of a fuzzy large preference relation R and the transitivity of the fuzzy strict pref- erence relation P obtained from R when every pair of elements can...

Interval orders play a significant role in preference modeling since they do not impose the transitivity of the in-difference relation. For crisp rela-tions this concept can be expressed in different equivalent ways. For fuzzy relations these definitions de-pend on the t-norm we employ and they are no longer equivalent. In this work we study the co...

The fitness value of a knowledge base (KB) can be unknown and only some imprecise in-formation about it can be obtained. In some cases this information is given by means of an interval where we know the fitness is con-tained. Thus, the comparison of two ran-domly distributed intervals is necessary in this context in order to be able to determine th...

RESUMEN El orden de intervalos es una conocida estructura de preferencia en la mode-lización clásica de la preferencia y su importancia en muchas situaciones es ampliamente reconocida. Sin embargo, la restricción a relaciones clásicas es demasiado restrictiva en la práctica, lo que llevó a introducir lo ordenes de intervalos multivaluados. Puesto q...

Interval orders admit five equivalent ways to be expressed in the case of dealing with crisp sets. Each one of those five definitions can be translated to fuzzy sets. However, the counterparts for fuzzy relations of those five properties are not equivalent. In this con-tribution we consider each one of those five notions and we study the connection...