Jaume Suñer

A characterization of residual implications derived from left-continuous uninorms

In this paper, a set of axioms is given that characterizes those functions I:[0,1]^2->[0,1] for which a left-continuous uninorm U exists in such a way that I is the residual implication derived from U. A characterization for the particular cases when U is representable, when U is continuous in ]0,1[^2 and when U is idempotent are also given.

Copula-like operations on finite settings

This paper deals with discrete copulas considered as a class of binary aggregation operators on a finite chain. A representation theorem by means of permutation matrices is given. From this characterization, we study the structure of associative discrete copulas and a theorem of decomposition of any discrete copula in terms of associative discrete copulas is obtained. Finally, some aspects concerning their extension to copulas are dealt with.

Generation of weighting triangles associated with aggregation functions

In this work, we present several ways to obtain different types of weighting triangles, due to these types characterize some interesting properties of Extended Ordered Weighted Averaging operators, EOWA, and Extended Quasi-linear Weighted Mean, EQLWM, as well as of their reverse functions. We show that any quantifier determines an EOWA operator which is also an Extended Aggregation Function, EAF, i.e., the weighting triangle generated by a quantifier is always regular. Moreover, we present different results about generation of weighting triangles by means of sequences and fractal structures. Finally, we introduce a degree of orness of a weighting triangle associated with an EOWA operator. After that, we mention some results on each class of triangle, considering each one of these triangles as triangles associated with their corresponding EOWA operator, and we calculate the ornessof some interesting examples.

Dual representable aggregation functions and their derived S-implications

In this paper dual representable aggregation functions (DRAFs) are introduced and studied. After giving a representation theorem for them, it is proved that they can be viewed as a nonassociative generalization of nilpotent t-conorms, some basic properties are proved and some examples are given. On the other hand, using DRAFs a new kind of strong implications are derived and some usual properties are studied for this new class of implications. In particular, it is shown that they have an easy structure always divided into three parts depending on the strong negation.

Sklar's Theorem in Finite Settings

This paper deals with the well-known Sklars theorem, which shows how joint distribution functions are related to their marginals by means of copulas. The main goal is to prove a discrete version of this theorem involving copula-like operators defined on a finite chain, that will be called discrete copulas. First, the idea of subcopulas in this finite setting is introduced and the problem of extending a subcopula to a copula is solved. This is precisely the key point which allows to state and prove the discrete version of Sklars theorem.

ON DISTRIBUTIVITY AND MODULARITY IN DE MORGAN TRIPLETS

We study some properties of De Morgan triplets. Firstly, we introduce submodular De Morgan triplets and we study its relationships with subdistributive ones. Moreover, we characterize them both in the strict and non strict archimedean cases. Secondly, we introduce the concepts of modularity, distributivity and (S, T)-distributivity degrees and we give some general results. Afterwards we apply these concepts to two particular cases, Lukasiewicz triplets and a kind of strict De Morgan triplets (Product triplets). For the Lukasiewicz ones we prove that all three degrees range over (0, 1/2]. For Product triplets, a recipe to calculate these degrees is given. In particular we present examples with distributivity degree r for all r ∈ (0, 1) and examples with (S, T)-distributivity and modularity degrees taking all values in (0, 3/4].

New types of contrapositivisation of fuzzy implications with respect to fuzzy negations

Among the most studied properties of fuzzy implication functions, the so-called contrapositive symmetry or law of contraposition with respect to a fuzzy negation is of special interest because it is a key property in approximate reasoning, deductive systems and formal methods of proof. Even when contraposition is not satisfied, there exist some methods for modifying an implication function with the aim that the new implication satisfies this property, these methods are known as contrapositivisations. In this paper some new methods of contrapositivisation with respect to any fuzzy negation are presented and the properties preserved by these new methods are studied. Along this study we will see that these new methods not only preserve the usual properties preserved by the already known methods, but they also have some additional properties.

Multi-dimensional aggregation of fuzzy numbers through the extension principle

In this paper we propose the problem of obtaining a procedure to aggregate fuzzy numbers in such a way that the output is also a fuzzy number. To do this, we use the Zadehs Extension Principle applied to multi-dimensional numerical functions which satisfy certain conditions, obtaining multi-dimensional aggregation functions on the lattice of fuzzy numbers. Special attention is given to the case of trapezoidal fuzzy numbers.

A New Look on Fuzzy Implication Functions: FNI -implications

Fuzzy implication functions are used to model fuzzy conditional and consequently they are essential in fuzzy logic and approximate reasoning. From the theoretical point of view, the study of how to construct new implication functions from old ones is one of the most important topics in this field. In this paper a construction method of implication functions from a t-conorm S (or any disjunctive aggregation function F), a fuzzy negation N and an implication function I is studied. Some general properties are analyzed and many illustrative examples are given. In particular, this method shows how to obtain new implications from old ones with additional properties not satisfied by the initial implication function.

On distances derived from t-norms

The construction of distances from t-norms is studied. The generation of distances from a t-norm and its dual t-conorm is extended to the more general case of any t-norm and t-conorm. The open problem of characterizing those t-norms that define distances is solved in the case that the t-norms have the same zero region as the Lukasiewicz one.