Pedro J. Burillo

Vague sets are intuitionistic fuzzy sets

Abstract We recapitulate the definition given by Atanassov (1983) of intuitionistic fuzzy sets as well as the definition of vague sets given by Gau and Byehrer (1993) and see that both definitions coincide.

Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets

Abstract We recall the definitions of intuitionistic fuzzy sets and interval-valued fuzzy sets with the relation between these sets established by K. Atanassov. We define the distance measure between intuitionistic fuzzy sets and we give an axiom definition of intuitionistic fuzzy entropy and a theorem which characterizes it. Finally, we study a very special entropy and recall that all we have done for intuitionistic fuzzy sets is also good for interval-valued fuzzy sets.

Correlation of interval-valued intuitionistic fuzzy sets

Abstract In this paper we introduce the concepts of correlation and correlation coefficient of interval-valued intuitionistic fuzzy sets and we study their first properties. We also introduce two decomposition theorems of the correlation of interval-valued intuitionistic fuzzy sets, one in terms of the correlation of interval-valued fuzzy sets and the entropy of the intuitionistic fuzzy sets, and the other theorem in terms of the correlation of intuitionistic fuzzy sets.

Automorphisms, negations and implication operators

In this paper we study the conditions in which the implication operators satisfy the property I(x,c(x)) = c(x) for all x ∈ [0, 1], c being any strong negation. This study has led us to present different implication operator characterization theorems from automorphisms, obtaining a theorem similar to the one presented by Smets and Magrez (Internat. J. Approx. Reason. 1 (1987) 327), in which the strong negation c used is not generated by the same automorphism that generates the implication.

Structures on intuitionistic fuzzy relations

Abstract In this paper we study the structures of the intuitionistic fuzzy relations. We analyse the existent relations between the structures of a relation and the structures of its complementary one. We finish characterizing certain structures of intuitionistic relations according to the structures of two concrete fuzzy relations.

Mathematical analysis of interval-valued fuzzy relations: application to approximate reasoning

Abstract In this paper, interval-valued fuzzy relations between sets X and Y are introduced as fuzzy subsets of the cartesian product X×Y , and t-norms and t-conorms are chosen in such a way that as many properties of relations in 2-valued logic are preserved. Besides, we will see that if we require certain reasonable properties, including distributivity, then we end up with the only possible choice: min and max. Finally, as an example, a method of inference in approximate reasoning for the one-dimensional case based on interval-valued fuzzy sets is considered and discussed using the idea of interval-valued fuzzy relations.

Construction theorems for intuitionistic fuzzy sets

Abstract We will begin presenting a point operator which allows us to associate a family of fuzzy sets with each intuitionistic fuzzy set. We will then expose two construction theorems of intuitionistic fuzzy sets from one fuzzy set and a theorem which allows us to construct an intuitionistic fuzzy set from two fuzzy sets. We will also prove that it is possible to recover the fuzzy sets used in the construction from the intuitionistic fuzzy set constructed by means of different operators. All the theorems shown let us generate intuitionistic fuzzy sets with fixed beforehand entropy and it is easy to construct algorithms to implement these processes. We conclude by proving a theorem relative to the way of recuperating from an intuitionistic fuzzy set built with two fuzzy sets, the two fuzzy sets used in its construction and a third fuzzy set included among them. Finally, we expose the importance the construction theorems will have in our future research relative to the obtainment of the conclusion of the generalized modus ponens when the antecedents of the generalized modus ponens are perturbed in the way indicated in the theorems developed.

Inclusion grade and fuzzy implication operators

In this paper we propose a family of fuzzy implication operators, which generalises the Luckasiewiczs one, and to analyse the impacts of Smets and Magrez properties on these operators. The result of this approach will be a characterisation of a proposed family of inclusion grade operators that satisfies the axioms of Divyendu and Dogherty.

Connecting fuzzy preorders, fuzzy consequence operators and fuzzy closure and co-closure systems

Abstract The relations between fuzzy preorders, fuzzy consequence operators, fuzzy closure systems and co-closure systems lattice structures are studied. Given a continuous t-norm ∗ , we will prove that the complete lattice of all fuzzy ∗ -preorders is isomorphic with a subfamily of the complete lattice of fuzzy consequence operators. Moreover, we will show that it is possible to embed both structures in the class of fuzzy closure systems by a lattice dual monomorphism between semilattices. A solution to the following open problem: “when two families of fuzzy possible valuations produce the same fuzzy implication relation” is given. Concept of ∗ -interior operator of a fuzzy consequence operator is suggested and a characterization of the previous open problem through ∗ -interior operator is shown.

On the relation between fuzzy preorders and fuzzy consequence operators

The purpose of this paper is to analyze the operators induced by relations and conversely the relations induced by operators in fuzzy logic. Given a t-norm * and given a non-empty universal set X, it is well known that if R is a fuzzy *-preorder on X then the operator induced by R, , is a fuzzy consequence operator (FCO). In fact, is a *-coherent FCO. It is also known that if C is a *-coherent FCO then the relation induced by C, RC, is a fuzzy *-preorder. We explore the *-coherence axiom because we do not know in the literature any example of non-coherent operator. Then, several families of these operators will be shown. Moreover we prove that the equivalence between fuzzy preorders and fuzzy consequence operators is held in only one way. As a result, a characterization of the *-preorder concept using the induced operator is given. Also some characterizations which show when an operator induces a *-preorder are proved. Finally, we will show that the characterization of the operators induced by relations given for finite universes cannot be generalized for infinite universes.