Hélida Salles Santos

T-Norms on Bounded Lattices: t-norm Morphisms and Operators

Triangular norms or t-norm, in short, and automorphisms are very useful to fuzzy logics in the narrow sense. Moreover, these notions are usually limited to the set [0,1]. In this paper we will consider a well known generalization of the t-norm for arbitrary bounded lattices and provide a generalization of automorphism notion for this same structure. We consider several typical lattice constructors and introduce versions of them for t-norms and morphisms. We also analyze some properties of these constructions.

A study of (T,N)-implications and its use to construct a new class of fuzzy subsethood measure

Abstract It is not hard to see that implications can be obtained by means of other operators. To name a few, we can mention (S,N)-, R- and QL-implications, which has been widely investigated. The aim of this work is to study a fuzzy implication called (T,N)-implication, obtained from the composition of a fuzzy negation and a t-norm. It is also discussed the conditions under which such functions preserve the main properties of fuzzy implications, such that some are related to the laws of contraposition. We also prove under what conditions our implication satisfies the following iterated boolean-like law in fuzzy logic: I ( x , I ( x , y ) ) = I ( x , y ) for all x and y in the unit interval. Finally, we provide a way to construct fuzzy subsethood measures by means of (T,N)-implications.

Bounded lattice t-norms as an interval category

Triangular norms or t-norms, in short, and automorphisms are very useful to fuzzy logics in the narrow sense. However, these notions are usually limited to the set [0, 1]. In this paper we will consider a generalization of the t-norm notion for arbitrary bounded lattices as a category where these generalized t-norms are the objects and generalizations of automorphisms are the morphisms of the category. We will prove that, this category is an interval category, which roughly means that it is a Cartesian category with an interval covariant functor.

(T, N)-implications

The aim of this work is to study a certain fuzzy implication, called (T, N)-implication, obtained by composition of a fuzzy negation and a t-norm. Thus, it discusses under which conditions such functions preserve the main properties of fuzzy implications, in which some are related to the laws of contraposition. Finally, we prove a result that ensure the necessary and sufficient conditions for a function I : [0,1]2 → [0,1] to be a (T, N)-implication.

Penalty functions constructed from QL subsethood measures

Penalty functions are widely used to measure disagreement or consensus. On the other hand, subsethood measures have been applied in several areas. In this paper, we introduce a method for constructing penalty functions by means of QL fuzzy subsethood measures, introduced by Dimuro et al., which are built from QL-operations derived from tuples (O, G, N), for overlap functions O, grouping functions G and fuzzy negations N.

A Generalized Class of T-norms From a Categorical Point of View

Triangular norms or t-norms, in short, and automorphisms are very useful to fuzzy logics in the narrow sense. However, these notions are usually limited to the set [0,1]. In this paper we will consider a generalization of the t-norm notion for arbitrary bounded lattices as a category, where these generalized t-norms are the objects, and a generalization of automorphism notion as the morphism of the category. We will prove that, this category is Cartesian and a subcategory of it is Cartesian closed. We show that the usual interval t-norms can be seen as a covariant functor for that category.

Crisp Fuzzy Implications

Implications play an important role in fuzzy logics as they can be used both in practical and theoretical works. There exist many works in the literature where fuzzy implications behave in a crisp manner, i.e., implications that map to either zero or one. In this sense, we call those implications as crisp fuzzy implications and our goal is to study some their main features.

(T, N)-Implications and Some Functional Equations

Fuzzy implications has drawn attention of many authors along the years, as their theoretical features seem to be a useful tool in a fair amount of applications. Meanwhile, functional equations are those in which the unknowns are functions instead of a traditional variable, and within the fuzzy logic, they can be considered generalizations of some tautologies of the classical logic. In this paper we investigate the validity of five functional equations for the class of (T, N)-implications, namely, we have selected the law of importation and four distributivity properties and have studied them in the context of the aforementioned operator.