Jaume Casasnovas

An axiomatic approach to fuzzy cardinalities of finite fuzzy sets

An axiomatic definition of fuzzy cardinalities for finite fuzzy sets, defined by means of a convex fuzzy set on the natural numbers, is presented in such a way that it includes the fuzzy cardinalities defined by authors like Zadeh, Ralescu, Dubois, Wygralak, and characterizes the cardinalities that fulfill the additivity property by means of the extended addition of fuzzy numbers. Such cardinalities result defined by two functions, one nondecreasing and the other nonincreasing in a similar way to the scalar cardinality (Fuzzy Sets and Systems 110 (2000) 175).

Discrete t-norms and operations on extended multisets

Multisets are set-like structures where an element can appear more than once. They are also called bags. A set means a collection of types of objects {x,y,...} rather than of concrete tokens{x,x,x,y,y,...} of them. The set of multisets on a universe is a partially ordered set for a functionally defined relationship of order. Moreover, it is a product of chains.Several pointwise defined operations as the addition, the union and the intersection between multisets have been defined and their properties investigated in several papers. The union and the intersection of two multisets is defined by means of the maximum and the minimum of the respective functions in N=N∪{∞} and a lattice structure is obtained for the previously defined poset.But, for multisets, the addition is an important operation because it corresponds to the simultaneous consideration of two multisets on a universe (or the consideration on a multiset twice). The addition can be defined as the disjoint union, i.e. A+B=A⊎B and of course is not idempotent. The addition and the union satisfy the properties of t-conorms in the same way that the intersection is a t-norm on the poset of the multisets and they are functionally, even pointwise, defined. In this paper we are concerned with some representation of all the possible functionally defined t-norms and t-conorms over the poset of the multisets that satisfy some interesting property like the divisibility. We distinguish the cases where the multisets are bounded or not.

Discrete Fuzzy Numbers Defined on a Subset of Natural Numbers

We introduce an alternative method to approach the addition of discrete fuzzy numbers when the application of the Zadeh’s extension principle does not obtain a convex membership function.

Scalar cardinalities of finite fuzzy sets for t-norms and t-conorms

An axiomatic approach to scalar cardinalities of finite fuzzy sets involving t-norms and t-conorms is presented. A characterization theorem for these cardinalities is proved and it is also proved that some standard properties remain true. On the other hand, properties like finite additivity, valuation property or finite subadditivity depend on the t-norm and the t-conorm.

Weighted Means of Subjective Evaluations

In this article, we recall different student evaluation methods based on fuzzy set theory. The problem arises is the aggregation of this fuzzy information when it is presented as a fuzzy number. Such aggregation problem is becoming present in an increasing number of areas: mathematics, physic, engineering, economy, social sciences, etc. In the previously quoted methods the fuzzy numbers awarded by each evaluator are not directly aggregated. They are previously defuzzycated and then a weighted mean or other type of aggregation function is often applied. Our aim is to aggregate directly the fuzzy awards (expressed as discrete fuzzy numbers) and to get like a fuzzy set (a discrete fuzzy number) resulting from such aggregation, because we think that in the defuzzification process a large amount of information and characteristics are lost. Hence, we propose a theoretical method to build n-dimensional aggregation functions on the set of discrete fuzzy number. Moreover, we propose a method to obtain the group consensus opinion based on discrete fuzzy weighted normed operators.

S-implications in the set of discrete fuzzy numbers

This paper is devoted to build S-implications on the bounded lattice A^L<inf>1</inf> of discrete fuzzy numbers whose support is a subset of consecutive natural numbers of the finite chain L = {0, 1, …, m}. Moreover, we propose a method to compare them.

AN APPROACH TO MEMBRANE COMPUTING UNDER INEXACTITUDE

In this paper we introduce a fuzzy version of symport/antiport membrane systems. Our fuzzy membrane systems handle possibly inexact copies of reactives and their rules are endowed with threshold functions that determine whether a rule can be applied or not to a given set of objects, depending of the degree of accuracy of these objects to the reactives specified in the rule. We prove that these fuzzy membrane systems generate exactly the recursively enumerable finite-valued fuzzy subsets of ℕ.

Probabilities of Fuzzy Events Based on Scalar Cardinalities

The standard set of Bell-type inequalities is satisfied by the extension of probabilities to fuzzy events based on the axiomatic definition of scalar cardinality of a fuzzy set, even though the lattice defined by the intersection, union and negation of fuzzy sets in the sense of Zadeh is not a boolean algebra.

Triangular Norms and Conorms on the Set of Discrete Fuzzy Numbers

In this paper, we study different methods to construct triangular operations (t-norms and t-conorms) on the bounded distributive lattice \(\mathcal{A}_1^{L}\), of discrete fuzzy numbers whose support is a subset of consecutive natural numbers on a finite chain L of consecutive natural numbers. Moreover, we propose a method to compare two t-norms (t-conorms) defined on \(\mathcal{A}_1^{L}\).

On the semilattice of inner extensions of a fuzzy partial algebra

In this paper, we establish a necessary and sufficient condition on a semilattice for it to be isomorphic to the semilattice of inner extensions of a L-fuzzy partial Σ-algebra, for some complete brouwerian lattice L and some signature Σ. We also consider the special case when L = [0, 1].