Benjamín R. C. Bedregal

A Historical Account of Types of Fuzzy Sets and Their Relationships

In this paper, we review the definition and basic properties of the different types of fuzzy sets that have appeared up to now in the literature. We also analyze the relationships between them and enumerate some of the applications in which they have been used.

A New Approach to Interval-Valued Choquet Integrals and the Problem of Ordering in Interval-Valued Fuzzy Set Applications

We consider the problem of choosing a total order between intervals in multiexpert decision making problems. To do so, we first start researching the additivity of interval-valued aggregation functions. Next, we briefly treat the problem of preserving admissible orders by linear transformations. We study the construction of interval-valued ordered weighted aggregation operators by means of admissible orders and discuss their properties. In this setting, we present the definition of an interval-valued Choquet integral with respect to an admissible order based on an admissible pair of aggregation functions. The importance of the definition of the Choquet integral, which is introduced by us here, lies in the fact that if the considered data are pointwise (i.e., if they are not proper intervals), then it recovers the classical concept of this aggregation. Next, we show that if we make use of intervals in multiexpert decision making problems, then the solution at which we arrive may depend on the total order between intervals that has been chosen. For this reason, we conclude the paper by proposing two new algorithms such that the second one allows us, by means of the Shapley value, to pick up the best alternative from a set of winning alternatives provided by the first algorithm.

On interval fuzzy S-implications

This paper presents an analysis of interval-valued S-implications and interval-valued automorphisms, showing a way to obtain an interval-valued S-implication from two S-implications, such that the resulting interval-valued S-implication is said to be obtainable. Some consequences of that are: (1) the resulting interval-valued S-implication satisfies the correctness property, and (2) some important properties of usual S-implications are preserved by such interval representations. A relation between S-implications and interval-valued S-implications is outlined, showing that the action of an interval-valued automorphism on an interval-valued S-implication produces another interval-valued S-implication.

Interval additive generators of interval t-norms and interval t-conorms

Abstract The aim of this paper is to introduce the concepts of interval additive generators of interval t-norms and interval t-conorms, as interval representations of additive generators of t-norms and t-conorms, respectively, considering both the correctness and the optimality criteria. The formalization of interval fuzzy connectives in terms of their interval additive generators provides a more systematic methodology for the selection of interval t-norms and interval t-conorms in the various applications of fuzzy systems. We also prove that interval additive generators satisfy the main properties of additive generators discussed in the literature.

New results on overlap and grouping functions

Abstract Overlap functions and grouping functions are special kinds of aggregation operators that have been recently proposed for applications in classification problems, like, e.g., imaging processing. Overlap and grouping functions can also be applied in decision making based on fuzzy preference relations, where the associativity property is not strongly required and the use of t-norms or t-conorms as the combination/separation operators is not necessary. The concepts of indifference and incomparability defined in terms of overlap and grouping functions may allow the application in several different contexts. This paper introduces new interesting results related to overlap and grouping functions, investigating important properties, such as migrativity, homogeneity, idempotency and the existence of generators. De Morgan triples are introduced in order to study the relationship between those dual concepts. In particular, we introduce important results related to the action of automorphisms on overlap and grouping functions, analyzing the preservation of those properties and also the Lipschitzianity condition.

The best interval representations of t-norms and automorphisms

This paper relates an interval generalization for t-norms given by the authors to the interval generalization of automorphism given by M. Gehrke et al. Both interval extensions can be seen as interval representations and therefore satisfy the correctness principle of interval computations. T-norms and automorphisms can be seen as objects and morphisms, respectively, of a category. Analogously, we will prove that interval t-norms supplied with interval automorphism is also a category, and we provide a functor between these categories that always returns the best interval representation of any t-norm and automorphism and therefore can be used to deal with optimality of interval fuzzy algorithms.

A class of fuzzy multisets with a fixed number of memberships

The main aim of this work is to present a generalization of Atanassovs operators to higher dimensions. To do so, we use the concept of fuzzy set, which can be seen as a special kind of fuzzy multiset, to define a generalization of Atanassovs operators for n-dimensional fuzzy values (called n-dimensional intervals). We prove that our generalized Atanassovs operators also generalize OWA operators of any dimension by allowing negative weights. We apply our results to a decision making problem. We also extend the notions of aggregating functions, in particular t-norms, fuzzy negations and automorphism and related notions for n-dimensional framework.

Interval valued QL-implications

The aim of this work is to analyze the interval canonical representation for fuzzy QL-implications and automorphisms. Intervals have been used to model the uncertainty of a specialists information related to truth values in the fuzzy propositional calculus: the basic systems are based on interval fuzzy connectives. Thus, using subsets of the real unit interval as the standard sets of truth degrees and applying continuous t-norms, t-conorms and negation as standard truth interval functions, the standard truth interval function of an QL-implication can be obtained. Interesting results on the analysis of interval canonical representation for fuzzy QL-implications and automorphisms are presented. In addition, commutative diagrams are used in order to understand how an interval automorphism acts on interval QL-implications, generating other interval fuzzy QL-implications.

Fuzzy rule-based hand gesture recognition

This paper introduces a fuzzy rule-based method for the recognition of hand gestures acquired from a data glove, with an application to the recognition of some sample hand gestures of LIBRAS, the Brazilian Sign Language. The method uses the set of angles of finger joints for the classification of hand configurations, and classifications of segments of hand gestures for recognizing gestures. The segmentation of gestures is based on the concept of monotonic gesture segment, sequences of hand configurations in which the variations of the angles of the finger joints have the same sign (non-increasing or non-decreasing). Each gesture is characterized by its list of monotonic segments. The set of all lists of segments of a given set of gestures determine a set of finite automata, which are able to recognize every such gesture.

Preaggregation Functions: Construction and an Application

In this paper, we introduce the notion of preaggregation function. Such a function satisfies the same boundary conditions as an aggregation function, but, instead of requiring monotonicity, only monotonicity along some fixed direction (directional monotonicity) is required. We present some examples of such functions. We propose three different methods to build preaggregation functions. We experimentally show that in fuzzy rule-based classification systems, when we use one of these methods, namely, the one based on the use of the Choquet integral replacing the product by other aggregation functions, if we consider the minimum or the Hamacher product t-norms for such construction, we improve the results obtained when applying the fuzzy reasoning methods obtained using two classical averaging operators such as the maximum and the Choquet integral.