Regivan H. N. Santiago

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.

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.

On (G,N)-implications derived from grouping functions

Overlap and grouping functions are special kinds of non-necessarily associative aggregation operators recently proposed for applications in classification problems involving the overlap problem and/or when the associativity property is not strongly required, as in image processing and decision making based on fuzzy preference relations, respectively. The concepts of indifference and incomparability defined in terms of overlap and grouping functions may allow their application in several different contexts. This paper introduces the concept of (G,N)-implication, for a grouping function G and a fuzzy negation N. (G,N)-implications are weaker then (S,N)-implications for positive and continuous t-conorms S, in the sense that (G,N)-implications do not necessarily satisfy certain properties, as the exchange and the left neutrality principles, which are not demanded for applications in decision making based on fuzzy preference relations. We analyze several related important properties, providing a characterization of (G,N)-implications.

Typical Hesitant Fuzzy Negations

Since the seminal paper of fuzzy set theory by Zadeh in 1965, many extensions have been proposed to overcome the difficulty for assigning the membership degrees. In recent years, a new extension, the hesitant fuzzy sets, has attracted a lot of interest due to its usefulness to handle those problems in which it is difficult to provide accurately a single membership value; since for hesitant sets, membership values are given by a whole set of values. On the other hand, since fuzzy negations have an important role in applications as well as in the theoretical approach to of fuzzy logics, it is important to study an extension of the concept of fuzzy negation for hesitant fuzzy degrees (elements). In this paper, we propose such a definition and we study some of the main properties of this new concept.

Interval representations, Łukasiewicz implicators and Smets-Magrez axioms

The Smets-Magrez axiomatic is usually used to define the class of fuzzy continuous implications which are both S and R-implications (Lukasiewicz implications). Another approach is the construction of such class starting from a basic implication and applying automorphisms. Literature has shown that there is a harmony between those approaches, however in this paper we show that the extension of the Lukasiewicz implication defined on [0,1] for interval values cannot be applied in a direct way. We show that the harmony between the Smets-Magrez axiomatic approach and the one that comes from the generation by automorphisms is not preserved when such extension is done. One of the main consequences lies on the fact that the automorphism approach induces the loss of R-implications from the resulting class of implicators. More precisely, we show that the interval version of such approaches produce two disjunct classes of implicators, meaning that, unlike the usual case, the choice of the respective approach is an important step.

Interval Additive Generators of Interval T-Norms

The aim of this paper is to introduce the notion of interval additive generators of interval t-norms as interval representations of additive generators of t-norms, considering both the correctness and the optimality criteria, in order to provide a more systematic methodology for the selection of interval t-norms in the various applications. We prove that interval additive generators satisfy the main properties of punctual additive generators discussed in the literature.

An Interval-Based Framework for Fuzzy Clustering Applications

The main goal of using data with interval nature is to represent numeric information endowed with impreciseness, which are normally captured from measures of real world. However, in order to do this, it is necessary to adapt real-valued techniques to be applied on interval-based data. For interval-based clustering applications, for instance, it is necessary to propose an interval-based distance and also to adapt clustering algorithms to be used in this context. Therefore, in this paper, we aim to provide a platform for performing clustering applications using interval-based data, including distance measure, clustering algorithms, and validation indexes. In this case, we adapt an interval-based distance called dkm, and we propose two interval-based fuzzy clustering algorithms: Interval-based FcM and interval-based ckMeans, and three interval-based validation indexes. In order to validate the proposed interval-based framework, an empirical analysis was conducted using seven clustering datasets, three real and four synthetic interval datasets. The empirical analysis is based on an external cluster validity index, corrected rand, and six internal-based validation indexes, in which three of them can be used in their original proposal and three are proposed in this paper. The obtained results show the usefulness of the proposed interval-based framework for interval-based clustering problems.

On fuzzy ideals of fuzzy lattice

We characterize a fuzzy lattice through a fuzzy partial order relation, define a fuzzy ideal and fuzzy filter of fuzzy lattice, characterize a fuzzy ideal of fuzzy lattice using its level set and its support and show that a subset of a fuzzy lattice is a fuzzy ideal if and only if its support is a crisp ideal. Similarly, we show the same for its level set.

Analyzing Properties of Fuzzy Implications Obtained via the Interval Constructor

This work considers an interval extension of fuzzy impli- cations based on the best interval representation of usual fuzzy implications. The related properties of fuzzy im- plications can be naturally extended and the interval re- presentation meets the optimality property and preserves the behaviors of the implications in the interval endpoints. Our discussion mainly focuses on the best interval repre- sentation of three important classes of fuzzy implications: S-implications, R-implications and QL-implications. We analyze sufficient and necessary conditions for these three classes of implications as inclusion-monotonic functions in both arguments satisfying the minimal properties of fuzzy implications.