Vladimír Janiš

Divergence Measures for Intuitionistic Fuzzy Sets

Characterization of dissimilarity/divergence between intuitionistic fuzzy sets (IFSs) is important as it has applications in different areas including image segmentation and decision making. This study deals with the problem of comparison of intuitionistic fuzzy sets. An axiomatic definition of divergence measures for IFSs is presented, which are particular cases of dissimilarities between IFSs. The relationships among IF-divergences, IF-dissimilarities, and IF-distances are studied. Finally, we propose a very general framework for comparison of IFSs, where depending on the conditions imposed on a particular function, we can realize measures of distance, dissimilarity, and divergence for IFSs. Some methods for building divergence measures for IFSs are also introduced, as well as some examples of IF-divergences. In particular, we have proved some results that can be used to generate measures of divergence for fuzzy sets as well as for intuitionistic fuzzy sets.

Non-standard cut classification of fuzzy sets

Abstract Several important non-standard cut sets of lattice-valued fuzzy sets are investigated. These are strong cuts, “not less” and “neither less nor equal” cuts. In each case it is proved that collection of all cuts of any lattice-valued fuzzy set form a complete lattice under inclusion. Decomposition theorem (representation by cuts) is proved for “neither less nor equal” cuts. Necessary and sufficient conditions under which two lattice-valued fuzzy sets with the same domain have equal families of corresponding cut sets are given.

An axiomatic definition of divergence for intuitionistic fuzzy sets

An axiomatic definition of divergence measure for intuitionistic fuzzy sets (IFSs, for short) is presented in this work, as a particular case of dissimilarity between IFSs. As the concept of divergence measure is more restrictive, it has particular properties which are studied. Furthermore, the relationships among IF-divergences, dissimilarities and distances are studied. We also provide some methods for building divergence measure for IFSs. They will allow us to conclude this work with a classification of the usual functions used in the literature for measuring the dierence between intuitionistic fuzzy sets in two classes: which are divergence measures between IFSs and which are not.

Aggregation operators preserving quasiconvexity

Quasiconvexity of a fuzzy set is the necessary and sufficient condition for its cuts to be convex. We study the class of those two variable aggregation operators that preserve quasiconvexity on a bounded lattice, i.e. A(@m,@n) is quasiconvex for quasiconvex lattice valued fuzzy sets @m, @n. The class of all such aggregation operators is characterized by a lattice identity that they have to fulfill. In case of a unit interval we show the construction of aggregation operators preserving quasiconvexity from a pair of real valued functions on the unit interval. As a consequence we get that the intersection of quasiconvex fuzzy sets is quasiconvex if and only if the intersection is based on the minimum triangular norm.

Resemblance is a nearness

In (Fuzzy Sets and Systems 133 (2) (2003)) the authors claim that the transitivity condition usually used in the definition of a fuzzy equivalence of fuzzy equality is not suitable in a lot of real situations. The fuzzy relation of a resemblance in a metric space is defined in a different way in (Fuzzy Sets and Systems 133 (2) (2003)). The present paper shows that the same relation (called fuzzy nearness) as in (Fuzzy Sets and Systems 133 (2) (2003)) has been used by several authors in several works to study mainly properties of real function in more general context. A brief review of a variety of results achieved using the relation of a fuzzy nearness is provided.

Aggregation of convex intuitionistic fuzzy sets

Aggregation of intuitionistic fuzzy sets is studied from the point of view of preserving various kinds of convexity. We focus on aggregation functions for intuitionistic fuzzy sets. These functions correspond to simultaneous separate aggregations of the membership as well as of the nonmembership indicators. It is performed by means of the so-called representable functions. Sufficient and necessary conditions are analyzed in order to guarantee that the composition of two intuitionistic fuzzy sets preserves convexity.

Local IF-Divergences

In this work a particular type of measure of comparison of intuitionistic fuzzy sets is introduced: the local IF-divergences. This measure appears as a generalization of the local divergences for fuzzy sets. Some properties of this concept are introduced. In particular, we show that two methods used to build divergences from IF-divergences, and conversely IF-divergence from divergences, preserve the local property.

Local Divergences for Atanassov Intuitionistic Fuzzy Sets

The comparison of Atanassov intuitionistic fuzzy sets (AIF-sets) is a topic that has been widely studied due to its several applications in image segmentation or decision making, among other fields. Divergences for AIF-sets (AIF-divergences) were introduced as an adequate measure of comparison for AIF-sets. This study investigates a family of AIF-divergences that satisfies a local property. Such a property allows us to compute the divergence between AIF-sets pointwise. A characterization of those AIF-divergences satisfying the local property is provided. Several interesting properties of local divergence are also discussed. Some applications of these AIF-divergences in pattern recognition and decision making illustrate their utility.

Young Adults’ Trait Affection Given and Received as Functions of Hofstede’s Dimensions of Cultures and National Origin

Abstract Based on the work conducted by trait psychologists, this cross-cultural investigation sought to examine young adults’ trait affection given and trait affection received in the U.S., Russia, and Slovakia as functions of (a) Hofstede’s four primary dimensions of national cultures (i.e. masculinity–femininity, individualism–collectivism, uncertainty avoidance, and power distance), and (b) national origin. Undergraduate students (N = 558) from the U.S. (n = 214), Russia (n = 169), and Slovakia (n = 210) completed a questionnaire in their native languages. The results of regression analyses and analyses of variances supported the notion that the four dimensions of national cultures influence people’s trait-like attributes and therefore also result in significant differences among the three countries examined in this investigation.

On cardinalities of finite interval-valued hesitant fuzzy sets

Abstract Certain extensions of the classical fuzzy sets have been studied in depth since they have a remarkable importance in many practical situations. We focus on finite interval-valued hesitant fuzzy sets, as they generalize the most usual sets (fuzzy sets, interval-valued fuzzy sets, intuitionistic fuzzy sets), so the results obtained can be immediately adapted to these types of sets. In addition, their membership functions are much more manageable than type-2 fuzzy sets. In this work, the cardinality of finite interval-valued hesitant fuzzy sets is studied from an axiomatic point of view, together with several properties that this definition satisfies, which enable to relate it to the classical definitions of cardinality given by other authors for fuzzy sets.