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In this paper, we define the concept of an ignorance function and use it to determine the best threshold with which to binarize an image. We introduce a method to construct such functions from t-norms and automorphisms. By means of these new measures, we represent the degree of ignorance of the expert when given one fuzzy set to represent the background and another to represent the object. From this ignorance degree, we assign interval-valued fuzzy sets to the image in such a way that the best threshold is given by the interval-valued fuzzy set with the lowest associated ignorance. We prove that the proposed method provides better thresholds than the fuzzy classical methods when applied to transrectal prostateultrasoundimages. The experimental results on ultrasound and natural images also allow us to determine the best choice of the function to represent the ignorance., GrantsTIN2006-06190, TIN2007-65981, Depto. de Estadística e Investigación Operativa, Fac. de Ciencias Matemáticas, TRUE, pub

In this paper we address a key problem in many fields: how a structured data set can be analyzed in order to take into account the neighborhood of each individual datum. We propose representing the dataset as a fuzzy relation, associating a membership degree with each element of the relation. We then introduce the concept of interval-contrast, a means of aggregating information contained in the immediate neighborhood of each element of the fuzzy relation. The interval-contrast measures the range of membership degrees present in each neighborhood. We use interval-contrasts to define the necessary properties of a contrast measure, construct several different local contrast and total contrast measures that satisfy these properties, and compare our expressions to other definitions of contrast appearing in the literature. Our theoretical results can be applied to several different fields. In an Appendix A, we apply our contrast expressions to photographic images., Depto. de Estadística e Investigación Operativa, Fac. de Ciencias Matemáticas, TRUE, pub

In this paper we introduce the definition of n-dimensional overlap functions and we justify the axiomatization proposed in its definition. Basically, these functions allow to measure the degree of overlapping of several classes in a given classification system and for any given object. We also show a construction method for this class of functions, studying its relationships with the properties of migrativity, homogeneity and Lipschitz continuity. Finally, we propose an example where the use of n-dimensional overlap functions provides better results than those obtained with the commonly used product t-norm., Depto. de Estadística e Investigación Operativa, Fac. de Ciencias Matemáticas, TRUE, inpress

We introduce a slight modification of the definition of migrativity for aggregation functions that allows useful characterization of this property. Among other things, in this context we prove that there are no t-conorms, uninorms or nullnorms that satisfy migrativity (with the product being the only migrative t-norm, as already shown by other authors) and that the only migrative idempotent aggregation function is the geometric mean. The k-Lipschitz migrative aggregation functions are also characterized and the product is shown to be the only 1-Lipschitz migrative aggregation function. Similarly, it is the only associative migrative aggregation function possessing a neutral element. Finally, the associativity and bisymmetry of migrative aggregation functions are discussed., GACR, Depto. de Estadística e Investigación Operativa, Fac. de Ciencias Matemáticas, TRUE, pub

In this paper we address a key issue in scenario classification, where classifying concepts show a natural overlapping. In fact, overlapping needs to be evaluated whenever classes are not crisp, in order to be able to check if a certain classification structure fits reality and still can be useful for our declared decision making purposes. In this paper we address an object recognition problem, where the best classification with respect to background is the one with less overlapping between the class object and the class background. In particular, in this paper we present the basic properties that must be fulfilled by overlap functions, associated to the degree of overlapping between two classes. In order to define these overlap functions we take as reference properties like migrativity, homogeneity of order 1 and homogeneity of order 2. Hence we define overlap functions, proposing a construction method and analyzing the conditions ensuring that t-norms are overlap functions. In addition, we present a characterization of migrative and strict overlap functions by means of automorphisms., TIN2007-65981, APVV-0012-07, VEGA 1/4209/07, GACR 402/08/0618, TIN2009-07901., Depto. de Estadística e Investigación Operativa, Fac. de Ciencias Matemáticas, TRUE, pub

In this paper we prove that, under suitable conditions, Atanassov’s Ka operators, which act on intervals, provide the same numerical results as OWA operators of dimension two. On one hand, this allows us to recover OWA operators from Ka operators. On the other hand, by analyzing the properties of Atanassov’s operators, we can generalize them. In this way, we introduce a class of aggregation functions – the generalized Atanassov operators – that,in particular, include two-dimensional OWA operators. We investigate under which conditions these generalized Atanassov operators satisfy some properties usually required for aggregation functions, such as bisymmetry, strictness, monotonicity, etc. We also show that if we apply these aggregation functions to interval-valued fuzzy sets, we obtain an ordered family of fuzzy sets., Depto. de Estadística e Investigación Operativa, Fac. de Ciencias Matemáticas, TRUE, pub

In this paper we consider aggregation operators satisfying non-decreasingness and some specific boundary conditions. We then analyze some properties of such a family of aggregation operators, introducing the semiautoduality condition, which is weaker than the standard autoduality condition (i.e., the standard self De Morgan identity). Particular families of aggregation operators will appear depending on the context., Depto. de Estadística e Investigación Operativa, Fac. de Ciencias Matemáticas, TRUE, pub

This paper brings a generalization of the migrativity property of aggregation functions, suggested in earlier work of some of the present authors by imposing the a-migrativity property of Durante and Sarkoci for all values of a instead of a single one. Replacing the algebraic product by an arbitrary aggregation function B naturally leads to the properties of a–B-migrativity and B-migrativity. This generalization establishes a link between migrativity and a particular case of Aczel’s general associativity equation, already considered by Cutello and Montero as a recursive formula for aggregation. Following a basic investigation, emphasis is put on aggregation functions that can be represented in terms of an additive generator, more specifically, strict t-norms, strict t-conorms and representable uninorms., Government of Spain, APVV, MSM VZ, Depto. de Estadística e Investigación Operativa, Fac. de Ciencias Matemáticas, TRUE, pub

In this paper we stress the relevance of a particular family of fuzzy sets, where each element can be viewed as the result of a classification problem. In particular, we assume that fuzzy sets are defined from a well-defined universe of objects into a valuation space where a particular graph is being defined, in such a way that each element of the considered universe has a degree of membership with respect to each state in the valuation space. The associated graph defines the structure of such a valuation space, where an ignorance state represents the beginning of a necessary learning procedure. Hence, every single state needs a positive definition, and possible queries are limited by such an associated graph. We then allocate this family of fuzzy sets with respect to other relevant families of fuzzy sets, and in particular with respect to Atanassov's intuitionistic fuzzy sets. We postulate that introducing this graph allows a natural explanation of the different visions underlying Atanassov's model and interval valued fuzzy sets, despite both models have been proven equivalent when such a structure in the valuation space is not assumed., National Science Foundation of Spain, Depto. de Estadística e Investigación Operativa, Fac. de Ciencias Matemáticas, TRUE, pub

Brain-computer interface (BCI) technologies are popular methods of communication between the human brain and external devices. One of the most popular approaches to BCI is motor imagery (MI). In BCI applications, the electroencephalography (EEG) is a very popular measurement for brain dynamics because of its noninvasive nature. Although there is a high interest in the BCI topic, the performance of existing systems is still far from ideal, due to the difficulty of performing pattern recognition tasks in EEG signals. This difficulty lies in the selection of the correct EEG channels, the signal-to-noise ratio of these signals, and how to discern the redundant information among them. BCI systems are composed of a wide range of components that perform signal preprocessing, feature extraction, and decision making. In this article, we define a new BCI framework, called enhanced fusion framework, where we propose three different ideas to improve the existing MI-based BCI frameworks. First, we include an additional preprocessing step of the signal: a differentiation of the EEG signal that makes it time invariant. Second, we add an additional frequency band as a feature for the system: the sensorimotor rhythm band, and we show its effect on the performance of the system. Finally, we make a profound study of how to make the final decision in the system. We propose the usage of both up to six types of different classifiers and a wide range of aggregation functions (including classical aggregations, Choquet and Sugeno integrals, and their extensions and overlap functions) to fuse the information given by the considered classifiers. We have tested this new system on a dataset of 20 volunteers performing MI-based brain-computer interface experiments. On this dataset, the new system achieved 88.80% accuracy. We also propose an optimized version of our system that is able to obtain up to 90.76%. Furthermore, we find that the pair Choquet/Sugeno integrals and overlap functions are th

In this work we develop moderate deviation functions to measure similarity and dissimilarity among a set of given interval-valued data to construct interval-valued aggregation functions, and we apply these functions in two Motor-Imagery Brain Computer Interface (MI-BCI) systems to classify electroencephalography signals. To do so, we introduce the notion of interval-valued moderate deviation function and, in particular, we study those interval-valued moderate deviation functions which preserve the width of the input intervals. In order to apply them in a MI-BCI system, we first use fuzzy implication operators to measure the uncertainty linked to the output of each classifier in the ensemble of the system, and then we perform the decision making phase using the new interval-valued aggregation functions. We have tested the goodness of our proposal in two MI-BCI frameworks, obtaining better results than those obtained using other numerical aggregation and interval-valued OWA operators, and obtaining competitive results versus some non aggregation-based frameworks.

[EN] Most of the strategies for boundary image evaluation involve the comparison of computer-generated images with ground truth solutions. While this can be done in different manners, recent years have seen a dominance of techniques based on the use of confusion matrices. That is, techniques that, at the evaluation stage, interpret boundary detection as a classification problem. These techniques require a correspondence between the boundary pixels in the candidate image and those in the ground truth; that correspondence is further used to create the confusion matrix, from which evaluation statistics can be computed. The correspondence between boundary images faces different challenges, mainly related to the matching of potentially displaced boundaries. Interestingly, boundary image comparison relates to many other fields of study in literature, from object tracking to biometrical identification. In this work, we survey all existing strategies for boundary matching, we propose a taxonomy to embrace them all, and perform a usability-driven quantitative analysis of their behaviour.

[EN] Most of the strategies for boundary image evaluation involve the comparison of computer-generated images with ground truth solutions. While this can be done in different manners, recent years have seen a dominance of techniques based on the use of confusion matrices. That is, techniques that, at the evaluation stage, interpret boundary detection as a classification problem. These techniques require a correspondence between the boundary pixels in the candidate image and those in the ground truth; that correspondence is further used to create the confusion matrix, from which evaluation statistics can be computed. The correspondence between boundary images faces different challenges, mainly related to the matching of potentially displaced boundaries. Interestingly, boundary image comparison relates to many other fields of study in literature, from object tracking to biometrical identification. In this work, we survey all existing strategies for boundary matching, we propose a taxonomy to embrace them all, and perform a usability-driven quantitative analysis of their behaviour.

In this contribution we study social network modelling by using human interaction as a basis. To do so, we propose a new set of functions, affinities, designed to capture the nature of the local interactions among each pair of actors in a network. By using these functions, we develop a new community detection algorithm, the Borgia Clustering, where communities naturally arise from the multi-agent interaction in the network. We also discuss the effects of size and scale for communities regarding this case, as well as how we cope with the additional complexity present when big communities arise. Finally, we compare our community detection solution with other representative algorithms, finding favourable results., Comment: Corrections in: Revamped affinity section, conclusions and minor changes in the introduction. Also, the dynamic delta section is expanded a bit

© 2019 Elsevier B.V. This paper introduces the theoretical framework for a generalization of CF1F2-integrals, a family of Choquet-like integrals used successfully in the aggregation process of the fuzzy reasoning mechanisms of fuzzy rule based classification systems. The proposed generalization, called by gCF1F2-integrals, is based on the so-called pseudo pre-aggregation function pairs (F1,F2), which are pairs of fusion functions satisfying a minimal set of requirements in order to guarantee that the gCF1F2-integrals to be either an aggregation function or just an ordered directionally increasing function satisfying the appropriate boundary conditions. We propose a dimension reduction of the input space, in order to deal with repeated elements in the input, avoiding ambiguities in the definition of gCF1F2-integrals. We study several properties of gCF1F2-integrals, considering different constraints for the functions F1 and F2, and state under which conditions gCF1F2-integrals present or not averaging behaviors. Several examples of gCF1F2-integrals are presented, considering different pseudo pre-aggregation function pairs, defined on, e.g., t-norms, overlap functions, copulas that are neither t-norms nor overlap functions and other functions that are not even pre-aggregation functions.

© 2019 Elsevier B.V. This paper introduces the theoretical framework for a generalization of CF1F2-integrals, a family of Choquet-like integrals used successfully in the aggregation process of the fuzzy reasoning mechanisms of fuzzy rule based classification systems. The proposed generalization, called by gCF1F2-integrals, is based on the so-called pseudo pre-aggregation function pairs (F1,F2), which are pairs of fusion functions satisfying a minimal set of requirements in order to guarantee that the gCF1F2-integrals to be either an aggregation function or just an ordered directionally increasing function satisfying the appropriate boundary conditions. We propose a dimension reduction of the input space, in order to deal with repeated elements in the input, avoiding ambiguities in the definition of gCF1F2-integrals. We study several properties of gCF1F2-integrals, considering different constraints for the functions F1 and F2, and state under which conditions gCF1F2-integrals present or not averaging behaviors. Several examples of gCF1F2-integrals are presented, considering different pseudo pre-aggregation function pairs, defined on, e.g., t-norms, overlap functions, copulas that are neither t-norms nor overlap functions and other functions that are not even pre-aggregation functions.

© 2005-2012 IEEE. Brain-computer interface technologies, such as steady-state visually evoked potential, P300, and motor imagery are methods of communication between the human brain and the external devices. Motor imagery-based brain-computer interfaces are popular because they avoid unnecessary external stimuli. Although feature extraction methods have been illustrated in several machine intelligent systems in motor imagery-based brain-computer interface studies, the performance remains unsatisfactory. There is increasing interest in the use of the fuzzy integrals, the Choquet and Sugeno integrals, that are appropriate for use in applications in which fusion of data must consider possible data interactions. To enhance the classification accuracy of brain-computer interfaces, we adopted fuzzy integrals, after employing the classification method of traditional brain-computer interfaces, to consider possible links between the data. Subsequently, we proposed a novel classification framework called the multimodal fuzzy fusion-based brain-computer interface system. Ten volunteers performed a motor imagery-based brain-computer interface experiment, and we acquired electroencephalography signals simultaneously. The multimodal fuzzy fusion-based brain-computer interface system enhanced performance compared with traditional brain-computer interface systems. Furthermore, when using the motor imagery-relevant electroencephalography frequency alpha and beta bands for the input features, the system achieved the highest accuracy, up to 78.81% and 78.45% with the Choquet and Sugeno integrals, respectively. Herein, we present a novel concept for enhancing brain-computer interface systems that adopts fuzzy integrals, especially in the fusion for classifying brain-computer interface commands.

© 2005-2012 IEEE. Brain-computer interface technologies, such as steady-state visually evoked potential, P300, and motor imagery are methods of communication between the human brain and the external devices. Motor imagery-based brain-computer interfaces are popular because they avoid unnecessary external stimuli. Although feature extraction methods have been illustrated in several machine intelligent systems in motor imagery-based brain-computer interface studies, the performance remains unsatisfactory. There is increasing interest in the use of the fuzzy integrals, the Choquet and Sugeno integrals, that are appropriate for use in applications in which fusion of data must consider possible data interactions. To enhance the classification accuracy of brain-computer interfaces, we adopted fuzzy integrals, after employing the classification method of traditional brain-computer interfaces, to consider possible links between the data. Subsequently, we proposed a novel classification framework called the multimodal fuzzy fusion-based brain-computer interface system. Ten volunteers performed a motor imagery-based brain-computer interface experiment, and we acquired electroencephalography signals simultaneously. The multimodal fuzzy fusion-based brain-computer interface system enhanced performance compared with traditional brain-computer interface systems. Furthermore, when using the motor imagery-relevant electroencephalography frequency alpha and beta bands for the input features, the system achieved the highest accuracy, up to 78.81% and 78.45% with the Choquet and Sugeno integrals, respectively. Herein, we present a novel concept for enhancing brain-computer interface systems that adopts fuzzy integrals, especially in the fusion for classifying brain-computer interface commands.

In many problems, it is crucial to find a relation between groups of data. Such relation can be expressed, for instance, in terms of an appropriate fuzzy measure or capacity([10, 21]) which reflects the way the different data are connected to each other [20]. In this chapter, taking into account this fact and following the developments in [8],we introduce a method to build capacities ([20, 21]) directly from the data (inputs) of a given problem. In order to do so, we make use of the notions of overlap function and overlap index ([5, 12, 13, 7, 4, 14, 16]) for constructing capacities which reflect how different data are related to each other. This paper is organized as follows: after providing some preliminaries, we analyse, in Section 3, some properties of overlap functions and indexes. In Sections 4 we discuss a method for constructing capacities from overlap functions and overlap indexes. Finally, we present some conclusions and references.

In this position paper we propose a consistent and unifying view to all those basic knowledge representation models that are based on the existence of two somehow opposite fuzzy concepts. A number of these basic models can be found in fuzzy logic and multi-valued logic literature. Here it is claimed that it is the semantic relationship between two paired concepts what determines the emergence of different types of neutrality, namely indeterminacy, ambivalence and conflict, widely used under different frameworks (possibly under different names). It will be shown the potential relevance of paired structures, generated from two paired concepts together with their associated neutrality, all of them to be modeled as fuzzy sets. In this way, paired structures can be viewed as a standard basic model from which different models arise. This unifying view should therefore allow a deeper analysis of the relationships between several existing knowledge representation formalisms, providing a basis from which more expressive models can be later developed., Ministerio de Ciencia e Innovación (MICINN), Comunidad de Madrid, UCM, Depto. de Estadística e Investigación Operativa, Fac. de Ciencias Matemáticas, TRUE, pub

In this chapter we review several properties of Atanassov’s intuitionistic fuzzy relations, recalling the main concepts related to Atanassov’s intuitionistic fuzzy relations and the main properties that can be demanded to such conepts. We also consider the use of Atanassov’s operators over such relations.

The necessity of dealing with uncertainty in real world problems has been a long-term research challenge which has originated different methodologies and theories. Recently, the concept of Hesitant Fuzzy Sets (HFSs) has been introduced to model the uncertainty that often appears when it is necessary to establish the membership degree of an element and there are some possible values that make to hesitate about which one would be the right one. Many researchers have paid attention on this concept who have proposed diverse extensions, relationships with other types of fuzzy sets, different types of operators to compute with this type of information, applications on information fusion and decision-making, etc. Nevertheless, some of these proposals are questionable, because they are straightforward extensions of previous works or they do not use the concept of HFSs in a suitable way. Therefore, this position paper studies the necessity of HFSs and provides a discussion about current proposals including a guideline that the proposals should follow and some challenges of HFSs.

In this position paper we propose a consistent and unifying view to all those basic knowledge representation models that are based on the existence of two somehow opposite fuzzy concepts. A number of these basic models can be found in fuzzy logic and multi-valued logic literature. Here it is claimed that it is the semantic relationship between two paired concepts what determines the emergence of different types of neutrality, namely indeterminacy, ambivalence and conflict, widely used under different frameworks (possibly under different names). It will be shown the potential relevance of paired structures, generated from two paired concepts together with their associated neutrality, all of them to be modeled as fuzzy sets. In this way, paired structures can be viewed as a standard basic model from which different models arise. This unifying view should therefore allow a deeper analysis of the relationships between several existing knowledge representation formalisms, providing a basis from which more expressive models can be later developed., In this position paper we propose a consistent and unifying view to all those basic knowledge representation models that are based on the existence of two somehow opposite fuzzy concepts. A number of these basic models can be found in fuzzy logic and multi-valued logic literature. Here it is claimed that it is the semantic relationship between two paired concepts what determines the emergence of different types of neutrality, namely indeterminacy, ambivalence and conflict, widely used under different frameworks (possibly under different names). It will be shown the potential relevance of paired structures, generated from two paired concepts together with their associated neutrality, all of them to be modeled as fuzzy sets. In this way, paired structures can be viewed as a standard basic model from which different models arise. This unifying view should therefore allow a deeper analysis of the relationships between several existing knowledge representation formalisms, providing a basis from which more expressive models can be later developed.

In a group decision making setting, we consider the potential impact an expert can have on the overall ranking by providing a biased assessment of the alternatives that differs substantially from the majority opinion. In the framework of similarity based averaging functions, we show that some alternative approaches to weighting the experts' inputs during the aggregation process can minimize the influence the biased expert is able to exert.

In this work we introduce a method for building interval-valued negations using the characterization theorem for strong negations which was proposed by Trillas in 1979. We also show that interval type-2 fuzzy sets are a three dimensional representation of interval-valued fuzzy sets and we analyze the problems to build complementation for such interval type-2 fuzzy sets. We finally propose a method to construct this complementation.

Many different kinds of sets have been defined within the framework of fuzzy sets . This paper focusses on those fuzzy set extensions that address the difficulties that experts find in order to build the membership values. In particular, we analyze type-2 fuzzy sets , interval-valued fuzzy sets , Atanassov’s intuitionistic fuzzy sets , or bipolar sets of type-2 and Atanassov’s interval-valued fuzzy sets. After stating a general approach to these extensions, we remark some structural problems in the extension problem and stress some applications for which the results obtained with extens

In 1979, Enric Trillas started to interest in in fuzzy connectives. His first paper on this topic was ”Funciones de negacin en la teora de subconjuntos difusos” ([9]) (Negation functions in the theory of fuzzy subsets), which appeared in Spanish in the Stochastica journal. This work, focused on the characterization of strong negations, has been so relevant for the development of fuzzy theory that it was translated into English and widely cited in the last 35 years

Monotonicity with respect to all arguments is fundamental to the definition of aggregation functions. Here we study means that are not necessarily monotone. Weak monotonicity was recently proposed as a relaxation of the monotonicity condition for averaging functions. We provide results for the weak monotonicity of some importantclasses of mixture functions. With these results we are able to extend and improve the understanding of this very important class of functions.

En este artículo se presentan los resultados de una experiencia realizada con dos grupos de alumnos,uno de Ciencias Sociales y otro de Ciencias Matemáticas, con la finalidad de observar las posibles disimilitudes en la percepción y uso de diferentes lógicas. El resultado de esta experiencia inicial mostrará, sobre todo, cómo el científico acepta más fácilmente que el humanista modelos que su propio sentido común rechaza, lo cual es un aspecto que se debe tomar en cuenta en estudios formales posteriores.

En este trabajo se analiza una cuestión clave respecto de la relación que debe existir entre operadores de una misma familia de operadores de agregación (FAO) {An}, a fin de comprender que ellos deben definir adecuadamente un todo consistente. Se extienden algunas de las ideas de estabilidad de una FAO con un enfoque más general,definiéndose formalmente las nociones de i−L y j−R estabilidad estricta para familias de operadores de agregación, e introduciendo la noción de estabilidad estricta de orden k. Finalmente,se muestra una aplicación de las condiciones de estabilidad estricta al problema de pérdida de datos en un proceso de agregación de información, utilizándose las familias de la media ponderada y de la media ponderada cuasi-aritmética.

We investigate the problem of combining or aggregating several color values given in coding scheme RGB. For this reason, we study the problem of averaging values on lattices, and in particular on discrete product lattices. We study the arithemtic mean and the median on product lattices. We apply these aggregation functions in image reduction and we present a new algorithm based on the minimization of penalty functions on discrete product lattices.

In this work we introduce the definition of restricted dissimilarity functions and we link it with some other notions, such as metrics. In particular, we also show how restricted dissimilarity functions can be used to build penalty functions.

In this work we present the concept of penalty function over a Cartesian product of lattices. To build these mappings, we make use of restricted dissimilarity functions and distances between fuzzy sets. We also present an algorithm that extends the weighted voting method for a fuzzy preference relation.

In this work we introduce a new construction method of Atanassov's intuitionistic fuzzy sets (A-IFSs) from fuzzy sets. We use A-IFSs in image processing. We propose a new image magnification algorithm using A-IFSs. This algorithm is characterized by its simplicity and its efficiency.

In this paper, we propose a new construction method for fuzzy and weak fuzzy subsethood measures based on the aggregation of implication operators. We study the desired properties of the implication operators in order to construct these measures. We also show the relationship between fuzzy entropy and weak fuzzy subsethood measures constructed by our method.

The main contribution of this paper is to obtain a polynomial algorithm that allows us to classify step by step a set of items that are related by means of a fuzzy graph extending the algorithm published in [8] to any class of graph.

Although consistency in information and actions is a major argument in any decision making problem, most available tools are crisp, while observation strongly suggests to consider different degrees of consistency, i.e.,consistency is fuzzy in nature. In this paper we propose to put together some of the works in the field, focusing on information consistency rather than on action consistency, showing in this way key classification features.

In this work we study some properties of overlap functions from an analytical point of view. We study the relation between overlap functions and continuous t-norms. We focus on homogeneous overlap functions and we get a characterization of them. Finally we completely identify homogeneous migrative overlap functions.

Extraction of rules for classification and decision tasks from databases is an issue of growing importance as automated processes based on data are being required in these fields. Interpretability of rules is improved by defining classes for independent variables. Moreover, though more complex, a more realistic and flexible framework is attained when fuzzy classes are considered. In this paper, an inductive approach is taken in order to develop a general methodology for building fuzzy rules from databases. Three types of rules are built in order to be able of dealing with both categorical and numerical data.

In this work we study overlap degrees expressed in terms of overlap functions. We present the basic properties that from our point of view must satisfy these overlap functions. We study a construction method, we analyze which t-norms are also overlap functions and we prove that if we apply particular aggregations to such functions we recover the overlap index between fuzzy sets as defined by Dubois, and the consistency index of Zadeh. We also consider some properties that can be required to overlap functions, as k-Lipschitzianity or migrativity

This paper stresses the relevance of taking into account that quite often information is structured, and therefore a standard aggregation focuss may produce misleading results, if such an approach assumes in fact that information does not present any particular structure, despite it is obvious such a structure exists. In this sense, this research is a consequence of a previous paper from the authors,already published in Fuzzy Sets and Systems (2007), where classical associative aggregation functions where linked to linear structures of data, meanwhile data quite often show a complex (non linear) structure moving within a spatial framework.

In this paper we shall stress the key relevance of addressing a specific structure for the valuation space in Atanassov's IFS, which may help to establish certain di®erences with other related models. In particular, we point out that each one of the valuation states proposed by Atanassov plays a different role, so they should be distinguished from their particular properties with respect to the other valuation states. In the same way, we realize that those three valuation states are not interchangeable and that each state should in principle allow a clear definition, as in any standard classification problem, always subject to potential direct estimation. Considering for example a transition (directed) graph in order to connect states provides in addition a new framework for useful more general models, taking simultaneously into account several logics when needed and avoiding the still binary heritage in Atanassov's model.

In this paper we study in depth certain properties of interval type 2 fuzzy sets. In particular we recall a method to construct different interval type 2 fuzzy connectives starting from an operator. We further study the law of contradiction and the law of excluded middle for these sets. Furthermore we analyze the properties: idempotency, absorption, and distributiveness.