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The idea of computable aggregation operators was introduced as a generalization of aggregation operators, allowing the replacement of the mathematical function usually considered for aggregation, by a program that performs the aggregation process. There are different reasons to justify this extension. One of them is the interest in exploring some computational properties not directly related to the aggregation itself but to its implementation (complexity, recursivity, parallelisation, etc). Another reason, the one driving to the present paper, is the need to define a framework where the quite common process of first sampling (over a large data set) and then aggregating the sample, could be analysed as a formal aggregation process. This process does not match with the idea of an aggregation function, due to its non-deterministic nature, but could easily be adapted to that of a (non-deterministic) computable aggregation. The idea of non-deterministic aggregation requires the extension of the concept of monotonicity (a key aspect of aggregation operators) to this new framework. The present paper will explore this kind of non-deterministic aggregation processes, first from an empirical point of view and then in terms of populations, adapting the idea of monotonicity to both of them and finally defining a common framework for its analysis.

Fuzzy relations R : A 2 → [ 0 , 1 ] are widely used in decision-making in order to represent degrees of preference between alternatives in A . Examining fuzzy relations, this paper first reflects on an inherent incompatibility inside the domain of reciprocal fuzzy preference relations and preference structures. Aiming at clarifying the semantics of reciprocal fuzzy relations by means of that of weak fuzzy relations, a general frame is then proposed for studying the transformation between them, based on the notion of reciprocal functions. Next, we show that reciprocal relations can be naturally endowed with a preference structure of their own, containing two relations respectively showing a semantics of strict preference and lack of preference. Finally, we study the compatibility between these reciprocal preference structures and the standard preference structures of weak fuzzy preference relations by means of an axiomatic approach, leading to a system of functional equations that is shown to admit different solutions, allowing to explicitly assign one of the available semantics for reciprocal preference relations.

This paper proposes a couple of criteria for evaluating the quality and relevance of a fuzzy partition. These criteria are established from a fuzzy classification system and its recursive De Morgan triplet. We propose a comparison process between the classes of a fuzzy partition, based on a translation invariant similarity relation. Therefore a classification process is carried out with the equivalence relations determined by the similarity relation. Such a relation is built on the commutative group structure formed by the elements of the fuzzy classification system. Our approach is illustrated through an example on image analysis by the fuzzy c-means algorithm.

In this paper we extend the notion of stability of each aggregation function as presented in a previous paper of the authors, where an aggregation function was denominated Family of Aggregation Operators (FAO) in order to stress that aggregation operators within an aggregation function should be consistent. We will show that the previous definition presents certain problems when dealing with a FAO that is not idempotent or continuous. Meanwhile the previous definition was based on the study of fixed points, without taking into account the environment of each point, as many applications demand, the new approach we propose now is more flexible, and at the end allows a more appropriate definition of the consistency of a FAO. It will be shown that under certain conditions our new proposal is equivalent to the previous definition of strict stability, but the new definition covers families of aggregation operators that are not strictly stable, but intuitively stable, and vice versa. In short, we will see that our new proposal fits much better the intuition of stability of a FAO. This reformulation of stability is based upon the concept of penalty function as a measure of proximity between a value and an array of values.

As a generalization of bivariate overlap functions, which measure the degree of overlapping (intersection for non-crisp sets) of n different classes, in this paper we introduce the concept of general overlap functions. We characterize the class of general overlap functions and include some construction methods by means of different aggregation and bivariate overlap functions. Finally, we apply general overlap functions to define a new matching degree in a classification problem. We deduce that the global behavior of these functions is slightly better than some other methods in the literature. The work has been supported by the Research Services of the Universidad Publica de Navarra, the research projects TIN2016-77356-P (AEI/FEDER, UE) and TIN2015-66471-P from the Government of Spain and by the Brazilian National Counsel of Technological and Scientific Development CNPq (Proc. 233950/2014-1, 306970/2013-9, 307781/2016-0) and by Caixa and Fundación Caja Navarra of Spain.

In this paper, different ways of aggregating color information in edge extraction task are explored. One of these approaches is based on a previous transformation of the RGB images into a new color space of 8 dimensions. This increases the number of dimensions, making differences between colors easier to detect, and in this way improving the performance of edge detection task. Sobel and Canny algorithms are employed over the RGB images and its transformed version of 8 dimensions -that we have named Super8 image-. The set of employed images is the one from Berkeley’s dataset. In order to evaluate the performance, precision, recall and F measure are computed. The way of aggregating the color information is showed to be relevant. For some well-known algorithms, the new 8 dimension color space overtakes RGB’s for edge detection problem.

Most supervised classification algorithms produce a soft score (either a probability, a fuzzy degree, a possibility, a cost, etc.) assessing the strength of the association between items and classes. After that, each item is assigned to the class with the highest soft score. In this paper, we show that this last step can be improved through alternative procedures more sensible to the available soft information. To this aim, we propose a general fuzzy bipolar approach that enables learning how to take advantage of the soft information provided by many classification algorithms in order to enhance the generalization power and accuracy of the classifiers. To show the suitability of the proposed approach, we also present some computational experiences for binary classification problems, in which its application to some well-known classifiers as random forest, classification trees and neural networks produces a statistically significant improvement in the performance of the classifiers.

The problem of missing data is common in real-world applications of supervised machine learning such as classification and regression. Such data often gives rise to the need for functions defined for varying dimension. Here we propose optimization methods for learning the weights of quasi-arithmetic means in the context of data with missing values. We investigate some alternative approaches depending on the number of variables that have missing values and show results for several numerical experiments.

Traditionally, the last step of edge detection algorithms, which is called scaling-evaluation, produces the final output classifying each pixel as edge or nonedge. This last step is usually done based on local evaluation methods. The local evaluation makes this classification based on measures obtained for every pixel. By contrast, in this work, we propose a global evaluation approach based on the idea of edge list to produce a solution that suits more with the human perception. In particular, we propose a new evaluation method that can be combined with any classical edge detection algorithm in an easy way to produce a novel edge detection algorithm. The new global evaluation method is divided in four steps: in first place we build the edge lists, that we have called edge segments. In second place we extract the characteristics associated to each segment: length, intensity, location, and so on. In the third step we learn the characteristics that make a segment good enough to become an edge. At the fourth step, we apply the classification task. In this work we have built the ground truth of edge list necessary for the supervised classification. Finally, we test the effectiveness of this algorithm against other classical algorithms based on local evaluation approach.

Most edge detection algorithms deal only with grayscale images, and the way of adapting them to use with RGB images is an open problem. In this work, we explore different ways of aggregating the color information of a RGB image for edges extraction, and this is made by means of well-known edge detection algorithms. In this research, it is been used the set of images from Berkeley. In order to evaluate the algorithm’s performance, F measure is computed. The way that color information -the different channels- is aggregated is proved to be relevant for the edge detection task. Moreover, post-aggregation of channels performed significatively better than the classic approach (pre-aggregation of channels).

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., This research has been partially supported by the Government of Spain (grants TIN2015-66471-P and TIN2013-40765-P), the Government of Madrid (grant S2013/ICCE-2845) and the UCM (Research Group 910149).

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.

In this paper, we introduce the concept of fuzzy interval-entropy. The main feature of this new entropy is that the provided value is a closed subinterval of [0, 1]. This fact has led us to define interval-valued fuzzy strong S-subsethood measures which we use to build fuzzy interval-entropies. As we require that the results of both measures are intervals, we should work with total orders to be able to compare the results. The problem of finding equilibrium points for interval-valued negations with respect to total orders has led us to introduce and study the concept of P-interval-entropy. Finally, in an illustrative example, we discuss an application of the different concepts we have considered. We also compare our results to those obtained in our application where we use the classical interval-valued entropy, which provides a real number instead of an interval as its result, as we know., This work was supported in part by the Spanish Ministry of Science and Technology under projects TIN2015-66471-P and TIN2016-77356-P (MINECO, AEI/FEDER, UE) and by grants VEGA 1/0420/15, VEGA 1/0614/18 and APVV-14-0013.

In this paper we analyze the notion of a finite mean from an axiomatic point of view. We discuss several axiomatic alternatives, with the aim of establishing a universal definition reconciling all of them and exploring theoretical links to some branches of Mathematics as well as to multidisciplinary applications. This work has been partially supported by the research projects S2013/ICE-2845, ECO2015-65031-R, MTM2015-63608-P (MINECO/ AEI-FEDER, UE), TIN2015-66471-P (MINECO/ AEI-FEDER, UE), TIN2016-77356-P (MINECO/ AEI-FEDER, UE).

Capturing aerial images by Unmanned Aerial Vehicles (UAV) allows gathering a general view of an agricultural site together with a detailed exploration of its relevant aspects for operational actions. Here we explore the challenging task of detecting cirsium arvense, a thistle-weed species, from aerial images of barley-cereal crops taken from 50 m above the ground, with the purpose of applying herbicide for site-specific weed treatment. The methods for automatic detection are based on object-based annotations, pointing out the RGB attributes of the Weed or Cereal classes for an entire group of pixels, referring to a crop area which will have to be treated if it is classified as being of the Weed class. In this way, an annotation belongs to the Weed class if more than half of its area is known to be covered by thistle weeds. Hence, based on object and pixel-level analysis, we compare the use of k-Nearest Neighbours (k-NN) and (feed-forward, one-hidden layer) neural networks, obtaining the best results for weed detection based on pixel-level analysis, based on a soft measure given by the proportion of predicted weed pixels per object, with a global accuracy of over 98%.

Edge detection problems try to identify those pixels that represent the boundaries of the objects in an image. The process for getting a solution is usually organized in several steps, producing at the end a set of pixels that could be edges (candidates to be edges). These pixels are then classified based on some local evaluation method, taking into account the measurements obtained in each pixel. In this paper, we propose a global evaluation method based on the idea of edge list to produce a solution. In particular, we propose an algorithm divided in four steps: in first place we build the edge list (that we have called segments); in second place we extract the characteristics associated to each segment (length, intensity, location,…); in the third step we learn which are the characteristics that make a segment good enough to be a boundary; finally, in the fourth place, we apply the classification task. In this work we have built the ground truth of edge list necessary for the supervised classification. Finally we test the effectiveness of this algorithm against other classical approaches.

This paper introduces a hierarchical clustering algorithm in networks based upon a first divisive stage to break the graph and a second linking stage which is used to join nodes. As a particular case, this algorithm is applied to the specific problem of community detection in social networks, where a betweenness measure is considered for the divisive criterion and a similarity measure associated to data is used for the linking criterion. We show that this algorithm is very flexible as well as quite competitive (from both a performance and a computational complexity point of view) in relation with a set of state-of-the-art algorithms. Furthermore, the output given by the proposed algorithm allows to show in a dynamic and interpretable way the evolution of how the groups are split in the network.

While classical image processing algorithms were designed for scalar-valued (binary or grayscale) images, new technologies have made it commonplace to work with vector-valued ones. These technologies can involve new types of sensors, as in remote sensing, but also mathematical models leading to an increased cardinality at each pixel. This work analyzes the role of first-order differentiation in vector-valued images; specifically, we explore a novel operator to produce a 2D vector from a Jacobian matrix, in order to represent the variation in a vector-valued image as a planar feature.

In this paper we discuss about graph approach in image segmentation. In first place, some main image processing techniques are classified based upon the output these methods provide. Then, a fuzzy image segmentation definition is presented because in the literature review was found that it was not clearly defined. This definition of fuzzy image segmentation is then related to a hierarchical image segmentation procedure, so this concept is also formally defined in this work. As every output of an image processing algorithm has to be evaluated, then a method to evaluate a hierarchical segmentation output is proposed in order to later propose a method to evaluate a fuzzy image segmentation output. Computational experiences point to some advantages of the proposed hierarchical image segmentation procedure over other algorithms.

This paper focuses on key issues related to aggregation tools within knowledge acquisition, and in particular for image processing. In particular, it is claimed the need of a number of simultaneous indices in order to evaluate the quality of a fuzzy classification (e.g., overlapping, covering and relevance), and that such a family of indices should allow learning, giving a hint on how such a classification can be improved. Our guess is that those indices can be obtained from certain families of well-known aggregation tools.

In this paper, we postulate computation as a key element in assuring the consistency of a family of aggregation functions so that such a family of operators can be considered an aggregation rule. In particular, we suggest that the concept of an aggregation rule should be defined from a computational point of view, focusing on the computational properties of such an aggregation, i.e., on the manner in which the aggregation values are computed. The new algorithmic definition of aggregation we propose provides an operational approach to aggregation, one that is based upon lists of variable length and that produces a solution even when portions of data are inserted or deleted. Among other advantages, this approach allows the construction of different classifications of aggregation rules according to the programming paradigms used for their computation or according to their computational complexity.

As recently proved in a previous work of Turunen, Tsoukias and Ozturk, starting from an evidence pair ( a , b ) on the real unit square and associated with a propositional statement α, we can construct evidence matrices expressed in terms of four values t, f, k, u that respectively represent the logical valuations true , false , contradiction (both true and false) and unknown (neither true nor false) regarding the statement α. The components of the evidence pair ( a , b ) are to be understood as evidence for and against α, respectively. Moreover, the set of all evidence matrices can be equipped with an injective MV-algebra structure. Thus, the set of evidence matrices can play the role of truth-values of a Lukasiewicz–Pavelka fuzzy logic, a rich and applicable mathematical foundation for fuzzy reasoning, and in such a way that the obtained new logic is paraconsistent. In this paper we show that a similar result can be also obtained when the evidence pair ( a , b ) is given on the real unit triangle. Since the real unit triangle does not admit a natural MV-structure, we introduce some mathematical results to show how this shortcoming can be overcome, and another injective MV-algebra structure in the corresponding set of evidence matrices is obtained. Also, we derive several formulas to explicitly calculate the evidence matrices for the operations associated to the usual connectives.

The interventions aimed at the early childhood are of a main interest in educational policy, since it is in this period when it is possible to produce a major impact in the subsequent human development. The quality of children's social environment is the main influence to consider in achieving sound child development, affecting throughout school life. For this reason, the development of child's environment indexes appears in a natural way in the evaluation of all kind of educational policy research and social programs. However, crisp measures and indexes, based on usual linear techniques, do not ensure an adequate representation of social reality, since this last has a fuzzy nature and a nonlinear behavior. The development of indexes can be seen as an aggregation problem. In this paper, we extend the notions of consistency and strict stability of a family of aggregation operators (FAO), proposed in a previous work of the authors for the case of an aggregation process in which the data have no particular structure, to the case in which the information has a prioritized hierarchical structure. This extended notion of strict stability is then used to address the construction of indexes. Particularly, we apply this approach to develop a construction method of child's home environment indexes in which a stable family of prioritized aggregation operators is used in order to ensure robustness of the aggregation process when the information has a lineal structure. These indexes are built using fuzzy data that fit into a hierarchical structure by means of a stable family of prioritized aggregation operators based on the prioritized operator formulated by Yager, where the order relationship over fuzzy information was defined by experts on child development.

Computing with words (CWW) explores the brain's ability to handle and evaluate perceptions through language, i.e., by means of the linguistic representation of information and knowledge. On the other hand, standard preference structures examine decision problems through the decomposition of the preference predicate into the simpler situations of strict preference, indifference and incomparability. Hence, following the distinctive cognitive/neurological features for perceiving positive and negative stimuli in separate regions of the brain, we consider two separate and opposite poles of preference and aversion, and obtain an extended preference structure named the Preference-aversion (P-A) structure. In this way, examining the meaning of words under an ordinal scale and using CWW's methodology, we are able to formulate the P-A model under a simple and purely linguistic approach to decision making, obtaining a solution based on the preference and non-aversion order.

In this paper we analyze the notion of family of aggregation operators (FAO), also refereed to as extended aggregation functions (EAF), i.e., a set of aggregation operators defined in the unit interval which aggregate several input values into a single output value. In particular, we address the key issue of the relationship that should hold between the operators in a family in order to understand that they properly define a consistent FAO. We focus on the idea of strict stability of a family of aggregation operators in order to propose an operative notion of consistency between operators of such a family. In this way, robustness of the aggregation process can be guaranteed. Some strict stability definitions for FAOs are proposed, leading to a classification of the main aggregation operators in terms of the properties they satisfy. Furthermore, we apply this approach to analyze the stability of some families of aggregation operators based on weights.

Fuzziness and bipolarity allow representing human knowledge, taking into account the gradual and the dialectic properties of language, focusing on the meaning of concepts. Under this cognitive and linguistic approach, we explore preference relations, examining their semantic decomposition through fuzzy preference structures and the specification of meaningful opposites. In particular, we introduce the Preference-Aversion (P-A) model, which allows analyzing, under an independent aggregation methodology, the possible gains and losses, like pros and cons, towards a given set of alternatives. As an attractive feature of this proposal, we show that the P-A model allows distinguishing between need and desire, contrary to common preference models where both notions are indistinguishable.

In this paper we stress the relevance of those fuzzy models that impose a couple of simultaneous views in order to represent concepts. In particular, we point out that the basic model to start with should contain at least two somehow opposite valuations plus a number of neutral concepts that are generated from the semantic relationship between those two opposites. Such a basic model should be distinguished from some other similar approaches that can be found in the literature, and that may bring some difficulties in intuition, partially because of their denomination. The general term “paired fuzzy sets” is then proposed together with the notion of sub-antonym, to be considered as a particular case of opposition relationship.

In this paper we discuss some main image processing techniques in order to propose a classification based upon the output these methods provide. Because despite a particular image analysis technique can be supervised or unsupervised, and can allow or not the existence of fuzzy information at some stage, each technique has been usually designed to focus on a specific objective, and their outputs are in fact different according to each objective. Thus, they are in fact different methods. But due to the essential relationship between them they are quite often confused. In particular, this paper pursues a clarification of the differences between image segmentation and edge detection, among other image processing techniques., This research has been partially supported by the Government of Spain, grant TIN2015-66471-P, and by the Government of the Community of Madrid, grant S2013/ICE-2845 (CASI-CAM-CM).

One of the main challenges of fuzzy community detection problems is to be able to measure the quality of a fuzzy partition. In this paper, we present an alternative way of measuring the quality of a fuzzy community detection output based on n-dimensional grouping and overlap functions. Moreover, the proposed modularity measure generalizes the classical Girvan–Newman (GN) modularity for crisp community detection problems and also for crisp overlapping community detection problems. Therefore, it can be used to compare partitions of different nature (i.e. those composed of classical, overlapping and fuzzy communities). Particularly, as is usually done with the GN modularity, the proposed measure may be used to identify the optimal number of communities to be obtained by any network clustering algorithm in a given network. We illustrate this usage by adapting in this way a well-known algorithm for fuzzy community detection problems, extending it to also deal with overlapping community detection problems and produce a ranking of the overlapping nodes. Some computational experiments show the feasibility of the proposed approach to modularity measures through n-dimensional overlap and grouping functions., Supported by the Government of Spain (grant TIN2012-32482), the Government of Madrid (grant S2013/ICCE-2845)

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.

This work has been supported by projects TIN2013-40765- P and TIN2012-32482 of the Spanish Ministry of Science and by the Brazilian funding agency CNPq (Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico), under Proc. No. 480832/2011-0 and No. 307681/2012-2., In this work 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., Spanish Government TIN2013-40765-P TIN2012-32482, National Council for Scientific and Technological Development (CNPq) 480832/2011-0 307681/2012-2

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.

In this paper, we present a method to obtain a Fuzzy Image Segmentation from the hierarchical clustering algorithm Divide and Link. The Divide and Link algorithm consists on treating a digital image as a graph, then building spanning forest through a Kruskal scheme to successively sort the edges while partitions are obtained. This process is driven until all the pixels of the image are segmented, that is, there are as many regions as pixels.

This short note is devoted to introduce the discussion carried out along this special issue on the concept of Soft Computing by key researchers in the field. We shall stress some aspects of the conception and origins of Soft Computing, supported by the scientific relevance of its participants. The contributors will show their own view about a single question, What is Soft Computing?, covering answers from a general historical approach to the role of some specific tools within their expertise. This discussion represents an extremely interesting view about the concept of Soft Computing, its meaning, its related techniques and its relationship with close fields.

In this paper we prove that, under suitable conditions, Atanassov's K"@a 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 K"@a 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.

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.

In this note, we stress the relevance of developing tools for modelling uncertainty in information management and decision aiding, conceived as previous stages for decision making. We discuss the general framework of modelling uncertainty in decision making problems and briefly introduce the specific models we have selected to illustrate these ideas, as developed by researchers.

Humanitarian Non-Governmental Organisations (NGOs) play a growing role in the response to natural disasters, but despite being largely demanded, there is no available decision support system (DSS) specifically designed to address their problem. In this paper we present a decision support system (DSS) to aid those Humanitarian NGOs concerned with the response to natural disasters. Such a DSS has been designed avoiding sophisticated methodologies that may exceed the infrastructural requirements and constraints of emergency management by NGOs. A data-based, two-level knowledge methodology which allows damage assessment of multiple disaster scenarios is presented in order to address that problem. Validation results show viability of our approach.

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 prostate ultrasound images. The experimental results on ultrasound and natural images also allow us to determine the best choice of the function to represent the ignorance.

In this paper 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.