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A characterization of what we call pseudo-projective aggregation functions is shown, thus generalizing existing recent results in the literature. The approach followed is then applied to obtain certain characterizations of projective and pseudo-projective aggregation operators as well as to develop a social choice voting model. [ABSTRACT FROM AUTHOR]

In this contribution, we fully characterize one- and two-dimensional positively homogeneous aggregation functions and provide a method of checking non-decreasingness of positively homogeneous functions in higher dimensions. We also examine the interaction of positive homogeneity with super-addivity and sub-additivity of functions and link it to the concavity and convexity, respectively, of their generalized opposite diagonal functions. Some remarks on the computation of a linear super-additive transformation of aggregation functions are added. [ABSTRACT FROM AUTHOR]

Overlap functions are a class of aggregation functions that measure the overlapping degree between two values. They have been successfully applied as a fuzzy conjunction operation in several problems in which associativity is not required, such as image processing and classification. Interval-valued overlap functions were defined as an extension to express the overlapping of interval-valued data, and they have been usually applied when there is uncertainty regarding the assignment of membership degrees, as in interval-valued fuzzy rule-based classification systems. In this context, the choice of a total order for intervals can be significant, which motivated the recent developments on interval-valued aggregation functions and interval-valued overlap functions that are increasing to a given admissible order, that is, a total order that refines the usual partial order for intervals. Also, width preservation has been considered on these recent works, in an intent to avoid the uncertainty increase and guarantee the information quality, but no deeper study was made regarding the relation between the widths of the input intervals and the output interval, when applying interval-valued functions, or how one can control such uncertainty propagation based on this relation. Thus, in this paper we: (i) introduce and develop the concepts of width-limited interval-valued functions and width limiting functions, presenting a theoretical approach to analyze the relation between the widths of the input and output intervals of bivariate interval-valued functions, with special attention to interval-valued aggregation functions; (ii) introduce the concept of (a , b) -ultramodular aggregation functions, a less restrictive extension of one-dimension convexity for bivariate aggregation functions, which have an important predictable behaviour with respect to the width when extended to the interval-valued context; (iii) define width-limited interval-valued overlap functions, taking into account a function that controls the width of the output interval and a new notion of increasingness with respect to a pair of partial orders (≤ 1 , ≤ 2) ; (iv) present and compare three construction methods for these width-limited interval-valued overlap functions, considering a pair of orders (≤ 1 , ≤ 2) , which may be admissible or not, showcasing the adaptability of our developments. [ABSTRACT FROM AUTHOR]

Conditional distributivity (also called restricted distributivity) is a form of relaxed distributivity on the restricted domain. There are three options for conditional distributivity in literature. In this paper, we focus on type-III conditional distributivity equation involving nullnorms over uninorms. Firstly, we show that it can be transformed into the other two types of conditional distributivity equations involving t-norms/t-conorms over uninorms. Secondly, type-I and type-II conditional distributivity equations involving t-norms/t-conorms over uninorms that are related to our study and remain unknown are investigated. Finally, we show that this new perspective results in not only all known solutions, but new solutions as well. [ABSTRACT FROM AUTHOR]

In this paper, we explore the concept of local contrast of a fuzzy relation, which can be perceived as a measure for distinguishing the degrees of membership of elements within a defined region of an image. We introduce four distinct methods for constructing fuzzy local contrast: one uses a similarity measure, the second relies on the aggregation of similarity, the third is based on the aggregation of restricted equivalence, and the fourth utilizes the notion of equivalence. We further divide the constructions using similarity measures into two categories based on the two known definitions of similarity: distance-based similarity and aggregation function-based similarity. These construction methods also incorporate fuzzy implications and negations. Aggregation functions, which can be manipulated to enhance the effectiveness of the constructed fuzzy local contrast, play a significant role in most of our proposed constructions. For each construction method, several examples of fuzzy local contrasts are provided. The usefulness of the new fuzzy local contrasts is examined by applying them in image processing for salient region detection. • Aggregation functions used in a few construction methods of a fuzzy local contrast. • Numerous examples of fuzzy local contrasts provided. • Effectivness of aggregation functions examined in fuzzy local contrast in image processing for salient region detection. [ABSTRACT FROM AUTHOR]

The enumeration of logical connectives and aggregation functions defined on a finite chain has been a hot topic in the literature for the last decades. Multiple advantages can be derived from knowing a general formula about their cardinality, for instance, the ability to anticipate the computational cost required for generating operators with different properties. This is of paramount importance in image processing and decision making scenarios, where the identification of the most optimal operator is essential. Furthermore, it facilitates the examination of how constraining a certain property is in relation to its parent class. As a consequence, this paper aims to compile the main existing formulas and the methodologies with which they have been derived. Additionally, we introduce some novel formulas for the number of smooth discrete aggregation functions with neutral element or absorbing element, idempotent conjunctions, and commutative and idempotent conjunctions. [ABSTRACT FROM AUTHOR]

The (conditional) distributivity plays an important role in the construction of integrals. In this work, we aim to characterize the conditional distributivity equation only involving uninorms, where the second uninorm belongs to the well-known classes. [ABSTRACT FROM AUTHOR]

Homogeneity, which plays an essential role in decision making, economics and image processing, reflects the regularity of aggregation functions with respect to the inputs with the same ratio. Quasi-homogeneity is a relaxed homogeneity that reflects the original output as well as the same ratio. This paper is devoted to the characterization of all homogeneous/quasi-homogeneous binary aggregation functions in terms of single-argument functions. [ABSTRACT FROM AUTHOR]

In this work, the structure of uninorms not locally internal on the boundary is discussed. We first study a conjunctive uninorm U with neutral element e , where the boundary function U (1 , ⋅) of U only has two points of discontinuity and one of them is 0 or e. We partially obtain the structure of the conjunctive uninorm U. Its complete structure, under some additional conditions, is achieved as well. For the structure of a disjunctive uninorm U , whose boundary function U (0 , ⋅) only has two points of discontinuity and one of them is 1 or e , we obtain similar results. [ABSTRACT FROM AUTHOR]

Fuzzy measures model interactions between the inputs in aggregation problems. Their complexity grows exponentially with the dimensionality of the problem, and elicitation of fuzzy measure coefficients either from domain experts or from empirical data poses a significant challenge. The notions of k -additivity and k -maxitivity simplify the fuzzy measures by limiting interactions to subsets of up to k elements. Learning fuzzy measures from data is an important elicitation technique which relies on solving an optimisation problem. A heuristic learning algorithm to identify k -maxitive fuzzy measures from the data on the basis of HLMS (Heuristic Least Mean Squares) was recently presented in Murillo et al. (2017) [11]. We present an alternative formulation of the fitting problem which delivers a globally optimal solution through the solution of a mixed integer programming (MIP) problem. To deal with high computational cost of MIP in moderate to large dimensions, we also propose a simple MIP relaxation technique which involves solving two related linear programming problems. We also provide a linear programming formulation for fitting k -tolerant fuzzy measures. We discuss implementations of the fitting methods and present the results of numerical experiments. [ABSTRACT FROM AUTHOR]

The goal of this article is to characterize the inner structure of a special class of uninorms belonging to the class of uninorms with continuous underlying operators, the results of which bring us a step closer to the structure of uninorms that have continuous underlying operators. [ABSTRACT FROM AUTHOR]

In the reliability literature, several results have been presented to compare binary (two states) systems. Often, such results are obtained from copula-based extensions of fuzzy measures , where a fuzzy measure describes the structure of a system and a copula describes the stochastic dependence among the lifetimes of its components. Other similar results have been obtained in terms of the concept of signature. Here, we extend all those results to multi-state systems made up from binary components by suitably constructing corresponding mixed binary systems. For such a construction, we show how any fuzzy measure can be decomposed as a convex combination of { 0 , 1 } -valued fuzzy measures and how such a decomposition extends to the corresponding aggregation function. For a mixed system we can furthermore consider its signature and so we can also define a signature for the multi-state system. For mixed systems associated to different multi-state systems, we can thus obtain different comparison results, which can be translated into the corresponding comparisons for the parent multi-state systems. Stochastic comparisons are obtained for the discrete random variables which represent the states of two systems at time t , as well. The arguments in the paper will be illustrated by means of examples and related remarks. [ABSTRACT FROM AUTHOR]

This paper introduces the theoretical framework for a generalization of C F 1 F 2 -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 g C F 1 F 2 -integrals, is based on the so-called pseudo pre-aggregation function pairs (F 1 , F 2) , which are pairs of fusion functions satisfying a minimal set of requirements in order to guarantee that the g C F 1 F 2 -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 g C F 1 F 2 -integrals. We study several properties of g C F 1 F 2 -integrals, considering different constraints for the functions F 1 and F 2 , and state under which conditions g C F 1 F 2 -integrals present or not averaging behaviors. Several examples of g C F 1 F 2 -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. [ABSTRACT FROM AUTHOR]

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 behaviour of these functions is slightly better than some other methods in the literature. [ABSTRACT FROM AUTHOR]

Abstract In this paper new notions of fuzzy equivalences interpreted as fuzzy connectives are introduced. One of the axioms of such fuzzy equivalences is transitivity defined by the use of a fuzzy conjunction or both a fuzzy conjunction and a parameter α from the unit interval, that is why they are called fuzzy α – C -equivalences. In particular, fuzzy C -equivalences, fuzzy semi- C -equivalences as well as fuzzy weak C -equivalences are considered. Dependencies between the new types of fuzzy equivalences are presented. Moreover, the problem of preservation of axioms of such defined fuzzy equivalences by aggregation functions is examined. As a conclusion, ways of generating new fuzzy equivalences of the considered type from given ones are indicated. [ABSTRACT FROM AUTHOR]

Abstract For certain purposes it would be of interest to find aggregation functions for which the aggregate value of the shortfalls coincides with the shortfall of the aggregate value of the original attainments, i.e. self-dual aggregation functions. It is crucial to know the scale type in which attainments are measured and how to reverse that scale to measure the corresponding shortfalls properly. In this paper we consider alternative definitions of shortfalls depending on the scale type and extend some characterisation results to this new framework. [ABSTRACT FROM AUTHOR]

Abstract Recently, Choquet-like Copula-based aggregation functions were introduced and discussed by Lucca et al. Afterwards, CMin-integral (i.e., a Choquet-like minimum-based operator) was examined by Dimuro et al. We show the link of these aggregation functions with universal integrals of Klement et al. In particular, CMin-integral is shown to be an alternative approach to the Sugeno integral. Assuming symmetry, an axiomatic characterization of the discussed aggregation functions is given and their coincidence with Ordered Modular Averages (in short, OMA) operators is shown. [ABSTRACT FROM AUTHOR]

Abstract Aggregations functions, that are obtainable as copula-based extensions of fuzzy measures, may emerge in several applied fields. We concentrate attention on the probabilistic interpretation of such objects. Starting from this issue, we point out the related meaning in the context of multi-attribute target-based utilities and multi-state reliability systems. Analogies and relations between the two fields of decisions under uncertainty and systems' reliability have been already remarked in the literature. Here, we prove in particular that aggregation functions, specifically emerging in the setting of multi-attribute (target-based) utilities and in the setting of multi-state systems, do exactly share the same analytic structure. We will show that this identification emerges naturally when the multi-state systems are formed with binary components. From a technical viewpoint, the comparison between two different representations for the considered aggregations, and the specific role of the Möbius transform, have a basic relevance in the proofs of our results and in the development of related discussions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study semi-t-operators on bounded lattices. A series of construction methods of semi-t-operators on bounded lattices are given by using semi-triangular (co)norms (semi-t-(co)norms for short) and order-preserving functions on bounded lattices, and it is shown that the methods obtained in this paper can cover several construction methods existing in the literature. By analyzing the properties of the constructed semi-t-operators, we further obtain the equivalent characterizations for their corresponding representations. The results of this paper will provide a larger operator selection space for practical applications. [ABSTRACT FROM AUTHOR]

This note focuses on ordering two families of rank-dependent poverty measures in terms of their distribution-sensitivity. It has been proved that a real value, between 1/2 and 1, called orness , which is assigned to every rank-dependent poverty measure, can be interpreted as a distribution-sensitivity indicator. Therefore, the rank-dependent poverty measures can be classified in terms of their distribution-sensitivity using the orness value assigned to them. This ranking has already been carried out for numerous poverty measures. However, two families of poverty measures, the Kakwani and the S-Gini families, which are defined for every real parameter larger than one, have only been ranked for natural values of their parameters. This note broadens the classification of these families for every real parameter larger than one, that is, for every member of these two families. It also provides a ranking between the two families for the same parameter. It concludes that for higher values of the parameter, the families will be more sensitive to the bottom part of the distribution. Furthermore, for the same value of the parameter, the Kakwani index will be more sensitive to poor incomes than the S-Gini index. In addition, we will see that the proposed ranking for the two families in terms of the orness value will be analogous to other distribution-sensitivity criteria existing in the literature. [ABSTRACT FROM AUTHOR]

We propose two distribution-sensitivity criteria to classify the rank-dependent welfare functions. These criteria compare the reaction of the welfare function to lossy transfers and lossy equalization transfers among individuals. We see that these classifications in terms of their distribution-sensitivity to these transfers can be established focusing only on the weights assigned to each welfare function. We also propose a criterion to sort the rank-dependent welfare functions and the rank-dependent poverty measures in terms of a mathematical value called orness . We provide a classification in terms of the orness value for the welfare functions of the S-Gini family, the Bonferroni index and the De Vergottini index. Another classification is provided for the poverty measures of the Poverty Gap Ratio, the Sen indices, the Thon index and the Thon family of indices, the Kakwani family of indices and the S-Gini family of indices. Finally, we prove that the orness classification for welfare functions and the orness classification for poverty measures can be interpreted as a distribution-sensitive classification since they have a direct link with the classifications proposed above. Moreover, we see that for a subset of welfare functions and another subset of poverty measures, the orness classification and the distribution-sensitivity classification based on lossy transfers and lossy equalization transfers are equivalent. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The migrativity equation with interesting applications in decision making and image processing has been extensively discussed involving different kinds of aggregation functions from t-norms and t-conorms to uninorms, nullnorms and some generalizations of them. In recent papers, the already known results concerning the migrativity of two uninorms are based on the assumption that both uninorms belong to one of the most studied classes of uninorms. In this paper we will explore the migrativity equation involving uninorms in a most general setting. Specifically, we will study the migrativity between two uninorms in the cases when the second uninorm lies in any of the most studied classes of uninorms, but the first one is any uninorm with no further assumptions. We will show along the paper that many new solutions appear from this new point of view that were not included in the previous approaches. [ABSTRACT FROM AUTHOR]

In this paper, we point out several problems in the different definitions (and related results) of penalty functions found in the literature. Then, we propose a new standard definition of penalty functions that overcomes such problems. Some results related to averaging aggregation functions, in terms of penalty functions, are presented, as the characterization of averaging aggregation functions based on penalty functions. Some examples are shown, as the penalty functions based on spread measures, which happen to be continuous. We also discuss the definition of quasi-penalty functions, in order to deal with non-monotonic (or weakly/directionally monotonic) averaging functions. [ABSTRACT FROM AUTHOR]

Fang and Hu studied the distributivity property for S -uninorms. However, they just discussed the distributivity equation under the condition that the involving S -uninorms are assumed to be in U min. In addition, some of their results are incorrect. In this work, we discuss the distributivity between two binary operations, where one of them is an arbitrary S -uninorm and the other is a given binary operation such as a t-norm, a t-conorm, a uninorm in U max (U min) and an S -uninorm with underlying uninorm being in U min , respectively. The results in the work generalize or revise some of the existing ones. [ABSTRACT FROM AUTHOR]

Drygaś et al. showed that Lemma 3.3(i) in [5] is false by a counterexample (see Example 2 in [4] for details). The goal of this short note is to indicate that Example 2 in [4] is wrong and Lemma 3.3(i) in [5] is valid. [ABSTRACT FROM AUTHOR]

In this paper, we review the role of self-duality in the theory of aggregation functions, the dual decomposition of aggregation functions into a self-dual core and an anti-self-dual remainder, and some applications to welfare, inequality, and poverty measures. [ABSTRACT FROM AUTHOR]

In this work we present an image reduction algorithm based on averaging aggregation functions. We axiomatically define the concepts of image reduction operator and local reduction operator. We study the construction of the latter by means of averaging functions and we propose an image reduction algorithm (image reduction operator). We analyze the properties of several averaging functions and their effect on the image reduction algorithm. Finally, we present experimental results where we apply our algorithm in two different applications, analyzing the best operators for each concrete application. [ABSTRACT FROM AUTHOR]

We consider flipping transformations of multivariate aggregation functions and we investigate the closure of these transformations with respect to the class of aggregation functions with annihilator element equal to 0. Moreover, the consecutive application of flipping transformations is also discussed. The results present interesting connections both with quasi-copulas and with suitable modifications of the n-increasing property of a multivariate probability distribution function. [ABSTRACT FROM AUTHOR]

Abstract: In the traditional framework, social welfare functions depend on the mean income and on the income inequality. An alternative illfare framework has been developed to take into account the disutility of unfavorable variables. The illfare level is assumed to increase with the inequality of the distribution. In some social and economic fields, such as those related to employment, health, education, or deprivation, the characteristics of the individuals in the population are represented by bounded variables, which encode either achievements or shortfalls. Accordingly, both the social welfare and the social illfare levels may be assessed depending on the framework we focus on. In this paper we propose a unified dual framework in which welfare and illfare levels can both be investigated and analyzed in a natural way. The dual framework leads to the consistent measurement of achievements and shortfalls, thereby overcoming one important difficulty of the traditional approach, in which the focus on achievements or shortfalls often leads to different inequality rankings. A number of welfare functions associated with inequality indices are OWA operators. Specifically this paper considers the welfare functions associated with the classical inequality measures due to Gini, Bonferroni, and De Vergottini. These three indices incorporate different value judgments in the measurement of inequality, leading to different behavior under income transfers between individuals in the population. In the bounded variables representation, we examine the dual decomposition and the orness degree of the three classical welfare/illfare functions in the standard framework of aggregation functions on the domain. The dual decomposition of each welfare/illfare function into a self-dual central index and an anti-self-dual inequality index leads to the consistent measurement of achievements and shortfalls. [Copyright &y& Elsevier]

Abstract: In many practical applications, the need arises to aggregate data of varying dimension. Following from the self-identity property, some recent studies have looked at the stability of aggregation operators in terms of their behavior as the dimensionality is increased from to n. We use the penalty-based representation of aggregation functions in order to investigate the conditions for weighting vectors associated with some important weighted families, extending on the results already established for quasi-arithmetic means. In particular, we obtain results for quasi-medians and functions that involve a reordering of the inputs such as the OWA and order statistics. [Copyright &y& Elsevier]

Abstract: Aggregation of real-valued evaluations of elements x, y from a set U is investigated in order to find an approximate evaluation of the composition of x and y performed by a binary operation defined on U. The notions of a lower bound and an upper bound of an evaluation function with respect to a binary operation are introduced and illustrated with examples. Approximate evaluations of composite concepts described by the union of granules of a finite fuzzy set are studied in detail. [Copyright &y& Elsevier]

Abstract: We examine the AHP in the framework of Choquet integration and we propose an extension of the standard AHP aggregation scheme on the basis of the Shapley values associated with the criteria. In our model a measure of dominance inconsistency between criteria is defined in terms of the totally inconsistent matrix associated with the main pairwise comparison matrix of the AHP. The measure of dominance inconsistency is then used to construct a non-additive capacity whose associated Shapley values reduce to the standard AHP priority weights in the consistency case. In the general inconsistency case, however, the extended aggregation scheme based on the Shapley weighted mean tends to attenuate (resp. emphasize) the priority weights of the criteria with higher (resp. lower) average dominance inconsistency with respect to the other criteria. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to analyze the behavior of aggregation functions when the inputs are contradictory. This may be a useful criterion helping to choose the most appropriate function for solving a given problem. With that goal, bivariate aggregation functions are classified depending on the output they associate to contradictory couples of the form , where N is a strong negation. The main properties of the newly defined classes are studied. Examples of functions in each class are provided, paying special attention to the most important families of aggregation functions, such as t-norms, copulas, symmetric sums, uninorms or nullnorms. [Copyright &y& Elsevier]

Abstract: We present a generalization of Choquet integral in which the capacity depends also on the value of the aggregated variables. We show that as particular cases of our generalization of Choquet integral there are the Sugeno integral, the Šipoš integral and the Cumulative Prospect Theory functional. We also show that many concepts such as Möbius transform, importance index, interaction index, -order capacities and OWA operators, introduced in the research about Choquet integral, can be generalized in the considered context. [Copyright &y& Elsevier]

Abstract: We review various representations of the median and related aggregation functions. An advantage of the median is that it discards extreme values of the inputs, and hence exhibits a better central tendency than the arithmetic mean. However, the value of the median depends on only one or two central inputs. Our aim is to design median-like aggregation functions whose value depends on several central inputs. Such functions will preserve the stability of the median against extreme values, but will take more inputs into account. A method based on graduation curves is presented. [Copyright &y& Elsevier]

Abstract: This article discusses Lipschitz properties of generated aggregation functions. Such generated functions include triangular norms and conorms, quasi-arithmetic means, uninorms, nullnorms and continuous generated functions with a neutral element. The Lipschitz property guarantees stability of aggregation operations with respect to input inaccuracies, and is important for applications. We provide verifiable sufficient conditions to determine when a generated aggregation function holds the k-Lipschitz property, and calculate the Lipschitz constants of power means. We also establish sufficient conditions which guarantee that a generated aggregation function is not Lipschitz. We found the only 1-Lipschitz generated function with a neutral element . [Copyright &y& Elsevier]

Abstract: We study graded properties (-properties) of fuzzy relations, which are parameterized versions of properties of a fuzzy relation defined by L.A. Zadeh. Namely, we take into account fuzzy relations which are: -reflexive, -irreflexive, -symmetric, -antisymmetric, -asymmetric, -connected, -transitive, where . We also pay our attention to the composed versions of these basic properties, e.g. an -equivalence, -orders. We also consider the so-called “weak” properties of fuzzy relations which are the weakest versions of the standard properties of fuzzy relations. We take into account the same types of properties as in the case of the graded ones. Using functions of n variables we consider an aggregated fuzzy relation of given fuzzy relations. We give conditions for functions to preserve graded and weak properties of fuzzy relations. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to choose diverse definitions of generalized logical connectives and to present them in a coherent order, from the weakest to the richest. Such a rich list of notions allows us to consider the problem of admissible aggregations in the presented classes of unary and binary operations. This gives a contribution to the discussion of tolerance analysis in soft computing, decision making, approximate reasoning, and fuzzy control. First, we present a short survey of the development of MV-logic''s connectives. Next, we discuss postulates used for generalized logical connectives. Finally, we describe families of weak connectives and indicate the preservation of their properties by some aggregation functions. [Copyright &y& Elsevier]

Abstract: The scalar evaluation of the aggregation of elements from a complete lattice via the operation meet (join) is compared with the aggregation of scalar evaluations of these elements via a t-norm (t-conorm). Based on this comparison, the notion of a T-evaluator (S-evaluator) is introduced and its basic properties are studied. The relationship between T-(S-) evaluators, fuzzy measures and generalized filters is investigated. An application of T-evaluators in fuzzy preference modelling is presented. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: The paper gives sufficient and necessary conditions of consensus requirement for eventwise aggregation of various families of lower probabilities, in particular, of coherent lower probabilities, and properties of the corresponding aggregation functions. [Copyright &y& Elsevier]