Considering the framework of weak concordance measures introduced by Liebscher in 2014, we propose and study convex weak concordance measures. This class of dependence measures contains as a proper subclass the class of all convex concordance measures, introduced and studied by Mesiar et al. in 2022, and thus it also covers the well-known concordance measures as Spearman's ρ , Gini's γ and Blomqvist's β . The class of all convex weak concordance measures also contains, for example, Spearman's footrule ϕ , which is not a concordance measure. In this paper, we first introduce basic convex weak concordance measures built in general by means of a single point ( u , v ) ∈ ▽ = { ( u , v ) ∈ ] 0 , 1 [ 2 | u ≥ v } and its transpose ( v , u ) only. Then, based on basic convex weak concordance measures and probability measures on the Borel subsets of ▽, two rather general constructions of convex weak concordance measures are proposed, discussed and exemplified. Inspired by Edwards et al., probability measures-based constructions are generalized to Borel measures on B ( ] 0 , 1 [ 2 ) -based constructions also allowing some infinite measures to be considered. Finally, it is shown that the presented constructions also cover the mentioned standard (convex weak) concordance measures ρ , γ , β , ϕ and provide alternative formulas for them.

Several extensions of the family of (bivariate) Eyraud-Farlie-Gumbel-Morgenstern copulas (EFGM copulas) are considered. Some of them are well-known from the literature, others have recently been suggested (copulas based on quadratic constructions, based on some forms of convexity, and polynomial copulas). For each of these extensions we analyze which properties of EFGM copulas are preserved (or even improved) and which are (partly) lost. Such properties can be structural (order theoretical or topological) in nature, or algebraic (symmetry or being a polynomial) or analytic (absolute continuity). Other examples are forms of convexity, quadrant dependence, and symmetry with respect to copula transformations. The last group of properties considered here is related to some dependence parameters.

In this paper, we study utility functions acting on score vectors from [0,1]n, and, in particular, we consider normed utility functions, which are also known as aggregation functions. We focus our study on the constrained additivity of the considered functions, i.e., on their additivity on special subdomains. Several well-known properties, such as homogeneity, unanimity, self-duality, shift-invariantness, etc., are shown to be particular instances of constrained additivity. Moreover, some non-trivial constrained additive utility functions are presented. We also open a problem concerning the existence of a minimal relation characterizing a considered constrained additivity and the standard additivity as well.

Inspired by the Zadeh approach to fuzzy connectives in fuzzy set theory and by some applications, we introduce and study set-based extended functions, and in particular, set-based extended aggregation functions. These functions reflect neither reordering nor repetition of input values, and, linking different arities, they introduce serious constraints for extended functions. A complete characterization of set-based extended (aggregation) functions is given, and some constructions of such functions are also proposed, including several examples.

In this paper we study some modifications and generalizations of additivity in the case of aggregation functions on the unit interval [0, 1]. We prove that some common properties of aggregation functions can be viewed as modifications of additivity — additivity with constraints.We also generalize additivity into pseudo-additivity and k-pseudo-additivity, and provide characterization of pseudo-additive and k-pseudo-additive aggregation functions, including maxitive and k-maxitive aggregation functions.

Admissible orders have brought the structure of chains in the framework of interval-valued fuzzy sets. However, a deeper study of functions monotone with respect to admissible orders is still missing in the literature. In this work, we consider the construction of negations and strong negations on intervals with respect to admissible orders, in particular, for the Xu and Yager and lexicographical orders, as well as for those based on Kαoperators. We introduce and discuss an approach to the construction of strong negations on intervals with respect to Kα,βorders based on an arbitrary couple of strong negations defined over the standard real interval [0, 1]. The introduced strong negations have a deep impact on all fields exploiting fuzzy methods dealing with intervals, allowing to introduce complements, dual aggregations, implications, entropies, etc.

In this paper, we introduce the notion of ordered directionally monotone function as a type of function which allows monotonicity along different directions in different points. In particular, these functions take into account the ordinal size of the coordinates of the inputs in order to fuse them. We show several examples of these functions and we study their properties. Finally, we present an illustrative example of an application which justifies the introduction and the study of the concept of ordered directional monotonicity.

In this contribution, we discuss the role and potential of pre-aggregation functions. This class of functions has the same boundary conditions as aggregation functions, but differs in the constraints related to function increase. Specifically, monotonicity is replaced by the so-called directional monotonicity. With this work, we attempt to shed light on the relationship between aggregation and pre-aggregation functions, as well as to discuss some construction methods of pre-aggregation functions.

In this work we propose a new generalization of the notion of monotonicity, the so-called ordered directionally monotonicity. With this new notion, the direction of increasingness or decreasingness at a given point depends on that specific point, so that it is not the same for every value on the domain of the considered function.

In this work we introduce the definition of pre-aggregation functions. These functions generalize aggregation functions, in the sense that they fulfill the same boundary conditions as the latter, but only directional monotonicity is demanded to them. We discuss a construction method which is inspired from the way the discrete Choquet integral is built and it replaces the product by other appropriate functions. Finally, we propose to apply them in fuzzy rule-based classification systems.

In this paper, we introduce the notion of preaggregation function. Such a function satisfies the same boundary conditions as an aggregation function, but, instead of requiring monotonicity, only monotonicity along some fixed direction (directional monotonicity) is required. We present some examples of such functions. We propose three different methods to build preaggregation functions. We experimentally show that in fuzzy rule-based classification systems, when we use one of these methods, namely, the one based on the use of the Choquet integral replacing the product by other aggregation functions, if we consider the minimum or the Hamacher product t-norms for such construction, we improve the results obtained when applying the fuzzy reasoning methods obtained using two classical averaging operators such as the maximum and the Choquet integral.

In this paper we axiomatically define weak homogeneity of a fuzzy subset, which means that its membership function fulfills at least the minimum properties required to represent the homogeneity of a region. We also provide several construction methods based on the homogeneity of an interval. Besides, we show an illustrative example of these functions applied to the problem of image segmentation.

We consider the problem of choosing a total order between intervals in multiexpert decision making problems. To do so, we first start researching the additivity of interval-valued aggregation functions. Next, we briefly treat the problem of preserving admissible orders by linear transformations. We study the construction of interval-valued ordered weighted aggregation operators by means of admissible orders and discuss their properties. In this setting, we present the definition of an interval-valued Choquet integral with respect to an admissible order based on an admissible pair of aggregation functions. The importance of the definition of the Choquet integral, which is introduced by us here, lies in the fact that if the considered data are pointwise (i.e., if they are not proper intervals), then it recovers the classical concept of this aggregation. Next, we show that if we make use of intervals in multiexpert decision making problems, then the solution at which we arrive may depend on the total order between intervals that has been chosen. For this reason, we conclude the paper by proposing two new algorithms such that the second one allows us, by means of the Shapley value, to pick up the best alternative from a set of winning alternatives provided by the first algorithm.

The stability of discrete universal integrals based on copulas is discussed and examined, both with respect to the norms L1 (Lipschitz stability) and L∞ (Chebyshev stability). Each of these integrals is shown to be 1-Lipschitz. Exactly the discrete universal integrals based on a copula which is stochastically increasing in its first coordinate turn out to be 1-Chebyshev. A new characterization of stochastically increasing Archimedean copulas is also given.

In this paper, we present some methods for constructing copulas with a given diagonal section that are not necessarily symmetric. An interesting application for the construction of copulas with given tail dependence coefficients is, hence, provided.

We define discrete copulas on a grid of the unit square and show that with each discrete copula there is associated, in a natural way, a bistochastic matrix. This is used in order to introduce the product of discrete copulas. Discrete copulas of order$n$are the smallest convex set containing the irreducible discrete copulas of order$n$introduced by Mayor .

Binary kernel aggregation operators are studied, especially construction of kernel aggregation operators from the given values at boundary points. Kernel aggregation operators uniquely determined by their marginal values are characterized. The importance of shift invariant aggregation operators and k–medians for construction of such aggregation operators is shown. Several examples are given.

Collapsed input-based aggregation leading to constrained aggregation operators is discussed. Constrained aggregation operators eliminating a non–distinguishing evaluation are studied. Moreover, collapse invariant aggregation operators are defined and a complete characterization of symmetric collapse invariant aggregation operators, which are independent of the position of the collapsed input, is given. Several examples are included.

In the paper the possibility of the construction of relevancy transformation operators introduced in 9 by means of triangular norms and copulas is studied. Several examples are included.

The k-order maxitive fuzzy measures on general spaces are discussed. The sum of ki-order maxitive measures is shown to be a ∑ ki-order maxitive fuzzy measure. The sum of two maxitive measures (possibility measures) is discussed and completely analyzed.