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The operations $C_{f,g}\colon I^{2}\rightarrow I$ of the form \begin{equation*} C_{f,g}\left( x,y\right) =\left ... [more ▼]Let $I\subset (0,\infty )$ be an interval that is closed with respect to the multiplication. The operations $C_{f,g}\colon I^{2}\rightarrow I$ of the form \begin{equation*} C_{f,g}\left( x,y\right) =\left( f\circ g\right) ^{-1}\left( f\left( x\right) \cdot g\left( y\right) \right) \text{,} \end{equation*} where $f,g$ are bijections of $I$ are considered. Their connections with generalized weighted quasi-geometric means is presented. It is shown that invariance\ question within the class of this operations leads to means of iterative type and to a problem on a composite functional equation. An application of the invariance identity to determine effectively the limit of the sequence of iterates of some generalized quasi-geometric mean-type mapping, and the form of all continuous functions which are invariant with respect to this mapping are given. The equality of two considered operations is also discussed. [less ▲]Detailed reference viewed: 84 (6 UL) Characterizations of idempotent discrete uninormsCouceiro, Miguel; Devillet, Jimmy ; Marichal, Jean-Luc in Fuzzy Sets and Systems (2018), 334In this paper we provide an axiomatic characterization of the idempotent discrete uninorms by means of three conditions only: conservativeness, symmetry, and nondecreasing monotonicity. We also provide an ... [more ▼]In this paper we provide an axiomatic characterization of the idempotent discrete uninorms by means of three conditions only: conservativeness, symmetry, and nondecreasing monotonicity. We also provide an alternative characterization involving the bisymmetry property. Finally, we provide a graphical characterization of these operations in terms of their contour plots, and we mention a few open questions for further research. [less ▲]Detailed reference viewed: 280 (54 UL) Relaxations of associativity and preassociativity for variadic functionsCouceiro, Miguel; Marichal, Jean-Luc ; Teheux, Bruno in Fuzzy Sets and Systems (2016), 299In this paper we consider two properties of variadic functions, namely associativity and preassociativity, that are pertaining to several data and language processing tasks. We propose parameterized ... [more ▼]In this paper we consider two properties of variadic functions, namely associativity and preassociativity, that are pertaining to several data and language processing tasks. We propose parameterized relaxations of these properties and provide their descriptions in terms of factorization results. We also give an example where these parameterized notions give rise to natural hierarchies of functions and indicate their potential use in measuring the degrees of associativeness and preassociativeness. We illustrate these results by several examples and constructions and discuss some open problems that lead to further directions of research. [less ▲]Detailed reference viewed: 206 (24 UL) Preassociative aggregation functionsMarichal, Jean-Luc ; Teheux, Bruno in Fuzzy Sets and Systems (2015), 268The classical property of associativity is very often considered in aggregation function theory and fuzzy logic. In this paper we provide axiomatizations of various classes of preassociative functions ... [more ▼]The classical property of associativity is very often considered in aggregation function theory and fuzzy logic. In this paper we provide axiomatizations of various classes of preassociative functions, where preassociativity is a generalization of associativity recently introduced by the authors. These axiomatizations are based on existing characterizations of some noteworthy classes of associative operations, such as the class of Aczélian semigroups and the class of t-norms. [less ▲]Detailed reference viewed: 304 (23 UL) Axiomatizations of Lovász extensions of pseudo-Boolean functionsCouceiro, Miguel ; Marichal, Jean-Luc in Fuzzy Sets and Systems (2011), 181(1), 28-38Three important properties in aggregation theory are investigated, namely horizontal min-additivity, horizontal max-additivity, and comonotonic additivity, which are defined by certain relaxations of the ... [more ▼]Three important properties in aggregation theory are investigated, namely horizontal min-additivity, horizontal max-additivity, and comonotonic additivity, which are defined by certain relaxations of the Cauchy functional equation in several variables. We show that these properties are equivalent and we completely describe the functions characterized by them. By adding some regularity conditions, these functions coincide with the Lovász extensions vanishing at the origin, which subsume the discrete Choquet integrals. We also propose a simultaneous generalization of horizontal min-additivity and horizontal max-additivity, called horizontal median-additivity, and we describe the corresponding function class. Additional conditions then reduce this class to that of symmetric Lovász extensions, which includes the discrete symmetric Choquet integrals. [less ▲]Detailed reference viewed: 134 (8 UL) Characterizations of discrete Sugeno integrals as polynomial functions over distributive latticesCouceiro, Miguel ; Marichal, Jean-Luc in Fuzzy Sets and Systems (2010), 161(5), 694-707We discuss several characterizations of discrete Sugeno integrals over bounded distributive lattices as particular cases of lattice polynomial functions, that is, functions which can be represented in the ... [more ▼]We discuss several characterizations of discrete Sugeno integrals over bounded distributive lattices as particular cases of lattice polynomial functions, that is, functions which can be represented in the language of bounded lattices using variables and constants. We also consider the subclass of term functions as well as the classes of symmetric polynomial functions and weighted infimum and supremum functions, and present their characterizations, accordingly. Moreover, we discuss normal form representations of these functions. [less ▲]Detailed reference viewed: 125 (9 UL) On the associativity functional equationMarichal, Jean-Luc in Fuzzy Sets and Systems (2000), 114(3), 381-389Let [a,b] be any bounded closed real interval. The class of all continuous, nondecreasing, associative functions M : [a,b]^2 --> [a,b] fulfilling the boundary conditions M(a,a)=a and M(b,b)=b is described.Detailed reference viewed: 167 (7 UL) On Sugeno integral as an aggregation functionMarichal, Jean-Luc in Fuzzy Sets and Systems (2000), 114(3), 347-365The Sugeno integral, for a given fuzzy measure, is studied under the viewpoint of aggregation. In particular, we give some equivalent expressions of it. We also give an axiomatic characterization of the ... [more ▼]The Sugeno integral, for a given fuzzy measure, is studied under the viewpoint of aggregation. In particular, we give some equivalent expressions of it. We also give an axiomatic characterization of the class of all the Sugeno integrals. Some particular subclasses, such as the weighted maximum and minimum functions are investigated as well. [less ▲]Detailed reference viewed: 134 (3 UL) Characterization of some aggregation functions stable for positive linear transformationsMarichal, Jean-Luc ; Mathonet, Pierre ; Tousset, Ericin Fuzzy Sets and Systems (1999), 102(2), 293-314This paper deals with the characterization of some classes of aggregation functions often used in multicriteria decision making problems. The common properties involved in these characterizations are ... [more ▼]This paper deals with the characterization of some classes of aggregation functions often used in multicriteria decision making problems. The common properties involved in these characterizations are "increasing monotonicity" and "stability for positive linear transformations". Additional algebraic properties related to associativity allow to completely specify the functions. [less ▲]Detailed reference viewed: 105 (5 UL) 1