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   Nonstandard n-distances based on certain geometric constructions; Marichal, Jean-Luc  in Beiträge zur Algebra und Geometrie (2023), 64(1), 107-126 The concept of n-distance was recently introduced to generalize the classical definition of distance to functions of n arguments. In this paper we investigate this concept through a number of examples ... [more ▼] The concept of n-distance was recently introduced to generalize the classical definition of distance to functions of n arguments. In this paper we investigate this concept through a number of examples based on certain geometrical constructions. In particular, our study shows to which extent the computation of the best constant associated with an n-distance may sometimes be difficult and tricky. It also reveals that two important graph theoretical concepts, namely the total length of the Euclidean Steiner tree and the total length of the minimal spanning tree constructed on n points, are instances of n-distances. [less ▲] Detailed reference viewed: 114 (20 UL)On the best constants associated with n-distances; Marichal, Jean-Luc in Acta Mathematica Hungarica (2020), 161(1), 341-365 We pursue the investigation of the concept of n-distance, an n-variable version of the classical concept of distance recently introduced and investigated by Kiss, Marichal, and Teheux. We especially focus ... [more ▼] We pursue the investigation of the concept of n-distance, an n-variable version of the classical concept of distance recently introduced and investigated by Kiss, Marichal, and Teheux. We especially focus on the challenging problem of computing the best constant associated with a given n-distance. In particular, we define and investigate the best constants related to partial simplex inequalities. We also introduce and discuss some subclasses of n-distances defined by considering some properties. Finally, we discuss an interesting link between the concepts of n-distance and multidistance. [less ▲] Detailed reference viewed: 132 (24 UL)On idempotent n-ary uninormsDevillet, Jimmy ; ; Marichal, Jean-Luc in Torra, Vicenç; Narukawa, Yasuo; Pasi, Gabriella (Eds.) et al Modeling Decisions for Artifical Intelligence (2019, July 24) In this paper we describe the class of idempotent n-ary uninorms on a given chain.When the chain is finite, we axiomatize the latter class by means of the following conditions: associativity ... [more ▼] In this paper we describe the class of idempotent n-ary uninorms on a given chain.When the chain is finite, we axiomatize the latter class by means of the following conditions: associativity, quasitriviality, symmetry, and nondecreasing monotonicity. Also, we show that associativity can be replaced with bisymmetry in this new axiomatization. [less ▲] Detailed reference viewed: 141 (10 UL)Characterizations of biselective operationsDevillet, Jimmy ; in Acta Mathematica Hungarica (2019), 157(2), 387-407 Let X be a nonempty set and let i,j in {1,2,3,4}. We say that a binary operation F:X^2 -> X is (i,j)-selective if F(F(x_1,x_2),F(x_3,x_4)) = F(x_i,x_j), for all x_1,x_2,x_3,x_4 in X. In this paper we ... [more ▼] Let X be a nonempty set and let i,j in {1,2,3,4}. We say that a binary operation F:X^2 -> X is (i,j)-selective if F(F(x_1,x_2),F(x_3,x_4)) = F(x_i,x_j), for all x_1,x_2,x_3,x_4 in X. In this paper we provide characterizations of the class of (i,j)-selective operations. We also investigate some subclasses by adding algebraic properties such as associativity or bisymmetry. [less ▲] Detailed reference viewed: 160 (29 UL)Characterizations of idempotent n-ary uninormsDevillet, Jimmy ; ; Marichal, Jean-Luc in 38th Linz Seminar on Fuzzy Set Theory (2019, February 05) In this paper we provide a characterization of the class of idempotent n-ary uninorms on a given chain. When the chain is finite, we also provide anaxiomatic characterization of the latter class by means ... [more ▼] In this paper we provide a characterization of the class of idempotent n-ary uninorms on a given chain. When the chain is finite, we also provide anaxiomatic characterization of the latter class by means of four conditions only: associativity, quasitriviality, symmetry, and nondecreasing monotonicity. In particular, we show that associativity can be replaced with bisymmetry in this axiomatization. [less ▲] Detailed reference viewed: 124 (19 UL)An n-ary generalization of the concept of distance; Marichal, Jean-Luc ; Teheux, Bruno Scientific Conference (2018, July 03) Detailed reference viewed: 103 (6 UL) |
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