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Characterizations of quasitrivial symmetric nondecreasing associative operations Devillet, Jimmy ; Kiss, Gergely ; Marichal, Jean-Luc in Semigroup Forum (in press) In this paper we are interested in the class of n-ary operations on an arbitrary chain that are quasitrivial, symmetric, nondecreasing, and associative. We first provide a description of these operations ... [more ▼] In this paper we are interested in the class of n-ary operations on an arbitrary chain that are quasitrivial, symmetric, nondecreasing, and associative. We first provide a description of these operations. We then prove that associativity can be replaced with bisymmetry in the definition of this class. Finally we investigate the special situation where the chain is finite. [less ▲] Detailed reference viewed: 36 (16 UL)Quasitrivial semigroups: characterizations and enumerations ; Devillet, Jimmy ; Marichal, Jean-Luc in Semigroup Forum (in press) We investigate the class of quasitrivial semigroups and provide various characterizations of the subclass of quasitrivial and commutative semigroups as well as the subclass of quasitrivial and order ... [more ▼] We investigate the class of quasitrivial semigroups and provide various characterizations of the subclass of quasitrivial and commutative semigroups as well as the subclass of quasitrivial and order-preserving semigroups. We also determine explicitly the sizes of these classes when the semigroups are defined on finite sets. As a byproduct of these enumerations, we obtain several new integer sequences. [less ▲] Detailed reference viewed: 133 (42 UL)An n-ary generalization of the concept of distance ; Marichal, Jean-Luc ; Teheux, Bruno Scientific Conference (2018, July 03) Detailed reference viewed: 34 (2 UL)Associative and quasitrivial operations on finite sets: characterizations and enumeration ; Devillet, Jimmy ; Marichal, Jean-Luc Scientific Conference (2018, July 02) We investigate the class of binary associative and quasitrivial operations on a given finite set. Here the quasitriviality property (also known as conservativeness) means that the operation always outputs ... [more ▼] We investigate the class of binary associative and quasitrivial operations on a given finite set. Here the quasitriviality property (also known as conservativeness) means that the operation always outputs one of its input values. We also examine the special situations where the operations are commutative and nondecreasing, in which cases the operations reduce to discrete uninorms (which are discrete fuzzy connectives playing an important role in fuzzy logic). Interestingly, associative and quasitrivial operations that are nondecreasing are characterized in terms of total and weak orderings through the so-called single-peakedness property introduced in social choice theory by Duncan Black. We also address and solve a number of enumeration issues: we count the number of binary associative and quasitrivial operations on a given finite set as well as the number of those operations that are commutative and/or nondecreasing. [less ▲] Detailed reference viewed: 34 (3 UL)A generalization of the concept of distance based on the simplex inequality Kiss, Gergely ; Marichal, Jean-Luc ; Teheux, Bruno in Beitraege zur Algebra und Geometrie = Contributions to Algebra and Geometry (2018), 59(2), 247266 We introduce and discuss the concept of \emph{$n$-distance}, a generalization to $n$ elements of the classical notion of distance obtained by replacing the triangle inequality with the so-called simplex ... [more ▼] We introduce and discuss the concept of \emph{$n$-distance}, a generalization to $n$ elements of the classical notion of distance obtained by replacing the triangle inequality with the so-called simplex inequality \[ d(x_1, \ldots, x_n)~\leq~K\, \sum_{i=1}^n d(x_1, \ldots, x_n)_i^z{\,}, \qquad x_1, \ldots, x_n, z \in X, \] where $K=1$. Here $d(x_1,\ldots,x_n)_i^z$ is obtained from the function $d(x_1,\ldots,x_n)$ by setting its $i$th variable to $z$. We provide several examples of $n$-distances, and for each of them we investigate the infimum of the set of real numbers $K\in\left]0,1\right]$ for which the inequality above holds. We also introduce a generalization of the concept of $n$-distance obtained by replacing in the simplex inequality the sum function with an arbitrary symmetric function. [less ▲] Detailed reference viewed: 95 (26 UL)Characterizations of idempotent discrete uninorms ; Devillet, Jimmy ; Marichal, Jean-Luc in Fuzzy Sets & Systems (2018), 334 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 ... [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: 152 (37 UL)Joint signature of two or more systems with applications to multistate systems made up of two-state components Marichal, Jean-Luc ; ; et al in European Journal of Operational Research (2017), 263(2), 559-570 The structure signature of a system made up of n components having continuous and i.i.d. lifetimes was defined in the eighties by Samaniego as the n-tuple whose k-th coordinate is the probability that the ... [more ▼] The structure signature of a system made up of n components having continuous and i.i.d. lifetimes was defined in the eighties by Samaniego as the n-tuple whose k-th coordinate is the probability that the k-th component failure causes the system to fail. More recently, a bivariate version of this concept was considered as follows. The joint structure signature of a pair of systems built on a common set of components having continuous and i.i.d. lifetimes is a square matrix of order n whose (k,l)-entry is the probability that the k-th failure causes the first system to fail and the l-th failure causes the second system to fail. This concept was successfully used to derive a signature-based decomposition of the joint reliability of the two systems. In the first part of this paper we provide an explicit formula to compute the joint structure signature of two or more systems and extend this formula to the general non-i.i.d. case, assuming only that the distribution of the component lifetimes has no ties. We also provide and discuss a necessary and sufficient condition on this distribution for the joint reliability of the systems to have a signature-based decomposition. In the second part of this paper we show how our results can be efficiently applied to the investigation of the reliability and signature of multistate systems made up of two-state components. The key observation is that the structure function of such a multistate system can always be additively decomposed into a sum of classical structure functions. Considering a multistate system then reduces to considering simultaneously several two-state systems. [less ▲] Detailed reference viewed: 80 (11 UL)Associative and quasitrivial operations on finite sets (invited lecture) Marichal, Jean-Luc ; ; Devillet, Jimmy Scientific Conference (2017, November 10) Detailed reference viewed: 50 (12 UL)On quasitrivial and associative operations Devillet, Jimmy ; ; Marichal, Jean-Luc Presentation (2017, October 25) Detailed reference viewed: 38 (12 UL)Enumerating quasitrivial semigroups Devillet, Jimmy ; ; Marichal, Jean-Luc Presentation (2017, October 03) We investigate the class of binary associative and quasitrivial operations on a given finite set. Here quasitriviality (also known as conserva-tiveness) means that the operation always outputs one of its ... [more ▼] We investigate the class of binary associative and quasitrivial operations on a given finite set. Here quasitriviality (also known as conserva-tiveness) means that the operation always outputs one of its input values. We also examine the special situations where the operations are commutative and nondecreasing. In the latter case, these operations reduce to discrete uninorms, which are discrete fuzzy connectives that play an important role in fuzzy logic. As we will see nondecreasing, associative and quasitrivial operations are chara-cterized in terms of total and weak orderings through the so-called single-peakedness property introduced in social choice theory by Duncan Black. This will enable visual interpretaions of the above mentioned algebraic properties. Motivated by these results, we will also address a number of counting issues: we enumerate all binary associative and quasitrivial operations on a given finite set as well as of those operations that are commutative, are nondecreasing, have neutral and/or annihilator elements. As we will see, these considerations lead to several, previously unknown, integer sequences. [less ▲] Detailed reference viewed: 49 (14 UL)Sur les uninormes discrètes idempotentes ; Devillet, Jimmy ; Marichal, Jean-Luc in Couceiro, Miguel; Devillet, Jimmy; Marichal, Jean-Luc (Eds.) LFA 2017 - Rencontres francophones sur la logique floue et ses applications (2017, October) In this paper we provide two axiomatizations of the class of idempotent discrete uninorms as conservative binary operations, where an operation is conservative if it always outputs one of its input values ... [more ▼] In this paper we provide two axiomatizations of the class of idempotent discrete uninorms as conservative binary operations, where an operation is conservative if it always outputs one of its input values. More precisely we first show that the idempotent discrete uninorms are exactly those operations that are conservative, symmetric, and nondecreasing. Then we show that, in this characterization, symmetry can be replaced with both bisymmetry and existence of a neutral element. [less ▲] Detailed reference viewed: 35 (6 UL)Integer sequence #A292933 Marichal, Jean-Luc Diverse speeches and writings (2017) Number of associative and quasitrivial binary operations on {1,...,n} that have neutral elements. Also: Number of associative and quasitrivial binary operations on {1,...,n} that have annihilator elements. Detailed reference viewed: 38 (6 UL)Integer sequence #A292932 Marichal, Jean-Luc Diverse speeches and writings (2017) Number of associative and quasitrivial binary operations on {1,...,n}. Convention a(0)=1. Detailed reference viewed: 43 (12 UL)Integer sequence #A292934 Marichal, Jean-Luc Diverse speeches and writings (2017) Number of associative and quasitrivial binary operations on {1,...,n} that have both neutral and annihilator elements. Detailed reference viewed: 36 (5 UL)A classification of polynomial functions satisfying the Jacobi identity over integral domains Marichal, Jean-Luc ; in Aequationes Mathematicae (2017), 91(4), 601-618 The Jacobi identity is one of the properties that are used to define the concept of Lie algebra and in this context is closely related to associativity. In this paper we provide a complete description of ... [more ▼] The Jacobi identity is one of the properties that are used to define the concept of Lie algebra and in this context is closely related to associativity. In this paper we provide a complete description of all bivariate polynomials that satisfy the Jacobi identity over infinite integral domains. Although this description depends on the characteristic of the domain, it turns out that all these polynomials are of degree at most one in each indeterminate. [less ▲] Detailed reference viewed: 76 (11 UL)Probability signatures of multistate systems made up of two-state components Marichal, Jean-Luc ; ; et al Scientific Conference (2017, July) The structure signature of a system made up of $n$ components having continuous and i.i.d. lifetimes was defined in the eighties by Samaniego as the $n$-tuple whose $k$-th coordinate is the probability ... [more ▼] The structure signature of a system made up of $n$ components having continuous and i.i.d. lifetimes was defined in the eighties by Samaniego as the $n$-tuple whose $k$-th coordinate is the probability that the $k$-th component failure causes the system to fail. More recently, a bivariate version of this concept was considered as follows. The joint structure signature of a pair of systems built on a common set of components having continuous and i.i.d. lifetimes is a square matrix of order $n$ whose $(k,l)$-entry is the probability that the $k$-th failure causes the first system to fail and the $l$-th failure causes the second system to fail. This concept was successfully used to derive a signature-based decomposition of the joint reliability of the two systems. In this talk we will show an explicit formula to compute the joint structure signature of two or more systems and extend this formula to the general non-i.i.d. case, assuming only that the distribution of the component lifetimes has no ties. Then we will discuss a condition on this distribution for the joint reliability of the systems to have a signature-based decomposition. Finally we will show how these results can be applied to the investigation of the reliability and signature of multistate systems made up of two-state components. [less ▲] Detailed reference viewed: 42 (3 UL)On conservative and associative operations on finite chains Devillet, Jimmy ; ; Marichal, Jean-Luc Scientific Conference (2017, June 16) See attached file Detailed reference viewed: 69 (24 UL)Recent results on conservative and symmetric n-ary semigroups Kiss, Gergely ; Devillet, Jimmy ; Marichal, Jean-Luc Scientific Conference (2017, June 16) See attached file Detailed reference viewed: 59 (16 UL)On idempotent discrete uninorms ; Devillet, Jimmy ; Marichal, Jean-Luc in De Baets, Bernard; Torra, Vicenç; Mesiar, Radko (Eds.) Aggregation Functions in Theory and in Practice (2017, June) In this paper we provide two axiomatizations of the class of idempotent discrete uninorms as conservative binary operations, where an operation is conservative if it always outputs one of its input values ... [more ▼] In this paper we provide two axiomatizations of the class of idempotent discrete uninorms as conservative binary operations, where an operation is conservative if it always outputs one of its input values. More precisely we first show that the idempotent discrete uninorms are exactly those operations that are conservative, symmetric, and nondecreasing in each variable. Then we show that, in this characterization, symmetry can be replaced with both bisymmetry and existence of a neutral element. [less ▲] Detailed reference viewed: 97 (22 UL)On the generalized associativity equation Marichal, Jean-Luc ; Teheux, Bruno in Aequationes Mathematicae (2017), 91(2), 265-277 The so-called generalized associativity functional equation G(J(x,y),z) = H(x,K(y,z)) has been investigated under various assumptions, for instance when the unknown functions G, H, J, and K are real ... [more ▼] The so-called generalized associativity functional equation G(J(x,y),z) = H(x,K(y,z)) has been investigated under various assumptions, for instance when the unknown functions G, H, J, and K are real, continuous, and strictly monotonic in each variable. In this note we investigate the following related problem: given the functions J and K, find every function F that can be written in the form F(x,y,z) = G(J(x,y),z) = H(x,K(y,z)) for some functions G and H. We show how this problem can be solved when any of the inner functions J and K has the same range as one of its sections. [less ▲] Detailed reference viewed: 134 (25 UL) |
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