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Associative, idempotent, symmetric, and order-preserving operations on chains Devillet, Jimmy ; Teheux, Bruno in Order: A Journal on the Theory of Ordered Sets and its Applications (in press) We characterize the associative, idempotent, symmetric, and order-preserving operations on (finite) chains in terms of properties of (the Hasse diagram of) their associated semilattice order. In ... [more ▼] We characterize the associative, idempotent, symmetric, and order-preserving operations on (finite) chains in terms of properties of (the Hasse diagram of) their associated semilattice order. In particular, we prove that the number of associative, idempotent, symmetric, and order-preserving operations on an n-element chain is the nth Catalan number. [less ▲] Detailed reference viewed: 187 (49 UL)Every quasitrivial n-ary semigroup is reducible to a semigroup ; Devillet, Jimmy in Algebra Universalis (in press) We show that every quasitrivial n-ary semigroup is reducible to a binary semigroup, and we provide necessary and sufficient conditions for such a reduction to be unique. These results are then refined in ... [more ▼] We show that every quasitrivial n-ary semigroup is reducible to a binary semigroup, and we provide necessary and sufficient conditions for such a reduction to be unique. These results are then refined in the case of symmetric n-ary semigroups. We also explicitly determine 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: 139 (21 UL)Reducibility of n-ary semigroups: from quasitriviality towards idempotency ; Devillet, Jimmy ; Marichal, Jean-Luc et al E-print/Working paper (2019) Let $X$ be a nonempty set. Denote by $\mathcal{F}^n_k$ the class of associative operations $F\colon X^n\to X$ satisfying the condition $F(x_1,\ldots,x_n)\in\{x_1,\ldots,x_n\}$ whenever at least $k$ of the ... [more ▼] Let $X$ be a nonempty set. Denote by $\mathcal{F}^n_k$ the class of associative operations $F\colon X^n\to X$ satisfying the condition $F(x_1,\ldots,x_n)\in\{x_1,\ldots,x_n\}$ whenever at least $k$ of the elements $x_1,\ldots,x_n$ are equal to each other. The elements of $\mathcal{F}^n_1$ are said to be quasitrivial and those of $\mathcal{F}^n_n$ are said to be idempotent. We show that $\mathcal{F}^n_1=\cdots =\mathcal{F}^n_{n-2}\varsubsetneq\mathcal{F}^n_{n-1}\varsubsetneq\mathcal{F}^n_n$. The class $\mathcal{F}^n_1$ was recently characterized by Couceiro and Devillet \cite{CouDev}, who showed that its elements are reducible to binary associative operations. However, some elements of $\mathcal{F}^n_n$ are not reducible. In this paper, we characterize the class $\mathcal{F}^n_{n-1}\setminus\mathcal{F}^n_1$ and show that its elements are reducible. In particular, we show that each of these elements is an extension of an $n$-ary Abelian group operation whose exponent divides $n-1$. [less ▲] Detailed reference viewed: 47 (5 UL)On idempotent n-ary uninorms Devillet, 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: 44 (7 UL)On the single-peakedness property Devillet, Jimmy Scientific Conference (2019, June 28) Detailed reference viewed: 40 (3 UL)Characterizations and enumerations of classes of quasitrivial n-ary semigroups Devillet, Jimmy ; Scientific Conference (2019, June 23) Detailed reference viewed: 64 (3 UL)Bisymmetric and quasitrivial operations: characterizations and enumerations Devillet, Jimmy in Aequationes Mathematicae (2019), 93(3), 501-526 We investigate the class of bisymmetric and quasitrivial binary operations on a given set and provide various characterizations of this class as well as the subclass of bisymmetric, quasitrivial, and ... [more ▼] We investigate the class of bisymmetric and quasitrivial binary operations on a given set and provide various characterizations of this class as well as the subclass of bisymmetric, quasitrivial, and order-preserving binary operations. We also determine explicitly the sizes of these classes when the set is finite. [less ▲] Detailed reference viewed: 122 (42 UL)Quasitrivial semigroups: characterizations and enumerations ; Devillet, Jimmy ; Marichal, Jean-Luc in Semigroup Forum (2019), 98(3), 472498 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: 262 (90 UL)Single-peakedness in aggregation function theory Devillet, Jimmy ; ; Marichal, Jean-Luc Presentation (2019, May 14) Due to their great importance in data fusion, aggregation functions have been extensively investigated for a few decades. Among these functions, fuzzy connectives (such as uninorms) play an important role ... [more ▼] Due to their great importance in data fusion, aggregation functions have been extensively investigated for a few decades. Among these functions, fuzzy connectives (such as uninorms) play an important role in fuzzy logic. We establish a remarkable connection between a family of associative aggregation functions, which includes the class of idempotent uninorms, and the concepts of single-peakedness and single-plateaudness, introduced in social choice theory by D. Black. Finally, we enumerate those orders when the underlying set is finite. [less ▲] Detailed reference viewed: 70 (8 UL)Characterizations of biselective operations Devillet, 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: 82 (25 UL)On quasitrivial semigroups Devillet, Jimmy Presentation (2019, March 27) Detailed reference viewed: 40 (6 UL)Characterizations and classifications of quasitrivial semigroups Devillet, Jimmy ; Marichal, Jean-Luc ; Teheux, Bruno Scientific Conference (2019, March 03) Detailed reference viewed: 75 (11 UL)Characterizations of idempotent n-ary uninorms Devillet, 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: 70 (18 UL)Characterizations of quasitrivial symmetric nondecreasing associative operations Devillet, Jimmy ; Kiss, Gergely ; Marichal, Jean-Luc in Semigroup Forum (2019), 98(1), 154-171 We provide a description of the class of n-ary operations on an arbitrary chain that are quasitrivial, symmetric, nondecreasing, and associative. We also prove that associativity can be replaced with ... [more ▼] We provide a description of the class of n-ary operations on an arbitrary chain that are quasitrivial, symmetric, nondecreasing, and associative. We also 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: 130 (50 UL)Generalizations of single-peakedness Devillet, Jimmy Scientific Conference (2019, January 30) We establish a surprising connection between a family of conservative semigroups, which includes the class of idempotent uninorms, and the concepts of single-peakedness and single-plateaudness, introduced ... [more ▼] We establish a surprising connection between a family of conservative semigroups, which includes the class of idempotent uninorms, and the concepts of single-peakedness and single-plateaudness, introduced in social choice theory by D. Black. We also introduce a generalization of single-peakedness to partial orders of join-semilattices and show how it is related to the class of idempotent and commutative semigroups. Finally, we enumerate those orders when the corresponding semigroups are finite. [less ▲] Detailed reference viewed: 68 (10 UL)Classifications of quasitrivial semigroups Devillet, Jimmy ; Marichal, Jean-Luc ; Teheux, Bruno E-print/Working paper (2018) We investigate classifications of quasitrivial semigroups defined by certain equivalence relations. The subclass of quasitrivial semigroups that preserve a given total ordering is also investigated. In ... [more ▼] We investigate classifications of quasitrivial semigroups defined by certain equivalence relations. The subclass of quasitrivial semigroups that preserve a given total ordering is also investigated. In the special case of finite semigroups, we address and solve several related enumeration problems. [less ▲] Detailed reference viewed: 87 (24 UL)Invariance in a class of operations related to weighted quasi-geometric means Devillet, Jimmy ; E-print/Working paper (2018) 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 ... [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: 38 (2 UL)A new invariance identity and means Devillet, Jimmy ; in Results in Mathematics (2018), 73(4), 130 The invariance identity involving three operations D_{f,g} : X x X -> X of the form D_{f,g} (x,y) = (f o g)^{-1} (f (x) + g (y)) , is proposed. The connections of these operations with means is ... [more ▼] The invariance identity involving three operations D_{f,g} : X x X -> X of the form D_{f,g} (x,y) = (f o g)^{-1} (f (x) + g (y)) , is proposed. The connections of these operations with means is investigated. The question when the invariance equality admits three means leads to a com- posite functional equation. Problem to determine its continuous solutions is posed [less ▲] Detailed reference viewed: 119 (49 UL)On associative, idempotent, symmetric, and nondecreasing operations Devillet, Jimmy ; Teheux, Bruno Scientific Conference (2018, July 02) see attached file Detailed reference viewed: 21 (1 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: 61 (5 UL) |
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