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Characterizations of quasitrivial symmetric nondecreasing associative operations Devillet, Jimmy ; Kiss, Gergely ; Marichal, Jean-Luc in Semigroup Forum (in press) 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: 52 (23 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: 145 (48 UL)Characterizations of biselective operations Devillet, Jimmy ; in Acta Mathematica Hungarica (in press) 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: 31 (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: 38 (5 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: 31 (1 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: 68 (36 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: 40 (3 UL)On associative, idempotent, symmetric, and nondecreasing operations Devillet, Jimmy ; Teheux, Bruno Scientific Conference (2018, July 02) see attached file Detailed reference viewed: 16 (1 UL)On bisymmetric and quasitrivial operations Devillet, Jimmy Scientific Conference (2018, June 21) See attachment Detailed reference viewed: 20 (0 UL)On biselective operations Devillet, Jimmy ; Kiss, Gergely Scientific Conference (2018, June 07) See attached file Detailed reference viewed: 36 (4 UL)Characterizations of nondecreasing semilattice operations on chains Devillet, Jimmy ; Teheux, Bruno Scientific Conference (2018, June 01) See attached file Detailed reference viewed: 47 (3 UL)Associative, idempotent, symmetric, and order-preserving operations on chains Devillet, Jimmy ; Teheux, Bruno E-print/Working paper (2018) 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: 19 (2 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: 154 (37 UL)Bisymmetric and quasitrivial operations: characterizations and enumerations Devillet, Jimmy E-print/Working paper (2018) We investigate the class of bisymmetric and quasitrivial binary operations on a given set X 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 X 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 X is finite. [less ▲] Detailed reference viewed: 44 (18 UL)Associative and quasitrivial operations on finite sets (invited lecture) Marichal, Jean-Luc ; ; Devillet, Jimmy Scientific Conference (2017, November 10) Detailed reference viewed: 53 (13 UL)On quasitrivial and associative operations Devillet, Jimmy ; ; Marichal, Jean-Luc Presentation (2017, October 25) Detailed reference viewed: 41 (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: 51 (15 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: 39 (6 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: 61 (16 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: 70 (24 UL) |
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