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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: 106 (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: 201 (54 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: 81 (10 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: 174 (50 UL)On associative, idempotent, symmetric, and nondecreasing operations Devillet, Jimmy ; Teheux, Bruno Scientific Conference (2018, July 02) see attached file Detailed reference viewed: 41 (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: 85 (5 UL)On bisymmetric and quasitrivial operations Devillet, Jimmy Scientific Conference (2018, June 21) See attachment Detailed reference viewed: 217 (3 UL)On biselective operations Devillet, Jimmy ; Kiss, Gergely Scientific Conference (2018, June 07) See attached file Detailed reference viewed: 61 (6 UL)Characterizations of nondecreasing semilattice operations on chains Devillet, Jimmy ; Teheux, Bruno Scientific Conference (2018, June 01) See attached file Detailed reference viewed: 82 (3 UL)Characterizations of idempotent discrete uninorms ; Devillet, Jimmy ; Marichal, Jean-Luc in Fuzzy Sets and 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: 277 (54 UL)Associative and quasitrivial operations on finite sets (invited lecture) Marichal, Jean-Luc ; ; Devillet, Jimmy Scientific Conference (2017, November 10) Detailed reference viewed: 84 (13 UL)On quasitrivial and associative operations Devillet, Jimmy ; ; Marichal, Jean-Luc Presentation (2017, October 25) Detailed reference viewed: 78 (13 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: 89 (16 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: 83 (8 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: 98 (25 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: 92 (18 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: 209 (39 UL) |
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