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Reducibility of n-ary semigroups: from quasitriviality towards idempotency ; Devillet, Jimmy ; Marichal, Jean-Luc et al in Beiträge zur Algebra und Geometrie (in press) 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 $\mathcal{F}^n_1=\cdots =\mathcal{F}^n_{n-2}\subseteq\mathcal{F}^n_{n-1}\subseteq\mathcal{F}^n_n$ and we give conditions on the set $X$ for the last inclusions to be strict. 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. We give a full description of the corresponding reductions and show how each of them is built from a quasitrivial semigroup and an Abelian group whose exponent divides $n-1$. [less ▲] Detailed reference viewed: 146 (21 UL)Invariance in a class of operations related to weighted quasi-geometric means Devillet, Jimmy ; in Fuzzy Sets and Systems (in press) 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: 77 (6 UL)Decomposition schemes for symmetric n-ary bands Devillet, Jimmy ; Scientific Conference (2020, August 27) We extend the classical (strong) semilattice decomposition scheme of certain classes of semigroups to the class of idempotent symmetric n-ary semigroups (i.e. symmetric n-ary bands) where n \geq 2 is an ... [more ▼] We extend the classical (strong) semilattice decomposition scheme of certain classes of semigroups to the class of idempotent symmetric n-ary semigroups (i.e. symmetric n-ary bands) where n \geq 2 is an integer. More precisely, we show that these semigroups are exactly the strong n-ary semilattices of n-ary extensions of Abelian groups whose exponents divide n-1. We then use this main result to obtain necessary and sufficient conditions for a symmetric n-ary band to be reducible to a semigroup. [less ▲] Detailed reference viewed: 78 (2 UL)On idempotent n-ary semigroups Devillet, Jimmy Doctoral thesis (2020) This thesis, which consists of two parts, focuses on characterizations and descriptions of classes of idempotent n-ary semigroups where n >= 2 is an integer. Part I is devoted to the study of various ... [more ▼] This thesis, which consists of two parts, focuses on characterizations and descriptions of classes of idempotent n-ary semigroups where n >= 2 is an integer. Part I is devoted to the study of various classes of idempotent semigroups and their link with certain concepts stemming from social choice theory. In Part II, we provide constructive descriptions of various classes of idempotent n-ary semigroups. More precisely, after recalling and studying the concepts of single-peakedness and rectangular semigroups in Chapters 1 and 2, respectively, in Chapter 3 we provide characterizations of the classes of idempotent semigroups and totally ordered idempotent semigroups, in which the latter two concepts play a central role. Then in Chapter 4 we particularize the latter characterizations to the classes of quasitrivial semigroups and totally ordered quasitrivial semigroups. We then generalize these results to the class of quasitrivial n-ary semigroups in Chapter 5. Chapter 6 is devoted to characterizations of several classes of idempotent n-ary semigroups satisfying quasitriviality on certain subsets of the domain. Finally, Chapter 7 focuses on characterizations of the class of symmetric idempotent n-ary semigroups. Throughout this thesis, we also provide several enumeration results which led to new integer sequences that are now recorded in The On-Line Encyclopedia of Integer Sequences (OEIS). For instance, one of these enumeration results led to a new definition of the Catalan numbers. [less ▲] Detailed reference viewed: 94 (8 UL)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 (2020), 37(1), 45-58 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: 373 (78 UL)Classifications of quasitrivial semigroups Devillet, Jimmy ; Marichal, Jean-Luc ; Teheux, Bruno in Semigroup Forum (2020), 100(3), 743-764 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: 273 (47 UL)Classifications of quasitrivial semigroups Devillet, Jimmy ; Marichal, Jean-Luc ; Teheux, Bruno in Semigroup Forum (2020), 100(3), 743-764 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: 273 (47 UL)Classifications of quasitrivial semigroups Devillet, Jimmy ; Marichal, Jean-Luc ; Teheux, Bruno in Semigroup Forum (2020), 100(3), 743-764 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: 273 (47 UL)Devillet, Jimmy ; Marichal, Jean-Luc ; Teheux, Bruno in Semigroup Forum (2020), 100(3), 743-764 On the structure of symmetric $n$-ary bands Devillet, Jimmy ; E-print/Working paper (2020) We study the class of symmetric $n$-ary bands. These are $n$-ary semigroups $(X,F)$ such that $F$ is invariant under the action of permutations and idempotent, i.e., satisfies $F(x,\ldots,x)=x$ for all $x ... [more ▼] We study the class of symmetric $n$-ary bands. These are $n$-ary semigroups $(X,F)$ such that $F$ is invariant under the action of permutations and idempotent, i.e., satisfies $F(x,\ldots,x)=x$ for all $x\in X$. We first provide a structure theorem for these symmetric $n$-ary bands that extends the classical (strong) semilattice decomposition of certain classes of bands. We introduce the concept of strong $n$-ary semilattice of $n$-ary semigroups and we show that the symmetric $n$-ary bands are exactly the strong $n$-ary semilattices of $n$-ary extensions of Abelian groups whose exponents divide $n-1$. Finally, we use the structure theorem to obtain necessary and sufficient conditions for a symmetric $n$-ary band to be reducible to a semigroup. [less ▲] Detailed reference viewed: 48 (5 UL)Every quasitrivial n-ary semigroup is reducible to a semigroup ; Devillet, Jimmy in Algebra Universalis (2019), 80(4), 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: 281 (62 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: 118 (10 UL)On the single-peakedness property Devillet, Jimmy Scientific Conference (2019, June 28) Detailed reference viewed: 73 (4 UL)Characterizations and enumerations of classes of quasitrivial n-ary semigroups Devillet, Jimmy ; Scientific Conference (2019, June 23) Detailed reference viewed: 103 (8 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: 201 (49 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: 353 (103 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: 105 (10 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: 147 (29 UL)On quasitrivial semigroups Devillet, Jimmy Presentation (2019, March 27) Detailed reference viewed: 59 (9 UL)Characterizations and classifications of quasitrivial semigroups Devillet, Jimmy ; Marichal, Jean-Luc ; Teheux, Bruno Scientific Conference (2019, March 03) Detailed reference viewed: 107 (11 UL) |
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