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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 (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: 366 (75 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: 261 (45 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: 261 (45 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: 261 (45 UL)Devillet, Jimmy ; Marichal, Jean-Luc ; Teheux, Bruno in Semigroup Forum (2020), 100(3), 743-764 Extending maps to profinite completions in finitely generated quasivarieties Teheux, Bruno ; in Beiträge zur Algebra und Geometrie (2020), 61(4), 627-647 We consider the problem of extending maps from algebras to their profinite completions in finitely generated quasivarieties. Our developments are based on the construction of the profinite completion of ... [more ▼] We consider the problem of extending maps from algebras to their profinite completions in finitely generated quasivarieties. Our developments are based on the construction of the profinite completion of an algebra as its natural extension. We provide an extension which is a multi-map and we study its continuity properties, and the conditions under which it is a map. [less ▲] Detailed reference viewed: 87 (6 UL)Characterizations and classifications of quasitrivial semigroups Devillet, Jimmy ; Marichal, Jean-Luc ; Teheux, Bruno Scientific Conference (2019, March 03) Detailed reference viewed: 101 (11 UL)Categories of coalgebras for modal extensions of Łukasiewicz logic Teheux, Bruno ; Scientific Conference (2018, August 27) The category of complete and completely distributive Boolean algebras with complete operators is dual to the category of frames. We lift this duality to the category of complete and completely ... [more ▼] The category of complete and completely distributive Boolean algebras with complete operators is dual to the category of frames. We lift this duality to the category of complete and completely distributive MV-algebras with complete operators. [less ▲] Detailed reference viewed: 57 (5 UL)An n-ary generalization of the concept of distance ; Marichal, Jean-Luc ; Teheux, Bruno Scientific Conference (2018, July 03) Detailed reference viewed: 79 (5 UL)On associative, idempotent, symmetric, and nondecreasing operations Devillet, Jimmy ; Teheux, Bruno Scientific Conference (2018, July 02) see attached file Detailed reference viewed: 38 (2 UL)Characterizations of nondecreasing semilattice operations on chains Devillet, Jimmy ; Teheux, Bruno Scientific Conference (2018, June 01) See attached file Detailed reference viewed: 75 (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: 210 (30 UL)Clones of pivotally decomposable operations ; Teheux, Bruno Scientific Conference (2018, June) We investigate the clones of operations that are pivotally decomposable. Detailed reference viewed: 47 (2 UL)Pivotal decomposition schemes inducing clones of operations ; Teheux, Bruno in Beitraege zur Algebra und Geometrie = Contributions to Algebra and Geometry (2018), 59(1), 25-40 We study pivotal decomposition schemes and investigate classes of pivotally decomposable operations. We provide sufficient conditions on pivotal operations that guarantee that the corresponding classes of ... [more ▼] We study pivotal decomposition schemes and investigate classes of pivotally decomposable operations. We provide sufficient conditions on pivotal operations that guarantee that the corresponding classes of pivotally decomposable operations are clones, and show that under certain assumptions these conditions are also necessary. In the latter case, the pivotal operation together with the constant operations generate the corresponding clone. [less ▲] Detailed reference viewed: 118 (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: 204 (28 UL)Generalized qualitative Sugeno integrals ; ; et al in Information Sciences (2017), 415-416 Sugeno integrals are aggregation operations involving a criterion weighting scheme based on the use of set functions called capacities or fuzzy measures. In this paper, we define generalized versions of ... [more ▼] Sugeno integrals are aggregation operations involving a criterion weighting scheme based on the use of set functions called capacities or fuzzy measures. In this paper, we define generalized versions of Sugeno integrals on totally ordered bounded chains, by extending the operation that combines the value of the capacity on each subset of criteria and the value of the utility function over elements of the subset. We show that the generalized concept of Sugeno integral splits into two functionals, one based on a general multiple-valued conjunction (we call integral) and one based on a general multiple-valued implication (we call cointegral). These fuzzy conjunction and implication connectives are related via a so-called semiduality property, involving an involutive negation. Sugeno integrals correspond to the case when the fuzzy conjunction is the minimum and the fuzzy implication is Kleene-Dienes implication, in which case integrals and cointegrals coincide. In this paper, we consider a very general class of fuzzy conjunction operations on a finite setting, that reduce to Boolean conjunctions on extreme values of the bounded chain, and are non-decreasing in each place, and the corresponding general class of implications (their semiduals). The merit of these new aggregation operators is to go beyond pure lattice polynomials, thus enhancing the expressive power of qualitative aggregation functions, especially as to the way an importance weight can affect a local rating of an object to be chosen. [less ▲] Detailed reference viewed: 114 (9 UL)Modal Extensions of Łukasiewicz Logic for Modeling Coalitional Power Teheux, Bruno ; in Journal of Logic & Computation (2017), 27(1), 129-154 Modal logics for reasoning about the power of coalitions capture the notion of effectivity functions associated with game forms. The main goal of coalition logics is to provide formal tools for modeling ... [more ▼] Modal logics for reasoning about the power of coalitions capture the notion of effectivity functions associated with game forms. The main goal of coalition logics is to provide formal tools for modeling the dynamics of a game frame whose states may correspond to different game forms. The two classes of effectivity functions studied are the families of playable and truly playable effectivity functions, respectively. In this paper we generalize the concept of effectivity function beyond the yes/no truth scale. This enables us to describe the situations in which the coalitions assess their effectivity in degrees, based on functions over the outcomes taking values in a finite Łukasiewicz chain. Then we introduce two modal extensions of Łukasiewicz finite-valued logic together with many-valued neighborhood semantics in order to encode the properties of many-valued effectivity functions associated with game forms. As our main results we prove completeness theorems for the two newly introduced modal logics. [less ▲] Detailed reference viewed: 131 (14 UL)Modal extensions of Ł_n-valued logics, coalgebraically ; Teheux, Bruno ; Scientific Conference (2017) Detailed reference viewed: 38 (2 UL)Strongly barycentrically associative and preassociative functions Teheux, Bruno ; Marichal, Jean-Luc Scientific Conference (2016, November 08) Detailed reference viewed: 117 (7 UL)Relaxations of associativity and preassociativity for variadic functions ; Marichal, Jean-Luc ; Teheux, Bruno in Fuzzy Sets & Systems (2016), 299 In this paper we consider two properties of variadic functions, namely associativity and preassociativity, that are pertaining to several data and language processing tasks. We propose parameterized ... [more ▼] In this paper we consider two properties of variadic functions, namely associativity and preassociativity, that are pertaining to several data and language processing tasks. We propose parameterized relaxations of these properties and provide their descriptions in terms of factorization results. We also give an example where these parameterized notions give rise to natural hierarchies of functions and indicate their potential use in measuring the degrees of associativeness and preassociativeness. We illustrate these results by several examples and constructions and discuss some open problems that lead to further directions of research. [less ▲] Detailed reference viewed: 191 (21 UL) |
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