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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: 269 (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: 269 (47 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: 351 (103 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: 195 (54 UL)A characterization of n-associative, monotone, idempotent functions on an interval that have neutral elements Kiss, Gergely ; in Semigroup Forum (2018) We investigate monotone idempotent n-ary semigroups and provide a generalization of the Czogala–Drewniak Theorem, which describes the idempotent monotone associative functions having a neutral element. We ... [more ▼] We investigate monotone idempotent n-ary semigroups and provide a generalization of the Czogala–Drewniak Theorem, which describes the idempotent monotone associative functions having a neutral element. We also present a complete characterization of idempotent monotone n-associative functions on an interval that have neutral elements. [less ▲] Detailed reference viewed: 116 (39 UL)Associative idempotent nondecreasing functions are reducible Kiss, Gergely ; in Semigroup Forum (2017) An n-variable associative function is called reducible if it can be written as a composition of a binary associative function. In this paper we summarize the known results when the function is defined on ... [more ▼] An n-variable associative function is called reducible if it can be written as a composition of a binary associative function. In this paper we summarize the known results when the function is defined on a chain and nondecreasing. The main result of this paper shows that associative idempotent and nondecreasing functions are uniquely reducible. [less ▲] Detailed reference viewed: 86 (9 UL)Associative and preassociative functions Marichal, Jean-Luc ; Teheux, Bruno in Semigroup Forum (2014), 89(2), 431-442 We investigate the associativity property for functions of multiple arities and introduce and discuss the more general property of preassociativity, a generalization of associativity which does not ... [more ▼] We investigate the associativity property for functions of multiple arities and introduce and discuss the more general property of preassociativity, a generalization of associativity which does not involve any composition of functions. [less ▲] Detailed reference viewed: 196 (26 UL)Associative Formal Power Series in Two Indeterminates ; ; et al in Semigroup Forum (2014), 88(3), 529-540 Investigating the associativity equation for formal power series in two variables we show that the transcendental associative formal power series are of order one or two and that they can be represented ... [more ▼] Investigating the associativity equation for formal power series in two variables we show that the transcendental associative formal power series are of order one or two and that they can be represented by an invertible formal power series in one variable. We also discuss the convergence of associative formal power series. [less ▲] Detailed reference viewed: 103 (5 UL)Generalized entropy in expanded semigroups and in algebras with neutral element Lehtonen, Erkko ; in Semigroup Forum (2014) The complex product of (non-empty) subalgebras of a given algebra from a variety V is again a subalgebra if and only if the variety V has the so-called generalized entropic property. This paper is devoted ... [more ▼] The complex product of (non-empty) subalgebras of a given algebra from a variety V is again a subalgebra if and only if the variety V has the so-called generalized entropic property. This paper is devoted to algebras with a neutral element or with a semigroup operation. We investigate relationships between the generalized entropic property and the commutativity of the fundamental operations of the algebra. In particular, we characterize the algebras with a neutral element that have the generalized entropic property. Furthermore, we show that, similarly as for n-monoids and n-groups, for inverse semigroups, the generalized entropic property is equivalent to commutativity. [less ▲] Detailed reference viewed: 122 (1 UL)Aczélian n-ary semigroups Couceiro, Miguel ; Marichal, Jean-Luc in Semigroup Forum (2012), 85(1), 81-90 We show that the real continuous, symmetric, and cancellative n-ary semigroups are topologically order-isomorphic to additive real n-ary semigroups. The binary case (n=2) was originally proved by Aczél ... [more ▼] We show that the real continuous, symmetric, and cancellative n-ary semigroups are topologically order-isomorphic to additive real n-ary semigroups. The binary case (n=2) was originally proved by Aczél (1949); there symmetry was redundant. [less ▲] Detailed reference viewed: 161 (19 UL)A description of n-ary semigroups polynomial-derived from integral domains Marichal, Jean-Luc ; Mathonet, Pierre in Semigroup Forum (2011), 83(2), 241-249 We provide a complete classification of the n-ary semigroup structures defined by polynomial functions over infinite commutative integral domains with identity, thus generalizing Glazek and Gleichgewicht ... [more ▼] We provide a complete classification of the n-ary semigroup structures defined by polynomial functions over infinite commutative integral domains with identity, thus generalizing Glazek and Gleichgewicht's classification of the corresponding ternary semigroups. [less ▲] Detailed reference viewed: 132 (15 UL) |
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