On associative operations on commutative integral domains

Lehtonen, Erkko; Starke, Florian

doi:10.1007/s00233-019-10044-x

Cited by 3 publications

(1 citation statement)

References 2 publications

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“…We also denote the set of neutral elements for an operation F ∶ X n → X by E F . The quest for conditions under which an associative n-ary operation is reducible to an associative binary operation gained an increasing interest since the pioneering work of Post [11] (see, e.g., [1,2,4,5,7,8]). For instance, Dudek and Mukhin [5, Theorem 1] proved that an associative operation F ∶ X n → X is reducible to an associative binary operation if and only if one can adjoin to X a neutral element e for F ; that is, there is an n-ary associative operation F ∶ (X ∪ {e}) n → X ∪ {e} such that e is a neutral element for F and F X n = F .…”

Section: Introductionmentioning

confidence: 99%

Reducibility of $n$-ary semigroups: from quasitriviality towards idempotency

Couceiro,

Devillet,

Marichal

et al. 2019

Preprint

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Let X be a nonempty set. Denote by F n k the class of associative operations F ∶ X n → X satisfying the condition F (x 1 , . . . , xn) ∈ {x 1 , . . . , xn} whenever at least k of the elements x 1 , . . . , xn are equal to each other. The elements of F n 1 are said to be quasitrivial and those of F n n are said to be idempotent. We show thatThe class F n 1 was recently characterized by Couceiro and Devillet [2], who showed that its elements are reducible to binary associative operations. However, some elements of F n n are not reducible. In this paper, we characterize the class F n n−1 ∖ F n 1 and show that its elements are reducible. In particular, we show that each of these elements is an extension of an n-ary Abelian group operation whose exponent divides n − 1.

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Section: Introductionmentioning

confidence: 99%

Reducibility of $n$-ary semigroups: from quasitriviality towards idempotency

Couceiro,

Devillet,

Marichal

et al. 2019

Preprint

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Reducibility of n-ary semigroups: from quasitriviality towards idempotency

2021

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On the structure of symmetric n-ary bands

Devillet

Mathonet

2021

Int. J. Algebra Comput.

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We study the class of symmetric [Formula: see text]-ary bands. These are [Formula: see text]-ary semigroups [Formula: see text] such that [Formula: see text] is invariant under the action of permutations and idempotent, i.e., satisfies [Formula: see text] for all [Formula: see text]. We first provide a structure theorem for these symmetric [Formula: see text]-ary bands that extends the classical (strong) semilattice decomposition of certain classes of bands. We introduce the concept of strong [Formula: see text]-ary semilattice of [Formula: see text]-ary semigroups and we show that the symmetric [Formula: see text]-ary bands are exactly the strong [Formula: see text]-ary semilattices of [Formula: see text]-ary extensions of Abelian groups whose exponents divide [Formula: see text]. Finally, we use the structure theorem to obtain necessary and sufficient conditions for a symmetric [Formula: see text]-ary band to be reducible to a semigroup.

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On associative operations on commutative integral domains

Abstract: We describe the associative multilinear polynomial functions over commutative integral domains. This extends Marichal and Mathonet's result on infinite integral domains and provides a new proof of Andres's classification of two-element n-semigroups.

Cited by 3 publications

References 2 publications

Reducibility of $n$-ary semigroups: from quasitriviality towards idempotency

Reducibility of $n$-ary semigroups: from quasitriviality towards idempotency

Reducibility of n-ary semigroups: from quasitriviality towards idempotency

On the structure of symmetric n-ary bands

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