Representations of Ordered Dimonoids by Binary Relations

Zhuchok, Yu. V.

doi:10.1142/s1793557114500065

Cited by 9 publications

(5 citation statements)

References 4 publications

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“…Thus, semigroups and doppelsemigroups can be characterized as n-tuple semigroups. The study of doppelsemigroups was initiated by the author in [28] and then it was continued in [5,6,25,30,33,35,36,41,44]. Note that doppelalgebras introduced by Richter [17] are linear analogs of doppelsemigroups.…”

Section: Introductionmentioning

confidence: 99%

Structure of relatively free n-tuple semigroups

Zhuchok

2023

ADM

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An n-tuple semigroup is an algebra defined on a set with n binary associative operations. This notion was considered by Koreshkov in the context of the theory of n-tuple algebras of associative type. The n>1 pairwise interassociative semigroups give rise to an n-tuple semigroup. This paper is a survey of recent developments in the study of free objects in the variety of n-tuple semigroups. We present the constructions of the free n-tuple semigroup, the free commutative n-tuple semigroup, the free rectangular n-tuple semigroup, the free left (right) k-nilpotent n-tuple semigroup, the free k-nilpotent n-tuple semigroup, and the free weakly k-nilpotent n-tuple semigroup. Some of these results can be applied to constructing relatively free cubical trialgebras and doppelalgebras.

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Section: Introductionmentioning

confidence: 99%

Structure of relatively free n-tuple semigroups

Zhuchok

2023

ADM

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“…Representations of ordered doppelsemigroups by binary relations were studied by Yu. Zhuchok and J. Koppitz, see [23].…”

Section: Introductionmentioning

confidence: 99%

Note on cyclic doppelsemigroups

Gavrylkiv

2022

ADM

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A doppelsemigroup (G,⊣,⊢) is calledcyclic if (G,⊣) is a cyclic group. In the paper, we describe up to isomorphism all cyclic (strong) doppelsemigroups. We prove that up to isomorphism there exist τ(n) finite cyclic (strong) doppelsemigroups of order n, where τ is the number of divisors function. Also there exist infinite countably many pairwise non-isomorphic infinite cyclic (strong) doppelsemigroups.

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“…Recall that an algebra (D, , ) with two binary associative operations and is called a dimonoid if for all x, y, z ∈ D the following conditions hold: Dimonoids play a prominent role in problems from the theory of Leibniz algebras. A more detailed information on dimonoids, free dimonoid constructions and examples of dimonoids can be found, e.g., in [2]- [5].…”

Section: Introductionmentioning

confidence: 99%

On two classes of digroups

Zhuchok

2016

São Paulo J. Math. Sci.

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The paper is devoted to studying two classes of digroups. We give new examples of digroups of such classes and construct an abelian digroup for which any mapping of its generating set to an arbitrary abelian digroup with one bar-unit can be uniquely extended to a homomorphism of these digroups. We also describe a congruence on a free dimonoid for which the quotient is an abelian digroup construction mentioned above.

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Representations of Ordered Dimonoids by Binary Relations

Cited by 9 publications

References 4 publications

Structure of relatively free n-tuple semigroups

Structure of relatively free n-tuple semigroups

Note on cyclic doppelsemigroups

On two classes of digroups

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