Automorphisms of the endomorphism semigroup of a free commutative dimonoid

Zhuchok, Yu. V.

doi:10.1080/00927872.2016.1248241

Cited by 7 publications

(4 citation statements)

References 11 publications

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“…Trioid theory found some applications in trialgebra theory (see, e.g., [1][2][3][4]). Trioids are also associated with n-tuple semigroups [13,15,22,26], dimonoids [5,9,10,12,14,17,19,32] and g-dimonoids [7,25,31,33]. Now trioid theory are developed quite intensively.…”

Section: Introductionmentioning

confidence: 99%

Structure of relatively free trioids

Zhuchok

2021

ADM

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Loday and Ronco introduced the notions of a~trioid and a trialgebra, and constructed the free trioid of rank 1 and the free trialgebra. This paper is a survey of recent developments in the study of free objects in the varieties of trioids and trialgebras. We present the constructions of the free trialgebra and the free trioid, the free commutative trioid, the free n-nilpotent trioid, the free left (right) n-trinilpotent trioid, and the free rectangular trioid. Some of these results can be applied to constructing relatively free trialgebras.

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Section: Introductionmentioning

confidence: 99%

Structure of relatively free trioids

Zhuchok

2021

ADM

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“…The problem of studying automorphisms of the endomorphism semigroup for free algebras in a certain variety was raised by B. I. Plotkin in his papers on universal algebraic geometry (see, e.g., [6], [7]). Now there are quite a lot papers devoted to studying automorphisms of endomorphism semigroups of free nitely generated algebras of dierent varieties (see, e.g., [8] [13]). In this paper, we investigate automorphisms of endomorphism semigroups for free algebras in the variety of abelian dimonoids [14].…”

Section: Introductionmentioning

confidence: 99%

On automorphisms of the semigroup of endomorphisms of a free abelian dimonoid

Zhuchok¹

2019

vmm

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“…Digroups are closely related to dimonoids [5] which also play an important role in problems from the theory of Leibniz algebras and they have been studied by many authors (see, e.g., [1,6,14,17]). One of the first results about digroups is the proof of the fact that Cayley's theorem for groups has an analogue in the class of all digroups [4].…”

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confidence: 99%

“…Digroup analogues of some structure results of group theory were obtained in [7]. Some properties of generalized digroups and also generalized dimonoids were investigated in [9,15]. The free digroup of an arbitrary rank was constructed in [11], where in particular the free monogenic digroup was separately described in a slightly cumbersome way.…”

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confidence: 99%

A new model of the free monogenic digroup

Zhuchok¹,

Pilz²

2023

Mat. Stud.

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It is well-known that one of open problems in the theory of Leibniz algebras is to find asuitable generalization of Lie’s third theorem which associates a (local) Lie group to any Liealgebra, real or complex. It turns out, this is related to finding an appropriate analogue of a Liegroup for Leibniz algebras. Using the notion of a digroup, Kinyon obtained a partial solution ofthis problem, namely, an analogue of Lie’s third theorem for the class of so-called split Leibnizalgebras. A digroup is a nonempty set equipped with two binary associative operations, aunary operation and a nullary operation satisfying additional axioms relating these operations.Digroups generalize groups and have close relationships with the dimonoids and dialgebras,the trioids and trialgebras, and other structures. Recently, G. Zhang and Y. Chen applied themethod of Grobner–Shirshov bases for dialgebras to construct the free digroup of an arbitraryrank, in particular, they considered a monogenic case separately. In this paper, we give a simplerand more convenient digroup model of the free monogenic digroup. We construct a new classof digroups which are based on commutative groups and show how the free monogenic groupcan be obtained from the free monogenic digroup by a suitable factorization.

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Automorphisms of the endomorphism semigroup of a free commutative dimonoid

Cited by 7 publications

References 11 publications

Structure of relatively free trioids

Structure of relatively free trioids

On automorphisms of the semigroup of endomorphisms of a free abelian dimonoid

A new model of the free monogenic digroup

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