Direct products, varieties, and compactness conditions

Shahryari, M.; Shevlyakov, Artem N.

doi:10.1515/gcc-2017-0011

Cited by 6 publications

(4 citation statements)

References 4 publications

(9 reference statements)

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“…For example, all finite algebraic structures, abelian groups, linear groups over a Noetherian ring, and torsion-free hyperbolic groups are equationally Noetherian. On the other hand, some infinitely generated nilpotent groups, wreath products of a non-abelian group and an infinite cyclic group, infinite direct products of non-abelian groups, and minimax algebraic structures are not equationally Noetherian [2,1,24,12,11].…”

Section: Introductionmentioning

confidence: 99%

On equationally Noetherian predicate structures

Buchinskiy,

Kotov,

Treier

2024

Journal of Groups, Complexity, Cryptology

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Section: Introductionmentioning

confidence: 99%

On equationally Noetherian predicate structures

Buchinskiy,

Kotov,

Treier

2024

Journal of Groups, Complexity, Cryptology

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“…Free groups, Abelian groups, linear groups over Noetherian rings and torsion-free hyperbolic groups are equationally Noetherian. To see interesting properties of this types of groups, the reader can consult [11] and [16]. This kind of groups have very important roles in algebraic geometry of groups.…”

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confidence: 99%

On the Equationally Artinian Groups

Shahryari

Tayyebi

2020

Journal of Siberian Federal University. Mathematics &Amp; Physics

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In this article, we study the property of being equationally Artinian in groups. We define the radical topology corresponding to such groups and investigate the structure of irreducible closed sets of these topologies. We prove that a finite extension of an equationally Artinian group is again equationally Artinian. We also show that a quotient of an equationally Artinian group of the form G[t] by a normal subgroup which is a finite union of radicals, is again equationally Artnian. A necessary and sufficient condition for an Abelian group to be equationally Artinian will be given as the last result. This will provide a large class of examples of equationally Artinian groups

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“…In the classical algebraic geometry A of type L is a field. Many articles already published about algebraic geometry over groups, see [1,8,16], and [10]. O. Kharlampovich and A. Miyasnikov developed algebraic geometry over free groups to give affirmative answer for an old problem of Alfred Tarski concerning elementary theory of free groups (see [7] and also [15] for the independent solution of Z. Sela).…”

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confidence: 99%