Geometric Equivalence of Algebras

Berzins, A.

doi:10.1142/s0218196701000668

Cited by 15 publications

(4 citation statements)

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“…Our primary interest to automorphisms of categories raised from the universal algebraic geometry (see [2][3][4]16,17,22,23,[26][27][28][29][30][31]37], etc.). The motivations we keep in mind are inspired by the following observations.…”

Section: Theorem 1 Every Automorphism Of the Category Of Free Lie Almentioning

confidence: 99%

Automorphisms of the category of free Lie algebras

Mashevitzky¹,

Plotkin²,

Plotkin³

2004

Journal of Algebra

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Section: Theorem 1 Every Automorphism Of the Category Of Free Lie Almentioning

confidence: 99%

Automorphisms of the category of free Lie algebras

Mashevitzky¹,

Plotkin²,

Plotkin³

2004

Journal of Algebra

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“…It was found [3,4,[11][12][13][14] that many problems in the universal algebraic geometry, such as geometric equivalence, geometric similarity, isomorphism and equivalence of categories of algebraic sets and varieties depend on structure of Aut(Θ 0 ) and Aut(End(W )), where Θ 0 is the category of algebras W = W (X) with the finite X that are free in Θ, and W is a free algebra in Θ. Structure of Aut(Com − P )) 0 (i.e, the classical case) was described in [2]. Here was introduced variant of the concept of quasiinner automorphism -basic concept for description of Aut(End(W )) and Aut(Θ 0 ).…”

Section: Introductionmentioning

confidence: 99%

The Group of Automorphisms of the Semigroup of Endomorphisms of Free Commutative and Free Associative Algebras

Berzins

2007

Int. J. Algebra Comput.

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“…All automorphisms of the category Θ 0 are known in the following cases (see [3], [14], [15], [13], [11], [24], [17]) .…”

Section: Aut (θ 0 )mentioning

confidence: 99%

Problems in algebra inspired by universal algebraic geometry

Plotkin

2006

J Math Sci

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Let Θ be a variety of algebras. In every Θ and every algebra H from Θ one can consider algebraic geometry in Θ over H. We consider also a special categorical invariant K Θ (H) of this geometry. The classical algebraic geometry deals with the variety Θ = Com − P of all associative and commutative algebras over the ground field of constants P . An algebra H in this setting is an extension of the ground field P . Geometry in groups is related to varieties Grp and Grp − G, where G is a group of constants. The case Grp − F where F is a free group, is related to Tarski's problems devoted to logic of a free group.The described general insight on algebraic geometry in different varieties of algebras inspires some new problems in algebra and algebraic geometry. The problems of such kind determine, to a great extent, the content of universal algebraic geometry.For example, a general and natural problem is:When do the algebras H 1 and H 2 have the same geometry?or more specifically,What are the conditions on algebras from a given variety Θ which provide coincidence of their algebraic geometries?We consider two variants of coincidence:2) These categories are equivalent. This problem is highly connected with the following general algebraic problem:Let Θ 0 be the category of all free in Θ algebras W = W (X), where X is finite. Consider the groups of automorphisms Aut (Θ 0 )

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Geometric Equivalence of Algebras

Abstract: In this paper, we study the geometric equivalence of algebras in several varieties of algebras. We solve some of the problems formulated in [2], in particular, that of geometric equivalence for real-closed fields and finitely generated commutative groups.

Cited by 15 publications

References 1 publication

Automorphisms of the category of free Lie algebras

Automorphisms of the category of free Lie algebras

The Group of Automorphisms of the Semigroup of Endomorphisms of Free Commutative and Free Associative Algebras

Problems in algebra inspired by universal algebraic geometry

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