Seven lectures on universal algebraic
                    geometry

Plotkin, B.

doi:10.1090/conm/726/14611

Cited by 19 publications

(16 citation statements)

References 29 publications

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“…Let Θ be a variety of linear algebras over a commutative-associative ring K and W = W(X) be a free algebra from Θ generated by a finite set X. Let H be an algebra from Θ and AG Θ (H) be the category of algebraic sets over H. Throughout the work, we refer to References [13,17] for definitions of the Universal Algebraic Geometry (UAG).…”

Section: Introductionmentioning

confidence: 99%

Nonstandard Analysis, Deformation Quantization and Some Logical Aspects of (Non)Commutative Algebraic Geometry

et al. 2020

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This paper surveys results related to well-known works of B. Plotkin and V. Remeslennikov on the edge of algebra, logic and geometry. We start from a brief review of the paper and motivations. The first sections deal with model theory. In the first part of the second section we describe the geometric equivalence, the elementary equivalence, and the isotypicity of algebras. We look at these notions from the positions of universal algebraic geometry and make emphasis on the cases of the first order rigidity. In this setting Plotkin’s problem on the structure of automorphisms of (auto)endomorphisms of free objects, and auto-equivalence of categories is pretty natural and important. The second part of the second section is dedicated to particular cases of Plotkin’s problem. The last part of the second section is devoted to Plotkin’s problem for automorphisms of the group of polynomial symplectomorphisms. This setting has applications to mathematical physics through the use of model theory (non-standard analysis) in the studying of homomorphisms between groups of symplectomorphisms and automorphisms of the Weyl algebra. The last sections deal with algorithmic problems for noncommutative and commutative algebraic geometry.The first part of it is devoted to the Gröbner basis in non-commutative situation. Despite the existence of an algorithm for checking equalities, the zero divisors and nilpotency problems are algorithmically unsolvable. The second part of the last section is connected with the problem of embedding of algebraic varieties; a sketch of the proof of its algorithmic undecidability over a field of characteristic zero is given.

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Section: Introductionmentioning

confidence: 99%

Nonstandard Analysis, Deformation Quantization and Some Logical Aspects of (Non)Commutative Algebraic Geometry

et al. 2020

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“…Our research is devoted to the studies in universal algebraic geometry. This direction was initiated in papers by B. Plotkin [7] and E. Daniyarova, A. Miasnikov, V. Remeslennikov [2], and it deals with equations over various algebraic structures (semigroups, groups, Lie algebras, etc.). In the current paper we study semigroup equations, but the problem we solved comes from group theory.…”

Section: Introductionmentioning

confidence: 99%

“…. + x j l−1 in terms t A (X), s A (X) (5,7) are not equal to the parts of the additive equation t B (X) = s B (X). New variables allow us to achieve this property, and we essentially use it in Lemma 6.5.…”

Section: Let Us Introduce New Variablesmentioning

confidence: 99%

Wreath products of semigroups in universal algebraic geometry and Plotkin's problem

Shevlyakov

2020

Preprint

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“…Let us just mention about the survey [1] for group equations. However the recent achievements of universal algebraic geometry (see the papers [2,3,4] by E.Daniyarova, A.Miasnikov, V.Remeslennikov, and B.Plotkin) allow us to pose new problems about equations (all required definitions may be found in Section 2 of the current paper). 3.…”

Section: Introductionmentioning

confidence: 99%