Algebra, categories and databases

Plotkin, B.

doi:10.1016/s1570-7954(00)80028-5

Cited by 11 publications

(10 citation statements)

References 36 publications

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“…The connections between algebraic geometry and mathematical logic are extremely important. First of all, notice a deep connection between algebra, category theory and model theory inspired by the results of Plotkin and his school (see References [1][2][3][4][5]. Note that this research is related to the one of most striking examples of interaction between model theory and geometry given by solutions of the famous Tarskii's problem, see References [6,7].…”

Section: Introductionmentioning

confidence: 80%

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: 80%

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

et al. 2020

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“…This invariant is a coordinate algebra of the set A. It can be represented as an algebra of regular functions determined on A (see [Pl2]).…”

Section: Preliminary Remarksmentioning

confidence: 99%

Seven lectures on universal algebraic geometry

Plotkin

2019

Groups, Algebras and Identities

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“…The same algebra plays an essential part in the algebraic geometry in the FOL in Â. In particular, the role of the algebras Hal Â ðGÞ is underlined by the following theorem [14]: Define the notion of the logical kernel of a homomorphism. Let the homomorphism " : WðXÞ !…”

Section: Algebras Of Formulasmentioning

confidence: 99%

“…This invariant is a coordinate algebra of the set A. It can be viewed as an algebra of regular functions determined on the variety A (see [14]). …”

Section: Preliminary Remarksmentioning

confidence: 99%

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An Algebraic Approach to Knowledge Base Models Informational Equivalence

Plotkin

2005

Acta Appl Math

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In this paper we study the notion of knowledge from the positions of universal algebra and algebraic logic. We consider first order knowledge which is based on first order logic. We define categories of knowledge and knowledge base models. These notions are defined for the fixed subject of knowledge. The key notion of informational equivalence of two knowledge base models is introduced. We use the idea of equivalence of categories in this definition. We prove that for finite models there is a clear way to determine whether the knowledge base models are informationally equivalent. (2000): 08A70, 03G25. Mathematics Subject Classifications

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Algebra, categories and databases

Cited by 11 publications

References 36 publications

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

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

Seven lectures on universal algebraic geometry

An Algebraic Approach to Knowledge Base Models Informational Equivalence

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