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

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1694
Author(s):  
Alexei Kanel-Belov ◽  
Alexei Chilikov ◽  
Ilya Ivanov-Pogodaev ◽  
Sergey Malev ◽  
Eugeny Plotkin ◽  
...  
Keyword(s):  
Algebraic Geometry ◽  
Model Theory ◽  
Weyl Algebra ◽  
First Order ◽  
Zero Divisors ◽  
Equivalence Of Categories ◽  
Algebra Logic ◽  
Algorithmic Problems ◽  
Universal Algebraic Geometry ◽  
Non Standard Analysis

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.

scholarly journals ALGEBRAIC LOGIC AND LOGICALLY-GEOMETRIC TYPES IN VARIETIES OF ALGEBRAS

2012 ◽  
Vol 12 (02) ◽  
pp. 1250146 ◽  
Author(s):  
BORIS PLOTKIN ◽  
ELENA ALADOVA ◽  
EUGENE PLOTKIN
Keyword(s):  
Algebraic Geometry ◽  
Model Theory ◽  
Affine Space ◽  
Algebraic Logic ◽  
Geometric Properties ◽  
First Order ◽  
Galois Correspondence ◽  
Universal Algebraic Geometry

The main objective of this paper is to show that the notion of type which was developed within the frames of logic and model theory has deep ties with geometric properties of algebras. These ties go back and forth from universal algebraic geometry to the model theory through the machinery of algebraic logic. We show that types appear naturally as logical kernels in the Galois correspondence between filters in the Halmos algebra of first order formulas with equalities and elementary sets in the corresponding affine space.

scholarly journals Notes on Quasiminimality and Excellence

2004 ◽  
Vol 10 (3) ◽  
pp. 334-366 ◽  
Author(s):  
John T. Baldwin
Keyword(s):  
Algebraic Geometry ◽  
Model Theory ◽  
Special Kind ◽  
Order Logic ◽  
First Order Logic ◽  
Structure Theory ◽  
Mathematical Structures ◽  
First Order ◽  
Elementary Class ◽  
Abstract Elementary Class

AbstractThis paper ties together much of the model theory of the last 50 years. Shelah's attempts to generalize the Morley theorem beyond first order logic led to the notion of excellence, which is a key to the structure theory of uncountable models. The notion of Abstract Elementary Class arose naturally in attempting to prove the categoricity theorem for Lω1,ω(Q). More recently, Zilber has attempted to identify canonical mathematical structures as those whose theory (in an appropriate logic) is categorical in all powers. Zilber's trichotomy conjecture for first order categorical structures was refuted by Hrushovski, by the introducion of a special kind of Abstract Elementary Class. Zilber uses a powerful and essentailly infinitary variant on these techniques to investigate complex exponentiation. This not only demonstrates the relevance of Shelah's model theoretic investigations to mainstream mathematics but produces new results and conjectures in algebraic geometry.

Boolean-valued structures

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Rules ◽  
Mathematical Concepts ◽  
Semantic Framework ◽  
First Order ◽  
Standard Models ◽  
Boolean Valued Model ◽  
Semantic Values

Chapters 6-12 are driven by questions about the ability to pin down mathematical entities and to articulate mathematical concepts. This chapter is driven by similar questions about the ability to pin down the semantic frameworks of language. It transpires that there are not just non-standard models, but non-standard ways of doing model theory itself. In more detail: whilst we normally outline a two-valued semantics which makes sentences True or False in a model, the inference rules for first-order logic are compatible with a four-valued semantics; or a semantics with countably many values; or what-have-you. The appropriate level of generality here is that of a Boolean-valued model, which we introduce. And the plurality of possible semantic values gives rise to perhaps the ‘deepest’ level of indeterminacy questions: How can humans pin down the semantic framework for their languages? We consider three different ways for inferentialists to respond to this question.

Weakly atomic-compact relational structures

1971 ◽  
Vol 36 (1) ◽  
pp. 129-140 ◽  
Author(s):  
G. Fuhrken ◽  
W. Taylor
Keyword(s):  
Abelian Group ◽  
Model Theory ◽  
Compact Abelian Group ◽  
Finite Subset ◽  
Relational Structure ◽  
Similarity Type ◽  
First Order ◽  
Relational Structures ◽  
Order Language ◽  
Weakly Atomic

A relational structure is called weakly atomic-compact if and only if every set Σ of atomic formulas (taken from the first-order language of the similarity type of augmented by a possibly uncountable set of additional variables as “unknowns”) is satisfiable in whenever every finite subset of Σ is so satisfiable. This notion (as well as some related ones which will be mentioned in §4) was introduced by J. Mycielski as a generalization to model theory of I. Kaplansky's notion of an algebraically compact Abelian group (cf. [5], [7], [1], [8]).

Model Theory of Epimorphisms

1974 ◽  
Vol 17 (4) ◽  
pp. 471-477 ◽  
Author(s):  
Paul D. Bacsich
Keyword(s):  
Model Theory ◽  
Order Theory ◽  
First Order ◽  
First Order Theory

Given a first-order theory T, welet be the category of models of T and homomorphisms between them. We shall show that a morphism A→B of is an epimorphism if and only if every element of B is definable from elements of A in a certain precise manner (see Theorem 1), and from this derive the best possible Cowell- power Theorem for .

scholarly journals Poisson algebras via model theory and differential-algebraic geometry

2017 ◽  
Vol 19 (7) ◽  
pp. 2019-2049 ◽  
Author(s):  
Jason Bell ◽  
Stéphane Launois ◽  
Omar León Sánchez ◽  
Rahim Moosa
Keyword(s):  
Algebraic Geometry ◽  
Model Theory ◽  
Poisson Algebras

Dimension in universal algebraic geometry

2014 ◽  
Vol 90 (1) ◽  
pp. 450-452 ◽  
Author(s):  
E. Yu. Daniyarova ◽  
A. G. Myasnikov ◽  
V. N. Remeslennikov
Keyword(s):  
Algebraic Geometry ◽  
Universal Algebraic Geometry

Some model theory for almost real closed fields

1996 ◽  
Vol 61 (4) ◽  
pp. 1121-1152 ◽  
Author(s):  
Françoise Delon ◽  
Rafel Farré
Keyword(s):  
Model Theory ◽  
Residue Field ◽  
Elementary Equivalence ◽  
First Order ◽  
Valuation Rings ◽  
Closed Fields ◽  
Theory Of Fields ◽  
Model Theory Of Fields ◽  
Elementary Inclusion ◽  
Henselian Valuation

AbstractWe study the model theory of fields k carrying a henselian valuation with real closed residue field. We give a criteria for elementary equivalence and elementary inclusion of such fields involving the value group of a not necessarily definable valuation. This allows us to translate theories of such fields to theories of ordered abelian groups, and we study the properties of this translation. We also characterize the first-order definable convex subgroups of a given ordered abelian group and prove that the definable real valuation rings of k are in correspondence with the definable convex subgroups of the value group of a certain real valuation of k.

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