The model theory of differential fields with finitely many commuting derivations

2000 ◽  
Vol 65 (2) ◽  
pp. 885-913 ◽  
Author(s):  
Tracey McGrail
Keyword(s):  
Model Theory ◽  
Stable Model ◽  
Basic Model ◽  
Differential Fields

AbstractIn this paper we set out the basic model theory of differential fields of characteristic 0, which have finitely many commuting derivations. We give axioms for the theory of differentially closed differential fields with m derivations and show that this theory is ω-stable, model complete, and quantifier-eliminable, and that it admits elimination of imaginaries. We give a characterization of forking and compute the rank of this theory to be ωm + 1.

scholarly journals ON THE HEIGHT AND RELATIONAL COMPLEXITY OF A FINITE PERMUTATION GROUP

2021 ◽  
pp. 1-40
Author(s):  
NICK GILL ◽  
BIANCA LODÀ ◽  
PABLO SPIGA
Keyword(s):  
Permutation Group ◽  
Model Theory ◽  
Independent Set ◽  
Maximum Size ◽  
Proper Subset ◽  
Permutation Groups ◽  
Finite Permutation Group ◽  
Pointwise Stabilizer ◽  
Relational Complexity

Abstract Let G be a permutation group on a set $\Omega $ of size t. We say that $\Lambda \subseteq \Omega $ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda $ . We define the height of G to be the maximum size of an independent set, and we denote this quantity $\textrm{H}(G)$ . In this paper, we study $\textrm{H}(G)$ for the case when G is primitive. Our main result asserts that either $\textrm{H}(G)< 9\log t$ or else G is in a particular well-studied family (the primitive large–base groups). An immediate corollary of this result is a characterization of primitive permutation groups with large relational complexity, the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\textrm{I}(G)$ , the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either $\textrm{I}(G)<7\log t$ or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).

A Course on Basic Model Theory

2017 ◽  
Author(s):  
Haimanti Sarbadhikari ◽  
Shashi Mohan Srivastava
Keyword(s):  
Model Theory ◽  
Basic Model

A simple fixed point characterization of three-valued stable model semantics

1991 ◽  
Vol 40 (2) ◽  
pp. 73-78 ◽  
Author(s):  
S. Bonnier ◽  
U. Nilsson ◽  
T. Näslund
Keyword(s):  
Fixed Point ◽  
Stable Model

Differential forms in the model theory of differential fields

2003 ◽  
Vol 68 (3) ◽  
pp. 923-945 ◽  
Author(s):  
David Pierce
Keyword(s):  
Differential Geometry ◽  
Model Theory ◽  
Characteristic Zero ◽  
Differential Forms ◽  
Main Tool ◽  
Frobenius Theorem ◽  
Lie Bracket ◽  
Existentially Closed ◽  
Differential Fields

AbstractFields of characteristic zero with several commuting derivations can be treated as fields equipped with a space of derivations that is closed under the Lie bracket. The existentially closed instances of such structures can then be given a coordinate-free characterization in terms of differential forms. The main tool for doing this is a generalization of the Frobenius Theorem of differential geometry.

Basic model theory

2011 ◽  
pp. 163-184
Author(s):  
Boris Zilber
Keyword(s):  
Model Theory ◽  
Basic Model

scholarly journals Imaginaries and invariant types in existentially closed valued differential fields

2019 ◽  
Vol 2019 (750) ◽  
pp. 157-196 ◽  
Author(s):  
Silvain Rideau
Keyword(s):  
Model Theory ◽  
Extension Property ◽  
Valued Fields ◽  
Invariant Extension ◽  
Open Questions ◽  
Existentially Closed ◽  
Differential Fields ◽  
Abstract Criterion

Abstract We answer three related open questions about the model theory of valued differential fields introduced by Scanlon. We show that they eliminate imaginaries in the geometric language introduced by Haskell, Hrushovski and Macpherson and that they have the invariant extension property. These two results follow from an abstract criterion for the density of definable types in enrichments of algebraically closed valued fields. Finally, we show that this theory is metastable.

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