SU\G(𝔸)/K·U
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All published works (23)

DIMENSION AND MEASURE IN PSEUDOFINITE <i>H</i>-STRUCTURES

2025-03-18
Alexander Berenstein , Darío García , Tingxiang Zou
Abstract We study H -structures associated with $SU$ -rank 1 measurable structures. We prove that the $SU$ -rank of the expansion is continuous and that it is uniformly definable in … Abstract We study H -structures associated with $SU$ -rank 1 measurable structures. We prove that the $SU$ -rank of the expansion is continuous and that it is uniformly definable in terms of the parameters of the formulas. We also introduce notions of dimension and measure for definable sets in the expansion and prove they are uniformly definable in terms of the parameters of the formulas.
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Pseudofiniteness and measurability of the everywhere infinite forest

2023-01-01
Darío García , Melissa Robles
In this paper we study the theories of the infinite-branching tree and the $r$-regular tree, and show that both of them are pseudofinite. Moreover, we show that they can be … In this paper we study the theories of the infinite-branching tree and the $r$-regular tree, and show that both of them are pseudofinite. Moreover, we show that they can be realized by infinite ultraproducts of polynomial exact classes of graphs, and so they are also generalised measurable.
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Model-theoretic dividing lines via posets

2022-01-01
Darío García , Rosario Mennuni
We show that for each property $\mathsf{P}\in \{\mathsf{OP}, \mathsf{IP}, \mathsf{TP}_1, \mathsf{TP}_2, \mathsf{ATP}, \mathsf{SOP}_3\}$ there is a poset $\Sigma_{\mathsf{P}}$ such that a theory has property $\mathsf{P}$ if and only if some … We show that for each property $\mathsf{P}\in \{\mathsf{OP}, \mathsf{IP}, \mathsf{TP}_1, \mathsf{TP}_2, \mathsf{ATP}, \mathsf{SOP}_3\}$ there is a poset $\Sigma_{\mathsf{P}}$ such that a theory has property $\mathsf{P}$ if and only if some model interprets a poset in which $\Sigma_{\mathsf{P}}$ can be embedded. We also introduce a new property $\mathsf{SUP}$, consistent with $\mathsf{NIP}_2$ and implying $\mathsf{ATP}$ and $\mathsf{SOP}$.
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A note on the definability of genus for Zariski geometries

2021-10-01
Darío García , Pedro Hernández Hernández , Joel Torres del Valle
In this work we propose a notion of genus in the context of Zariski geometries and we obtain natural generalizations of the Riemann--Hurwitz Theorem and the Hurwitz Theorem in the … In this work we propose a notion of genus in the context of Zariski geometries and we obtain natural generalizations of the Riemann--Hurwitz Theorem and the Hurwitz Theorem in the context of very ample Zariski geometries. As a corollary, we show that such notion of genus cannot be first-order definable in the full language of a Zariski geometry.
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A note on the definability of genus for Zariski geometries

2021-01-01
Darío García , Pedro Hernández Hernández , Joel Torres del Valle
In this work we propose a notion of genus in the context of Zariski geometries and we obtain natural generalizations of the Riemann--Hurwitz Theorem and the Hurwitz Theorem in the … In this work we propose a notion of genus in the context of Zariski geometries and we obtain natural generalizations of the Riemann--Hurwitz Theorem and the Hurwitz Theorem in the context of very ample Zariski geometries. As a corollary, we show that such notion of genus cannot be first-order definable in the full language of a Zariski geometry.
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Ordered asymptotic classes of finite structures

2020-01-07
Darío García
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Dimension and measure in pseudofinite H-structures

2020-01-01
Alexander Berentein , Darío García , Tingxiang Zou
We study H-structures associated to SU-rank 1 measurable structures. We prove that the SU-rank of the expansion is continuous and that it is uniformly definable in terms of the parameters … We study H-structures associated to SU-rank 1 measurable structures. We prove that the SU-rank of the expansion is continuous and that it is uniformly definable in terms of the parameters of the formulas. We also introduce notions of dimension and measure for definable sets in the expansion and prove they are uniformly definable in terms of the parameters of the formulas.
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UNIMODULARITY UNIFIED

2017-05-08
Darío García , Frank Olaf Wagner
Abstract Unimodularity is localized to a complete stationary type, and its properties are analysed. Some variants of unimodularity for definable and type-definable sets are introduced, and the relationship between these … Abstract Unimodularity is localized to a complete stationary type, and its properties are analysed. Some variants of unimodularity for definable and type-definable sets are introduced, and the relationship between these different notions is studied. In particular, it is shown that all notions coincide for non-multidimensional theories where the dimensions are associated to strongly minimal types.

Unimodularity unified

2017-01-01
Darío García , Frank Olaf Wagner
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Unimodularity unified

2017-01-01
Darío García , Frank Olaf Wagner
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Unimodularity unified

2017-01-01
Darío García , Frank Olaf Wagner
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Unimodularity unified

2017-01-01
Darío García , Frank Olaf Wagner
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Unimodularity unified

2016-01-01
Darío García , Frank Olaf Wagner
Unimodularity is localized to a complete stationary type, and its properties are analysed. Some variants of unimodularity for definable and type-definable sets are introduced, and the relationship between these different … Unimodularity is localized to a complete stationary type, and its properties are analysed. Some variants of unimodularity for definable and type-definable sets are introduced, and the relationship between these different notions is studied. In particular, it is shown that all notions coincide for non-multidimensional theories where the dimensions are associated to strongly minimal types.
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Pseudofinite structures and simplicity

2015-04-16
Darío García , Dugald Macpherson , Charles Steinhorn
We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or … We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or supersimplicity of the underlying theory, and that a drop in pseudofinite dimension is equivalent to forking. Under a suitable assumption, a measure-theoretic condition is shown to be equivalent to local stability. Many examples are explored, including vector spaces over finite fields viewed as 2-sorted finite structures, and homocyclic groups. Connections are made to products of sets in finite groups, in particular to word maps, and a generalization of Tao's Algebraic Regularity Lemma is noted.
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Ordered asymptotic classes of finite structures

2014-10-09
Darío García
In this paper we introduce the concept of O-asymptotic classes of finite structures, melding ideas coming from 1-dimensional asymptotic classes and o-minimality. The results we present here include a cell-decomposition … In this paper we introduce the concept of O-asymptotic classes of finite structures, melding ideas coming from 1-dimensional asymptotic classes and o-minimality. The results we present here include a cell-decomposition result for O-asymptotic classes and a classification of their infinite ultraproducts: Every infinite ultraproduct of structures in an O-asymptotic class is superrosy of U-thorn-rank 1, and NTP2 (in fact, inp-minimal).
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O-asymptotic classes of finite structures

2014-10-09
Darío García
In this paper we introduce the concept of O-asymptotic classes of finite structures, melding ideas coming from 1-dimensional asymptotic classes and o-minimality. The results we present here include a cell-decomposition … In this paper we introduce the concept of O-asymptotic classes of finite structures, melding ideas coming from 1-dimensional asymptotic classes and o-minimality. The results we present here include a cell-decomposition result for O-asymptotic classes and a classification of their infinite ultraproducts: Every infinite ultraproduct of structures in an O-asymptotic class is superrosy of U-thorn-rank 1, and NTP2 (in fact, inp-minimal).
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Pseudofinite structures and simplicity

2014-09-30
Darío García , Dugald Macpherson , Charles Steinhorn
We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or … We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or supersimplicity of the underlying theory, and that a drop in pseudofinite dimension is equivalent to forking. Under a suitable assumption, a measure-theoretic condition is shown to be equivalent to local stability. Many examples are explored, including vector spaces over finite fields viewed as 2-sorted finite structures, and homocyclic groups. Connections are made to products of sets in finite groups, in particular to word maps, and a generalization of Tao's algebraic regularity lemma is noted.
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A note on pseudofinite dimension and forking

2014-02-21
Darío García
In this paper we show that an instance of dividing in pseudofinite structures can be witnessed by a drop of the pseudofinite dimension. As an application of this result we … In this paper we show that an instance of dividing in pseudofinite structures can be witnessed by a drop of the pseudofinite dimension. As an application of this result we give new proofs of known results for asymptotic classes of finite structures.
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A note on pseudofinite dimension and forking

2014-01-01
Darío García
In this paper we show that an instance of dividing in pseudofinite structures can be witnessed by a drop of the pseudofinite dimension. As an application of this result we … In this paper we show that an instance of dividing in pseudofinite structures can be witnessed by a drop of the pseudofinite dimension. As an application of this result we give new proofs of known results for asymptotic classes of finite structures.
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Ordered asymptotic classes of finite structures

2014-01-01
Darío García
In this paper we introduce the concept of O-asymptotic classes of finite structures, melding ideas coming from 1-dimensional asymptotic classes and o-minimality. The results we present here include a cell-decomposition … In this paper we introduce the concept of O-asymptotic classes of finite structures, melding ideas coming from 1-dimensional asymptotic classes and o-minimality. The results we present here include a cell-decomposition result for O-asymptotic classes and a classification of their infinite ultraproducts: Every infinite ultraproduct of structures in an O-asymptotic class is superrosy of U-thorn-rank 1, and NTP2 (in fact, inp-minimal).
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Generic stability, forking, and þ-forking

2012-07-24
Darío García , Alf Onshuus , Alexander Usvyatsov
Abstract notions of “smallness” are among the most important tools that model theory offers for the analysis of arbitrary structures. The two most useful notions of this kind are forking … Abstract notions of “smallness” are among the most important tools that model theory offers for the analysis of arbitrary structures. The two most useful notions of this kind are forking (which is closely related to certain measure zero ideals) and thorn-forking (which generalizes the usual topological dimension). Under certain mild assumptions, forking is the finest notion of smallness, whereas thorn-forking is the coarsest. In this paper we study forking and thorn-forking, restricting ourselves to the class of generically stable types. Our main conclusion is that in this context these two notions coincide. We explore some applications of this equivalence.

A remark on locally determining sequences in infinite dimensional spaces

1990-01-01
F. Galindo , Darío García , Manuel Maestre
We obtain a real version of a result of J. M. Ansemil and S. Dineen [2],i.e., every locally determining at 0 set for analytic functions in a metrizable separable real … We obtain a real version of a result of J. M. Ansemil and S. Dineen [2],i.e., every locally determining at 0 set for analytic functions in a metrizable separable real locally convex space contains a locally determining at 0 null sequence.

Stationary axisymmetric solutions of the Jordan-Brans-Dicke equations with electromagnetic sources

1988-10-01
Darío García , H. Guadalupe Mart�nez

Common Coauthors

Coauthor Papers Together
Frank Olaf Wagner 6
Dugald Macpherson 2
Charles Steinhorn 2
Tingxiang Zou 2
Joel Torres del Valle 2
Pedro Hernández Hernández 2
Alexander Berenstein 1
H. Guadalupe Mart�nez 1
Melissa Robles 1
Rosario Mennuni 1
F. Galindo 1
Alexander Berentein 1
Alexander Usvyatsov 1
Manuel Maestre 1
Alf Onshuus 1

Commonly Cited References

One-dimensional asymptotic classes of finite structures

2007-09-25
Dugald Macpherson , Charles Steinhorn
A collection ${\mathcal C}$ of finite $\mathcal {L}$-structures is a 1-dimensional asymptotic class if for every $m \in {\mathbb N}$ and every formula $\varphi (x,\bar {y})$, where $\bar {y}=(y_1,\ldots ,y_m)$: … A collection ${\mathcal C}$ of finite $\mathcal {L}$-structures is a 1-dimensional asymptotic class if for every $m \in {\mathbb N}$ and every formula $\varphi (x,\bar {y})$, where $\bar {y}=(y_1,\ldots ,y_m)$: [(i)] There is a positive constant $C$ and a finite set $E\subset {\mathbb R}^{>0}$ such that for every $M\in {\mathcal C}$ and $\bar {a}\in M^m$, either $|\varphi (M,\bar {a})|\leq C$, or for some $\mu \in E$, \[ \big ||\varphi (M,\bar {a})|-\mu |M|\big | \leq C|M|^{\frac {1}{2}}.\] [(ii)] For every $\mu \in E$, there is an $\mathcal {L}$-formula $\varphi _{\mu }(\bar {y})$, such that $\varphi _{\mu }(M^m)$ is precisely the set of $\bar {a}\in M^m$ with \[ \big ||\varphi (M,\bar {a})|-\mu |M|\big | \leq C|M|^{\frac {1}{2}}.\] One-dimensional asymptotic classes are introduced and studied here. These classes come equipped with a notion of dimension that is intended to provide for the study of classes of finite structures a concept that is central in the development of model theory for infinite structures. Connections with the model theory of infinite structures are also drawn.
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Properties and consequences of Thorn-independence

2006-03-01
Alf Onshuus
Abstract We develop a new notion of independence (ϸ-independence, read “thorn”-independence) that arises from a family of ranks suggested by Scanlon (ϸ-ranks). We prove that in a large class of … Abstract We develop a new notion of independence (ϸ-independence, read “thorn”-independence) that arises from a family of ranks suggested by Scanlon (ϸ-ranks). We prove that in a large class of theories (including simple theories and o-minimal theories) this notion has many of the properties needed for an adequate geometric structure. We prove that ϸ-independence agrees with the usual independence notions in stable, supersimple and o-minimal theories. Furthermore, we give some evidence that the equivalence between forking and ϸ-forking in simple theories might be closely related to one of the main open conjectures in simplicity theory, the stable forking conjecture. In particular, we prove that in any simple theory where the stable forking conjecture holds, ϸ-independence and forking independence agree.

Stable group theory and approximate subgroups

2011-06-15
Ehud Hrushovski
We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group <inline-formula content-type="math/mathml"> <mml:math … We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we show that a finite subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper X upper X Superscript negative 1 Baseline upper X EndAbsoluteValue slash StartAbsoluteValue upper X EndAbsoluteValue"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>X</mml:mi> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">|X X ^{-1}X |/ |X|</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Model-theoretically we prove the independence theorem and the stabilizer theorem in a general first-order setting.
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Asymptotic classes of finite structures

2007-06-01
Richard Elwes
In this paper we consider classes of finite structures where we have good control over the sizes of the definable sets. The motivating example is the class of finite fields: … In this paper we consider classes of finite structures where we have good control over the sizes of the definable sets. The motivating example is the class of finite fields: it was shown in [1] that for any formula in the language of rings, there are finitely many pairs ( d , μ ) ∈ ω × Q &gt;0 so that in any finite field F and for any ā ∈ F m the size | ø ( F n ,ā)| is “approximately” μ | F | d . Essentially this is a generalisation of the classical Lang-Weil estimates from the category of varieties to that of the first-order-definable sets.

A survey of asymptotic classes and measurable structures

2008-05-22
Richard Elwes , Dugald Macpherson
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
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Remarks on unimodularity

2011-10-11
Charlotte Kestner , Anand Pillay
Abstract We clarify and correct some statements and results in the literature concerning unimodularity in the sense of Hrushovski [7], and measurability in the sense of Macpherson and Steinhorn [8], … Abstract We clarify and correct some statements and results in the literature concerning unimodularity in the sense of Hrushovski [7], and measurability in the sense of Macpherson and Steinhorn [8], pointing out in particular that the two notions coincide for strongly minimal structures and that another property from [7] is strictly weaker, as well as “completing” Elwes' proof [5] that measurability implies 1-basedness for stable theories.
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On NIP and invariant measures

2011-05-13
Ehud Hrushovski , Anand Pillay
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are … We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = \mathrm{tp}(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over \mathrm{bdd}(A) , (ii) analogous statements for Keisler measures and definable groups, including the fact that G^{000} = G^{00} for G definably amenable, (iii) definitions, characterizations and properties of “generically stable” types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o -minimal expansions of real closed fields.
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Theories without the tree property of the second kind

2013-10-28
Artem Chernikov
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Unimodular Minimal Structures

1992-12-01
Ehud Hrushovski

Definable sets over finite fields.

1992-05-01
Zoé Chatzidakis , Lou van den Dries , Angus Macintyre

Quasi-o-minimal structures

2000-09-01
Oleg Belegradek , Ya’acov Peterzil , Frank Olaf Wagner
Abstract A structure ( M , &lt;, …) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets … Abstract A structure ( M , &lt;, …) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets and intervals. We give a series of natural examples of quasi-o-minimal structures which are not o-minimal; one of them is the ordered group of integers. We develop a technique to investigate quasi-o-minimality and use it to study quasi-o-minimal ordered groups (possibly with extra structure). Main results: any quasi-o-minimal ordered group is abelian; any quasi-o-minimal ordered ring is a real closed field, or has zero multiplication; every quasi-o-minimal divisible ordered group is o-minimal; every quasi-o-minimal archimedian densely ordered group is divisible. We show that a counterpart of quasi-o-minimality in stability theory is the notion of theory of U -rank 1.

Finite subgroups of algebraic groups

2011-04-28
Michael Larsen , Richard Pink
Generalizing a classical theorem of Jordan to arbitrary characteristic, we prove that every finite subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L Subscript n"> <mml:semantics> <mml:msub> <mml:mi>GL</mml:mi> <mml:mi>n</mml:mi> … Generalizing a classical theorem of Jordan to arbitrary characteristic, we prove that every finite subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L Subscript n"> <mml:semantics> <mml:msub> <mml:mi>GL</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\operatorname {GL}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over a field of any characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> possesses a subgroup of bounded index which is composed of finite simple groups of Lie type in characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, a commutative group of order prime to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-group. While this statement can be deduced from the classification of finite simple groups, our proof is self-contained and uses methods only from algebraic geometry and the theory of linear algebraic groups. We believe that our results can serve as a viable substitute for classification in a range of applications in various areas of mathematics.

From Stability to Simplicity

1998-03-01
Byunghan Kim , Anand Pillay
§1. Introduction . In this report we wish to describe recent work on a class of first order theories first introduced by Shelah in [32], the simple theories. Major progress … §1. Introduction . In this report we wish to describe recent work on a class of first order theories first introduced by Shelah in [32], the simple theories. Major progress was made in the first author's doctoral thesis [17]. We will give a survey of this, as well as further works by the authors and others. The class of simple theories includes stable theories, but also many more, such as the theory of the random graph. Moreover, many of the theories of particular algebraic structures which have been studied recently (pseudofinite fields, algebraically closed fields with a generic automorphism, smoothly approximable structures) turn out to be simple. The interest is basically that a large amount of the machinery of stability theory, invented by Shelah, is valid in the broader class of simple theories. Stable theories will be defined formally in the next section. An exhaustive study of them is carried out in [33]. Without trying to read Shelah's mind, we feel comfortable in saying that the importance of stability for Shelah lay partly in the fact that an unstable theory T has 2 λ many models in any cardinal λ ≥ ω 1 + | T | (proved by Shelah). (Note that for λ ≥ | T | 2 λ is the maximum possible number of models of cardinality λ.)

Generic stability, regularity, and quasiminimality

2011-09-07
Anand Pillay , Predrag Tanović
We study the notions generic stability, regularity, homogeneous pregeometries, quasiminimality, and their mutual relations, in arbitrary first order theories. We prove that “infinite-dimensional homogeneous pregeometries” coincide with generically stable strongly … We study the notions generic stability, regularity, homogeneous pregeometries, quasiminimality, and their mutual relations, in arbitrary first order theories. We prove that “infinite-dimensional homogeneous pregeometries” coincide with generically stable strongly regular types (p(x), x = x). We prove that in a theory without the strict order property, regular types are generically stable, and prove analogous results for quasiminimal structures. We prove that the “generic type” of a quasiminimal structure is “locally strongly regular”.
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Weakly Normal Groups

1987-01-01
U. Hrushovski , Anthony L. Pillay

Generic trivializations of geometric theories

2014-08-01
Alexander Berenstein , Evgueni Vassiliev
We study the theory of the structure induced by parameter free formulas on a “dense” algebraically independent subset of a model of a geometric theory T . We show that … We study the theory of the structure induced by parameter free formulas on a “dense” algebraically independent subset of a model of a geometric theory T . We show that while being a trivial geometric theory, inherits most of the model theoretic complexity of T related to stability, simplicity, rosiness, the NIP and the NTP 2 . In particular, we show that T is strongly minimal, supersimple of SU‐rank 1, has the NIP or the NTP 2 exactly when has these properties. We show that if T is superrosy of thorn rank 1, then so is , and that the converse holds if T satisfies acl = dcl.
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Remarks on Tao's algebraic regularity lemma

2013-10-28
Anand Pillay , Sergei Starchenko
We give a stability-theoretic proof of the algebraic regularity lemma from [6], in a slightly strengthened form. We also point out that the underlying lemmas hold at a greater level … We give a stability-theoretic proof of the algebraic regularity lemma from [6], in a slightly strengthened form. We also point out that the underlying lemmas hold at a greater level of generality, namely \measurable theories and structures in the sense of Elwes-MacphersonSteinhorn.
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Pseudofinite structures and simplicity

2015-04-16
Darío García , Dugald Macpherson , Charles Steinhorn
We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or … We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or supersimplicity of the underlying theory, and that a drop in pseudofinite dimension is equivalent to forking. Under a suitable assumption, a measure-theoretic condition is shown to be equivalent to local stability. Many examples are explored, including vector spaces over finite fields viewed as 2-sorted finite structures, and homocyclic groups. Connections are made to products of sets in finite groups, in particular to word maps, and a generalization of Tao's Algebraic Regularity Lemma is noted.
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Generic structures and simple theories

1998-11-01
Zoé Chatzidakis , Anthony L. Pillay

Geometric structures with a dense independent subset

2015-10-07
Alexander Berenstein , Evgueni Vassiliev
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Product decompositions of quasirandom groups and a Jordan type theorem

2011-05-13
Nikolay Nikolov , László Pyber
We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If k is the minimal degree of a representation of the finite group … We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If k is the minimal degree of a representation of the finite group G , then for every subset B of G with |B| &gt; |G| / k^{\frac13} we have B^3 = G . We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan's theorem which implies that if k \geq 2 , then G has a proper subgroup of index at most c_0 k^2 for some constant c_0 , hence a product-free subset of size at least |G| / ck . This answers a question of Gowers.
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Valued difference fields and NTP2

2014-06-24
Artem Chernikov , Martin Hils
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On lovely pairs of geometric structures

2009-11-29
Alexander Berenstein , Evgueni Vassiliev

Groups and fields with $\operatorname {NTP}_{2}$

2014-08-19
Artem Chernikov , Itay Kaplan , Pierre Simon
$\operatorname {NTP}_{2}$ is a large class of first-order theories defined by Shelah generalizing simple and NIP theories. Algebraic examples of $\operatorname {NTP}_{2}$ structures are given by ultra-products of $p$-adics and … $\operatorname {NTP}_{2}$ is a large class of first-order theories defined by Shelah generalizing simple and NIP theories. Algebraic examples of $\operatorname {NTP}_{2}$ structures are given by ultra-products of $p$-adics and certain valued difference fields (such as a non-standard Frobenius automorphism living on an algebraically closed valued field of characteristic 0). In this note we present some results on groups and fields definable in $\operatorname {NTP}_{2}$ structures. Most importantly, we isolate a chain condition for definable normal subgroups and use it to show that any $\operatorname {NTP}_{2}$ field has only finitely many Artin-Schreier extensions. We also discuss a stronger chain condition coming from imposing bounds on burden of the theory (an appropriate analogue of weight) and show that every strongly dependent valued field is Kaplansky.

The Elementary Theory of the Frobenius Automorphisms

2004-06-25
Ehud Hrushovski
A Frobenius difference field is an algebraically closed field of characteristic $p>0$, enriched with a symbol for $x \mapsto x^{p^m}$. We study a sentence or formula in the language of … A Frobenius difference field is an algebraically closed field of characteristic $p>0$, enriched with a symbol for $x \mapsto x^{p^m}$. We study a sentence or formula in the language of fields with a distinguished automorphism, interpreted in Frobenius difference fields with $p$ or $m$ tending to infinity. In particular, a decision procedure is found to determine when a sentence is true in almost every Frobenius difference field. This generalizes Cebotarev's density theorem and Weil's Riemann hypothesis for curves (both in qualitative versions), but hinges on a result going slightly beyond the latter. The setting for the proof is the geometry of difference varieties of transformal dimension zero; these generalize algebraic varieties, and are shown to have a rich structure, only partly explicated here. Some applications are given, in particular to finite simple groups, and to the Jacobi bound for difference equations.
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Quasirandom Groups

2008-01-07
W. T. Gowers
Babai and Sós have asked whether there exists a constant c &gt; 0 such that every finite group G has a product-free subset of size at least c | G … Babai and Sós have asked whether there exists a constant c &gt; 0 such that every finite group G has a product-free subset of size at least c | G |: that is, a subset X that does not contain three elements x , y and z with xy = z . In this paper we show that the answer is no. Moreover, we give a simple sufficient condition for a group not to have any large product-free subset.

An introduction to forking

1979-09-01
Daniel Lascar , Bruno Poizat
The notion of forking has been introduced by Shelah, and a full treatment of it will appear in his book on stability [S1]. The principal aim of this paper is … The notion of forking has been introduced by Shelah, and a full treatment of it will appear in his book on stability [S1]. The principal aim of this paper is to show that it is an easy and natural notion. Consider some well-known examples of ℵ 0 -stable theories: vector spaces over Q , algebraically closed fields, differentially closed fields of characteristic 0; in each of these cases, we have a natural notion of independence: linear, algebraic and differential independence respectively. Forking gives a generalization of these notions. More precisely, if are subsets of some model and c a point of this model, the fact that the type of c over does not fork over means that there are no more relations of dependence between c and than there already existed between c and . In the case of the vector spaces, this means that c is in the space generated by only if it is already in the space generated by . In the case of differentially closed fields, this means that the minimal differential equations of c with coefficient respectively in and have the same order. Of course, these notions of dependence are essential for the study of the above mentioned structures. Forking is no less important for stable theories. A glance at Shelah's book will convince the reader that this is the case. What we have to do is the following. Assuming T stable and given and p a type on , we want to distinguish among the extensions of p to some of them that we shall call the nonforking extensions of p .

Groups in supersimple and pseudofinite theories

2011-07-14
Richard Elwes , Éric Jaligot , Dugald Macpherson +1 more
We consider groups G interpretable in a supersimple finite rank theory T such that Teq eliminates ∃∞. It is shown that G has a definable soluble radical. If G has … We consider groups G interpretable in a supersimple finite rank theory T such that Teq eliminates ∃∞. It is shown that G has a definable soluble radical. If G has rank 2, then if G is pseudofinite, it is soluble-by-finite, and partial results are obtained under weaker hypotheses, such as ‘functional unimodularity’ of the theory. A classification is obtained when T is pseudofinite and G has a definable and definably primitive action on a rank 1 set.
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Groups, measures, and the NIP

2007-02-02
Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay
We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating … We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce the new notion of "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.
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Measurable groups of low dimension

2008-07-01
Richard Elwes , Mark Ryten
Abstract We consider low‐dimensional groups and group‐actions that are definable in a supersimple theory of finite rank. We show that any rank 1 unimodular group is (finite‐by‐Abelian)‐by‐finite, and that any … Abstract We consider low‐dimensional groups and group‐actions that are definable in a supersimple theory of finite rank. We show that any rank 1 unimodular group is (finite‐by‐Abelian)‐by‐finite, and that any 2‐dimensional asymptotic group is soluble‐by‐finite. We obtain a field‐interpretation theorem for certain measurable groups, and give an analysis of minimal normal subgroups and socles in groups definable in a supersimple theory of finite rank where infinity is definable. We prove a primitivity theorem for measurable group actions. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

ELIMINATION OF QUANTIFIERS OVER VECTORS IN SOME THEORIES OF VECTOR SPACES

1992-01-01
Andrey A. Kuzichev
Abstract We consider two‐sorted theories of vector spaces and prove a criterion for the assertion that such a theory allows elimination of quantifiers over vector variables. Abstract We consider two‐sorted theories of vector spaces and prove a criterion for the assertion that such a theory allows elimination of quantifiers over vector variables.

Forking geometry on theories with an independent predicate

2014-11-03
Juan Felipe Carmona

Eine Methode zur Konstruktion stationärer EINSTEIN‐MAXWELL‐Felder

1969-01-01
Gernot Neugebauer , Dietrich Krämer
Abstract General methods are developed for constructing stationary solutions of the EINSTEIN‐MAXWELL equations from known EINSTEIN‐MAXWELL fields or vacuum fields. These methods are based on the investigation of an abstract … Abstract General methods are developed for constructing stationary solutions of the EINSTEIN‐MAXWELL equations from known EINSTEIN‐MAXWELL fields or vacuum fields. These methods are based on the investigation of an abstract RIEMANNian V 4 characteristic of the EINSTEIN‐MAXWELL equations. The application to KERR'S solution demonstrates the practical use of this procedure (part 3 A).

Pseudofinite groups with NIP theory and definability in finite simple groups

2012-01-01
Dugald Macpherson , Katrin Tent
We show that any pseudofinite group with NIP theory and with a finite upper bound on the length of chains of centralisers is soluble-by-finite. In particular, any NIP rosy pseudofinite … We show that any pseudofinite group with NIP theory and with a finite upper bound on the length of chains of centralisers is soluble-by-finite. In particular, any NIP rosy pseudofinite group is soluble-by-finite. This gener- alises, and shortens the proof of, an earlier result for stable pseudofinite groups. An example is given of an NIP pseudofinite group which is not soluble-by-finite. However, if C is a class of finite groups such that all infinite ultraproducts of members of C have NIP theory, then there is a bound on the index of the soluble radical of any member of C. We also survey some ways in which model theory gives information on families of finite simple groups, particularly concerning products of images of word maps.
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Counting and dimensions

2008-05-22
Ehud Hrushovski , Frank Olaf Wagner
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
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TAME TOPOLOGY AND O-MINIMAL STRUCTURES (London Mathematical Society Lecture Note Series 248)

2000-05-01
A. J. Wilkie
By Lou van den Dries: 180 pp., £24.95 (US$39.95, LMS Members' price £18.70), isbn 0 521 59838 9 (Cambridge University Press 1998). By Lou van den Dries: 180 pp., £24.95 (US$39.95, LMS Members' price £18.70), isbn 0 521 59838 9 (Cambridge University Press 1998).

CLASSIFICATION THEORY FOR ELEMENTARY CLASSES WITH THE DEPENDENCE PROPERTY-A MODEST BEGINNING

2004-03-01
Saharon Shelah
Our thesis is that for the family of classes of the form EC(T),T a com- plete first order theory with the dependence property (which is just the negation of the … Our thesis is that for the family of classes of the form EC(T),T a com- plete first order theory with the dependence property (which is just the negation of the independence property) there is a substantial theory which means: a substantial body of basic results for all such classes and some complimentary results for the first order theories with the independence property, as for the family of stable (and the family of simple) first order theories. We examine some properties.

An approximate logic for measures

2011-06-15
Isaac Goldbring , Henry Towsner
We present a logical framework for formalizing connections between finitary combinatorics and measure theory or ergodic theory that have appeared various places throughout the literature. We develop the basic syntax … We present a logical framework for formalizing connections between finitary combinatorics and measure theory or ergodic theory that have appeared various places throughout the literature. We develop the basic syntax and semantics of this logic and give applications, showing that the method can express the classic Furstenberg correspondence and to give a short proof of the Szemeredi Regularity Lemma. We also derive some connections between the model-theoretic notion of stability and the Gowers uniformity norms from combinatorics.

Definably amenable NIP groups

2017-12-27
Artem Chernikov , Pierre Simon
We study definably amenable NIP groups. We develop a theory of generics showing that various definitions considered previously coincide, and we study invariant measures. As applications, we characterize ergodic measures, … We study definably amenable NIP groups. We develop a theory of generics showing that various definitions considered previously coincide, and we study invariant measures. As applications, we characterize ergodic measures, give a proof of the conjecture of Petrykowski connecting existence of bounded orbits with definable amenability in the NIP case, and prove the Ellis group conjecture of Newelski and Pillay connecting the model-theoretic connected component of an NIP group with the ideal subgroup of its Ellis enveloping semigroup.

Supersimple fields and division rings

1998-01-01
Anthony L. Pillay , Thomas Scanlon , Frank Olaf Wagner
It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types … It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types in groups and fields whose theory is simple.

Ample thoughts

2013-05-15
Daniel Palacín , Frank Olaf Wagner
Abstract Non- n -ampleness as denned by Pillay [20] and Evans [5] is preserved under analysability. Generalizing this to a more general notion of Σ-ampleness, this gives an immediate proof … Abstract Non- n -ampleness as denned by Pillay [20] and Evans [5] is preserved under analysability. Generalizing this to a more general notion of Σ-ampleness, this gives an immediate proof for all simple theories of a weakened version of the Canonical Base Property (CBP) proven by Chatzidakis [4] for types of finite SU-rank. This is then applied to the special case of groups.
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Zeta functions from definable equivalence relations

2006-12-31
Ehud Hrushovski , Benjamin Martin
We prove that the theory of the $p$-adics ${\mathbb Q}_p$ admits elimination of imaginaries provided we add a sort for ${\rm GL}_n({\mathbb Q}_p)/{\rm GL}_n({\mathbb Z}_p)$ for each $n$. We also … We prove that the theory of the $p$-adics ${\mathbb Q}_p$ admits elimination of imaginaries provided we add a sort for ${\rm GL}_n({\mathbb Q}_p)/{\rm GL}_n({\mathbb Z}_p)$ for each $n$. We also prove that the elimination of imaginaries is uniform in $p$. Using $p$-adic and motivic integration, we deduce the uniform rationality of certain formal zeta functions arising from definable equivalence relations. This also yields analogous results for definable equivalence relations over local fields of positive characteristic. The appendix contains an alternative proof, using cell decomposition, of the rationality (for fixed $p$) of these formal zeta functions that extends to the subanalytic context. As an application, we prove rationality and uniformity results for zeta functions obtained by counting twist isomorphism classes of irreducible representations of finitely generated nilpotent groups; these are analogous to similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald for subgroup zeta functions of finitely generated nilpotent groups.
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Pseudofinite difference fields and counting dimensions

2020-03-30
Tingxiang Zou
We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension … We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we also discuss the possible connection between coarse dimension and transformal transcendence degree in these difference fields.
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Geometric Stability Theory

1996-09-12
Anand Pillay
Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine … Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine structure of models of stable theories, using the geometry of forking; this often achieves global results relevant to classification theory. Topics range from Zilber-Cherlin classification of infinite locally finite homogenous geometries, to regular types, their geometries, and their role in superstable theories. The structure and existence of definable groups is featured prominently, as is work by Hrushovski. The book is unique in the range and depth of material covered and will be invaluable to anyone interested in modern model theory.

Studies in Logic and the Foundations of Mathematics

1990-01-01

PSEUDOFINITE <i>H</i>-STRUCTURES AND GROUPS DEFINABLE IN SUPERSIMPLE <i>H</i>-STRUCTURES

2019-04-02
Tingxiang Zou
Abstract In this article we explore some properties of H -structures which are introduced in [2]. We describe a construction of H -structures based on one-dimensional asymptotic classes which preserves … Abstract In this article we explore some properties of H -structures which are introduced in [2]. We describe a construction of H -structures based on one-dimensional asymptotic classes which preserves pseudofiniteness. That is, the H -structures we construct are ultraproducts of finite structures. We also prove that under the assumption that the base theory is supersimple of SU -rank one, there are no new definable groups in H -structures. This improves the corresponding result in [2].
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The Notre Dame Lectures

2017-03-29
Peter Cholak
Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been … Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In the fall of 2000, the logic community at the University of Notre Dame, Indiana hosted Greg Hjorth, Rodney G. Downey, Zoé Chatzidakis and Paola D'Aquino as visiting lecturers. Each of them presented a month-long series of expository lectures at the graduate level. This volume, the eighteenth publication in the Lecture Notes in Logic series, contains refined and expanded versions of those lectures. The four articles are entitled 'Countable models and the theory of Borel equivalence relations', 'Model theory of difference fields', 'Some computability-theoretic aspects of reals and randomness' and 'Weak fragments of Peano arithmetic'.

Definability in classes of finite structures

2011-03-10
Dugald Macpherson , Charles Steinhorn
This paper provides an overview of recent work by the authors and others on two topics in the model theory of finite structures. The point of view here differs from … This paper provides an overview of recent work by the authors and others on two topics in the model theory of finite structures. The point of view here differs from that usually associated with the term 'finite model theory', as presented for example in [21] or [46], in which the emphasis and motivation come primarily from computer science. Instead, the inspiration for this work has its origins in contemporary (infinite) model theoretic themes such as dimension, independence, and various measures of the complexity of definable sets. Each of the topics deals with classes of finite structures for first-order logic that are isolated by conditions that are drawn from these model-theoretic considerations. Moreover, in both cases, connections exist to areas in infinite model theory such as stability and simplicity theory, and o-minimality. This survey is intended for both mathematical logicians and computer scientists whose work focuses on logical aspects of the subject.

Finite Structures with Few Types

1993-01-01
Ehud Hrushovski

Model theory of difference fields

1999-04-08
Zoé Chatzidakis , Ehud Hrushovski
A difference field is a field with a distinguished automorphism<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"><mml:semantics><mml:mi>σ</mml:mi><mml:annotation encoding="application/x-tex">\sigma</mml:annotation></mml:semantics></mml:math></inline-formula>. This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory … A difference field is a field with a distinguished automorphism<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"><mml:semantics><mml:mi>σ</mml:mi><mml:annotation encoding="application/x-tex">\sigma</mml:annotation></mml:semantics></mml:math></inline-formula>. This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory on formulas, and in particular on difference equations. We show that an arbitrary formula may be reduced into one-dimensional ones, and analyze the possible internal structures on the one-dimensional formulas when the characteristic is<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"><mml:semantics><mml:mn>0</mml:mn><mml:annotation encoding="application/x-tex">0</mml:annotation></mml:semantics></mml:math></inline-formula>.