On NIP and invariant measures

Authors

Ehud Hrushovski, Anand Pillay

Type: Article

Publication Date: 2011-05-13

Citations: 177

DOI: https://doi.org/10.4171/jems/274

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Abstract

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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arXiv (Cornell University)
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Journal of the European Mathematical Society
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We study the model theory of covers of groups definable in o-minimal structures. This includes the case of covers of compact real Lie groups. In particular we study categoricity questions, … We study the model theory of covers of groups definable in o-minimal structures. This includes the case of covers of compact real Lie groups. In particular we study categoricity questions, pointing out some notable differences with the case of covers of complex algebraic groups studied by Zilber and his students. We also discuss from a model-theoretic point of view the following question, related to Milnor's conjecture: is a finite central extension (as an abstract group) of a compact Lie group also a topological extension?

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GROUP COVERS, o-MINIMALITY, AND CATEGORICITY

2010-12-01

Alessandro Berarducci , Ya’acov Peterzil , Anthony L. Pillay

We study the model theory of "covers" of groups H definable in an o-minimal structure M. We pose the question of whether any finite central extension G of H is … We study the model theory of "covers" of groups H definable in an o-minimal structure M. We pose the question of whether any finite central extension G of H is interpretable in M, proving some cases (such as when H is abelian) as well as stating various equivalences. When M is an o-minimal expansion of the reals (so H is a definable Lie group) this is related to Milnor's conjecture [15], and many cases are known. We also prove a strong relative Lω1, ω-categoricity theorem for universal covers of definable Lie groups, and point out some notable differences with the case of covers of complex algebraic groups (studied by Zilber and his students).

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G-linear sets and torsion points in definably compact groups

2009-04-15

Margarita Otero , Ya’acov Peterzil

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A note on fsg$\text{fsg}$ groups in p‐adically closed fields

2023-02-01

Will Johnson

Abstract Let G be a definable group in a p ‐adically closed field M . We show that G has finitely satisfiable generics () if and only if G is … Abstract Let G be a definable group in a p ‐adically closed field M . We show that G has finitely satisfiable generics () if and only if G is definably compact. The case was previously proved by Onshuus and Pillay.

Cohomology of algebraic varieties over non-archimedean fields

2022-01-01

Pablo Cubides Kovacsics , Mário J. Edmundo , Jinhe Ye

Abstract We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed nontrivially valued nonarchimedean field K based on Hrushovski-Loeser’s stable completion. In parallel, we develop a sheaf … Abstract We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed nontrivially valued nonarchimedean field K based on Hrushovski-Loeser’s stable completion. In parallel, we develop a sheaf cohomology of definable subsets in o-minimal expansions of the tropical semi-group $\Gamma _{\infty }$ , where $\Gamma $ denotes the value group of K . For quasi-projective varieties, both cohomologies are strongly related by a deformation retraction of the stable completion homeomorphic to a definable subset of $\Gamma _{\infty }$ . In both contexts, we show that the corresponding cohomology theory satisfies the Eilenberg-Steenrod axioms, finiteness and invariance, and we provide natural bounds of cohomological dimension in each case. As an application, we show that there are finitely many isomorphism types of cohomology groups in definable families. Moreover, due to the strong relation between the stable completion of an algebraic variety and its analytification in the sense of V. Berkovich, we recover and extend results on the singular cohomology of the analytification of algebraic varieties concerning finiteness and invariance.

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Topological dynamics and NIP fields

2021-06-02

Grzegorz Jagiella

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Around definable types in p-adically closed fields

2024-06-05

Pablo Andújar Guerrero , Will Johnson

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Density of compressible types and some consequences

2024-02-18

Martin Bays , Itay Kaplan , Pierre Simon

We study compressible types in the context of (local and global) NIP. By extending a result in machine learning theory (the existence of a bound on the recursive teaching dimension), … We study compressible types in the context of (local and global) NIP. By extending a result in machine learning theory (the existence of a bound on the recursive teaching dimension), we prove density of compressible types. Using this, we obtain explicit uniform honest definitions for NIP formulas (answering a question of Eshel and the second author), and build compressible models in countable NIP theories.

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Pseudo real closed fields, pseudo p-adically closed fields and NTP2

2016-10-01

Samaria Montenegro

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Consistent amalgamation for þ-forking

2013-07-23

Clifton Ealy , Alf Onshuus

On Stably Pointed Varieties and Generically Stable Groups in ACVF

2018-09-21

Yatir Halevi

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Connected components of definable groups, and o-minimality II

2015-05-04

Annalisa Conversano , Anand Pillay

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Non-forking and preservation of NIP and dp-rank

2021-02-05

Pedro Andrés Estevan , Itay Kaplan

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Remarks on generic stability in independent theories

2019-09-05

Gabriel Conant , Kyle Gannon

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Some model theory and topological dynamics of $p$-adic algebraic groups

2019-01-01

Davide Penazzi , Anand Pillay , Ningyuan Yao

We initiate the study of $p$-adic algebraic groups $G$ from the stability-theoretic and definable topological-dynamical points of view, that is, we consider invariants of the action of $G$ on its … We initiate the study of $p$-adic algebraic groups $G$ from the stability-theoretic and definable topological-dynamical points of view, that is, we consider invariants of the action of $G$ on its space of types over ${\mathbb Q}_p$ in the language of fiel

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Model theoretic connected components of groups

2011-07-30

Jakub Gismatullin

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A Vietoris-Smale mapping theorem for the homotopy of hyperdefinable sets

2017-06-07

Alessandro Achille , Alessandro Berarducci

Results of Smale (1957) and Dugundji (1969) allow to compare the homotopy groups of two topological spaces $X$ and $Y$ whenever a map $f:X\to Y$ with strong connectivity conditions on … Results of Smale (1957) and Dugundji (1969) allow to compare the homotopy groups of two topological spaces $X$ and $Y$ whenever a map $f:X\to Y$ with strong connectivity conditions on the fibers is given. We apply similar techniques in o-minimal expansions of fields to compare the o-minimal homotopy of a definable set $X$ with the homotopy of some of its bounded hyperdefinable quotients $X/E$. Under suitable assumption, we show that $\pi_{n}(X)^{\rm def}\cong\pi_{n}(X/E)$ and $\dim(X)=\dim_{\mathbb R}(X/E)$. As a special case, given a definably compact group, we obtain a new proof of Pillay's group conjecture $\dim(G)=\dim_{\mathbb R}(G/G^{00}$) largely independent of the group structure of $G$. We also obtain different proofs of various comparison results between classical and o-minimal homotopy.

Structure and regularity for subsets of groups with finite VC-dimension

2021-06-14

Gabriel Conant , Anand Pillay , C. Terry

Suppose G is a finite group and A\subseteq G is such that \{gA:g\in G\} has VC-dimension strictly less than k . We find algebraically well-structured sets in G which, up … Suppose G is a finite group and A\subseteq G is such that \{gA:g\in G\} has VC-dimension strictly less than k . We find algebraically well-structured sets in G which, up to a chosen \epsilon>0 , describe the structure of A and behave regularly with respect to translates of A . For the subclass of groups with uniformly fixed finite exponent r , these algebraic objects are normal subgroups with index bounded in terms of k , r , and \epsilon . For arbitrary groups, we use Bohr neighborhoods of bounded rank and width inside normal subgroups of bounded index. Our proofs are largely model-theoretic, and heavily rely on a structural analysis of compactifications of pseudofinite groups as inverse limits of Lie groups. The introduction of Bohr neighborhoods into the nonabelian setting uses model-theoretic methods related to the work of Breuillard, Green, and Tao [8] and Hrushovski [28] on approximate groups, as well as a result of Alekseev, Glebskiĭ, and Gordon [1] on approximate homomorphisms.

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On minimal flows, definably amenable groups, and o-minimality

2015-12-30

Anand Pillay , Ningyuan Yao

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On ω -categorical, generically stable groups and rings

2013-01-29

Jan Dobrowolski , Krzysztof Krupiński

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On minimal flows and definable amenability in some distal NIP theories

2023-04-19

Ningyuan Yao , Zhentao Zhang

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Definable groups in models of Presburger Arithmetic

2020-03-09

Alf Onshuus , Mariana Vicaría

Minimal stable types in Banach spaces

2019-08-19

Saharon Shelah , Alexander Usvyatsov

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Forking and dividing in fields with several orderings and valuations

2021-01-29

Will Johnson

We consider existentially closed fields with several orderings, valuations, and [Formula: see text]-valuations. We show that these structures are NTP 2 of finite burden, but usually have the independence property. … We consider existentially closed fields with several orderings, valuations, and [Formula: see text]-valuations. We show that these structures are NTP 2 of finite burden, but usually have the independence property. Moreover, forking agrees with dividing, and forking can be characterized in terms of forking in ACVF, RCF, and [Formula: see text]CF.

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Amenability, definable groups, and automorphism groups

2019-01-29

Krzysztof Krupiński , Anand Pillay

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References (39)

Classification theory for theories with NIP - a modest beginning

2000-01-01

Saharon Shelah

A relevant thesis is that for the family of complete first order theories with NIP (i.e. without the independence property) there is a substantial theory, like the family of stable … A relevant thesis is that for the family of complete first order theories with NIP (i.e. without the independence property) there is a substantial theory, like the family of stable (and the family of simple) first order theories. We examine some properties.

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Definably simple groups in o-minimal structures

2000-02-24

Ya’acov Peterzil , Anthony L. Pillay , Sergei Starchenko

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G equals mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G equals mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}=\langle G, \cdot \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a group definable in an o-minimal structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is definable in the structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\langle G,\cdot \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (while <italic>definable</italic> means definable in the structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). Assume <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable proper subgroup of finite index. In this paper we prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no nontrivial abelian normal subgroup, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the direct product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable subgroups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 1 comma ellipsis comma upper H Subscript k Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">H_1,\ldots ,H_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript i"> <mml:semantics> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">H_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin’s conjecture.

Simple algebraic and semialgebraic groups over real closed fields

2000-06-13

Ya’acov Peterzil , Anand Pillay , Sergei Starchenko

We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In<italic>Definably simple groups in o-minimal structures</italic>, we showed that every such group is a semialgebraic … We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In<italic>Definably simple groups in o-minimal structures</italic>, we showed that every such group is a semialgebraic group over a real closed field. Our main result here, stated in a model theoretic language, is that every such group is either bi-interpretable with an algebraically closed field of characteristic zero (when the group is stable) or with a real closed field (when the group is unstable). It follows that every abstract isomorphism between two unstable groups as above is a composition of a semialgebraic map with a field isomorphism. We discuss connections to theorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.

Stable Domination and Independence in Algebraically Closed Valued Fields

2007-12-10

Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson

This book addresses a gap in the model-theoretic understanding of valued fields that had limited the interactions of model theory with geometry. It contains significant developments in both pure and … This book addresses a gap in the model-theoretic understanding of valued fields that had limited the interactions of model theory with geometry. It contains significant developments in both pure and applied model theory. Part I of the book is a study of stably dominated types. These form a subset of the type space of a theory that behaves in many ways like the space of types in a stable theory. This part begins with an introduction to the key ideas of stability theory for stably dominated types. Part II continues with an outline of some classical results in the model theory of valued fields and explores the application of stable domination to algebraically closed valued fields. The research presented here is made accessible to the general model theorist by the inclusion of the introductory sections of each part.

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Generic sets in definably compact groups

2007-01-01

Ya’acov Peterzil , Anand Pillay

A subset $X$ of a group $G$ is called left generic if finitely many left translates of $X$ cover $G$. Our main result is that if $G$ is a definably … A subset $X$ of a group $G$ is called left generic if finitely many left translates of $X$ cover $G$. Our main result is that if $G$ is a definably compact group in an o-minimal structure and a definable $X\subseteq G$ is not right generic then i

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Definable Compactness and Definable Subgroups of o-Minimal Groups

1999-06-01

Ya’acov Peterzil , Charles Steinhorn

The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an … The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an o-minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the study of groups and rings in o-minimal structures. The main result proved is that any infinite definable group in an o-minimal structure that is not definably compact contains a definable torsion-free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings without zero divisors that are definable in o-minimal structures. The paper concludes with several examples illustrating some limitations on extending the theorem.

Strongly determined types and G-compactness

2006-01-01

А. А. Иванов

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On groups and fields definable in o-minimal structures

1988-09-01

Anand Pillay

Forking and independence in o-minimal theories

2004-03-01

Alfred Dolich

In the following we try to answer a simple question, “what does forking look like in an o-minimal theory”, or more generally, “what kinds of notions of independence with what … In the following we try to answer a simple question, “what does forking look like in an o-minimal theory”, or more generally, “what kinds of notions of independence with what kinds of properties are admissible in an o-minimal theory?” The motivation of these question begin with the study of simple theories and generalizations of simple theories. In [3] Kim and Pillay prove that the class of simple theories may be described exactly as those theories bearing a notion of independence satisfying various axioms. Thus it is natural to ask, if we weaken the assumptions as to which axioms must hold, what kind of theories do we get? Another source of motivation, also stemming from the study of simple theories, comes from the work of Shelah in [8] and [7]. Here Shelah addresses a “classification” type problem for class of models of a theory, showing that a theory will have the appropriate “structure” type property if one can construct a partially ordered set, satisfying various properties, of models of the theory. Using this criterion Shelah shows that the class of simple theories has this “structure” property, yet also that several non-simple examples do as well (though it should be pointed out that o-minimal theories can not be among these since any theory with the strict order property will have the corresponding “non-structure” property [8]). Thus one is lead to ask, what are the non-simple theories meeting this criterion, and one is once again led to study the types of independence relation a theory might bear. Finally, Shelah in [6] provides some possible definitions of what axioms for a notion of independence one should possibly look for in order to hope that theories bearing such a notion of independence should be amenable closer analysis. In studying all of the above mentioned situations it readily becomes clear that dividing and forking play a central role in all of them, even though we are no longer dealing with the simple case where we know that dividing and forking are very well behaved. All of these considerations lead one to look for classes of non-simple theories of which something is known where one can construct interesting notions of independence and consequently also say something about the nature of forking and dividing in these contexts. Given this one is naturally lead to one of the most well behaved classes of non-simple theories, namely the o-minimal theories.

Hyperimaginaries and automorphism groups

2001-03-01

Daniel Lascar , Anthony L. Pillay

A hyperimaginary is an equivalence class of a type-definable equivalence relation on tuples of possibly infinite length. The notion was recently introduced in [1], mainly with reference to simple theories. … A hyperimaginary is an equivalence class of a type-definable equivalence relation on tuples of possibly infinite length. The notion was recently introduced in [1], mainly with reference to simple theories. It was pointed out there how hyperimaginaries still remain in a sense within the domain of first order logic. In this paper we are concerned with several issues: on the one hand, various levels of complexity of hyperimaginaries, and when hyperimaginaries can be reduced to simpler hyperimaginaries. On the other hand the issue of what information about hyperimaginaries in a saturated structure M can be obtained from the abstract group Aut(M) . In Section 2 we show that if T is simple and canonical bases of Lascar strong types exist in M eq then hyperimaginaries can be eliminated in favour of sequences of ordinary imaginaries. In Section 3, given a type-definable equivalence relation with a bounded number of classes, we show how the quotient space can be equipped with a certain compact topology. In Section 4 we study a certain group introduced in [5], which we call the Galois group of T , develop a Galois theory and make the connection with the ideas in Section 3. We also give some applications, making use of the structure of compact groups. One of these applications states roughly that bounded hyperimaginaries can be eliminated in favour of sequences of finitary hyperimaginaries. In Sections 3 and 4 there is some overlap with parts of Hrushovski's paper [2].

Model theoretic connected components of groups

2011-07-30

Jakub Gismatullin

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A trichotomy theorem for o-minimal structures

1998-11-01

Ya’acov Peterzil , Sergei Starchenko

Let M = 〈M, <, …〉 be alinearly ordered structure. We define M to be o-minimal if every definable subset of M is a finite union of intervals. Classical examples … Let M = 〈M, <, …〉 be alinearly ordered structure. We define M to be o-minimal if every definable subset of M is a finite union of intervals. Classical examples are ordered divisible abelian groups and real closed fields. We prove a trichotomy theorem for the structure that an arbitraryo-minimal M can induce on a neighbourhood of any a in M. Roughly said, one of the following holds: (i) a is trivial (technical term), or (ii) a has a convex neighbourhood on which M induces the structure of an ordered vector space, or (iii) a is contained in an open interval on which M induces the structure of an expansion of a real closed field. The proof uses 'geometric calculus' which allows one to recover a differentiable structure by purely geometric methods. 1991 Mathematics Subject Classification: primary 03C45; secondary 03C52, 12J15, 14P10.

Strongly determined types

1999-08-01

А. А. Иванов , Dugald Macpherson

On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities

2015-01-01

Vladimir Vapnik , Alexey Chervonenkis

Measures and forking

1987-01-01

H. Jerome Keisler

Dependent first order theories, continued

2009-09-01

Saharon Shelah

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O-minimal spectra, infinitesimal subgroups and cohomology

2007-12-01

Alessandro Berarducci

Abstract By recent work on some conjectures of Pillay, each definably compact group G in a saturated o-minimal expansion of an ordered field has a normal “infinitesimal subgroup” G 00 … Abstract By recent work on some conjectures of Pillay, each definably compact group G in a saturated o-minimal expansion of an ordered field has a normal “infinitesimal subgroup” G 00 such that the quotient G/G 00 , equipped with the “logic topology”, is a compact (real) Lie group. Our first result is that the functor G ↦ G/G 00 sends exact sequences of definably compact groups into exact sequences of Lie groups. We then study the connections between the Lie group G/G 00 and the o-minimal spectrum of G . We prove that G/G 00 is a topological quotient of . We thus obtain a natural homomorphism Ψ* from the cohomology of G/G 00 to the (Čech-)cohomology of . We show that if G 00 satisfies a suitable contractibility conjecture then is acyclic in Čech cohomology and Ψ is an isomorphism. Finally we prove the conjecture in some special cases.

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Definable sets in algebraically closed valued fields: elimination of imaginaries

2006-01-01

Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson

It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain … It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain cosets in Kn of certain definable R-submodules of Kn (for all ). The proof involves the development of a theory of independence for unary types, which play the role of 1-types, followed by an analysis of germs of definable functions from unary sets to the sorts.

Minimal bounded index subgroup for dependent theories

2007-11-09

Saharon Shelah

For a dependent theory <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper C Subscript upper T"> <mml:semantics> <mml:msub> … For a dependent theory <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper C Subscript upper T"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>T</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">{\mathfrak {C}}_{T}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for every type definable 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>, the intersection of type definable subgroups with bounded index is a type definable subgroup with bounded index.

Definable groups and compact p -adic Lie groups

2008-05-19

Alf Onshuus , Anthony L. Pillay

We formulate p-adic analogues of the o-minimal group conjectures from the works of Hrushovski, Peterzil and Pillay [J. Amer. Math. Soc., to appear] and Pillay [J. Math. Log. 4 (2004) … We formulate p-adic analogues of the o-minimal group conjectures from the works of Hrushovski, Peterzil and Pillay [J. Amer. Math. Soc., to appear] and Pillay [J. Math. Log. 4 (2004) 147–162]; that is, we formulate versions that are appropriate for groups G definable in (saturated) P-minimal fields. We then restrict our attention to saturated models K of Th(ℚp) and Th(ℚp, an), record some elementary observations when G is defined over the standard model ℚp, and then make a detailed analysis of the case where G = E(K) for E an elliptic curve over K. Essentially, our P-minimal conjectures hold in these contexts and, moreover, our case study of elliptic curves yields counterexamples to a more naive direct translation of the o-minimal conjectures.

Introduction to theories without the independence property

2008-01-01

Hans H. Adler

We present an updated exposition of the classical theory of complete first order theories without the independence property (also called NIP theories or dependent theories). We present an updated exposition of the classical theory of complete first order theories without the independence property (also called NIP theories or dependent theories).

G-linear sets and torsion points in definably compact groups

2009-04-15

Margarita Otero , Ya’acov Peterzil

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Integration in valued fields

2007-12-31

Ehud Hrushovski , David Kazhdan

We develop a theory of integration over valued fields of residue characteristic zero. In particular, we obtain new and base-field independent foundations for integration over local fields of large residue … We develop a theory of integration over valued fields of residue characteristic zero. In particular, we obtain new and base-field independent foundations for integration over local fields of large residue characteristic, extending results of Denef, Loeser, and Cluckers. The method depends on an analysis of definable sets up to definable bijections. We obtain a precise description of the Grothendieck semigroup of such sets in terms of related groups over the residue field and value group. This yields new invariants of all definable bijections, as well as invariants of measure-preserving bijections.

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On central extensions and definably compact groups in o-minimal structures

2008-01-01

Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay

We prove several structural results on definably compact groups G in o-minimal expansions of real closed fields, such as (i) G is definably an almost direct product of a semisimple … We prove several structural results on definably compact groups G in o-minimal expansions of real closed fields, such as (i) G is definably an almost direct product of a semisimple group and a commutative group, and (ii) the group (G, .) is elementarily equivalent to (G/G^00, .). We also prove results on the internality of finite covers of G in an o-minimal environment, as well as the full compact domination conjecture. These results depend on key theorems about the interpretability of central and finite extensions of definable groups, in the o-minimal context. These methods and others also yield interpretability results for universal covers of arbitrary definable real Lie groups, from which we can deduce the semialgebraicity of finite covers of Lie groups such as SL(2,R).

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Groups, measures, and the NIP

2007-02-02

Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay

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Paires de structures O-minimales

1998-06-01

Yerzhan Baisalov , Bruno Poizat

Resumo Ni montras propecon de el j eteco de la kvantoro (∃y ∈ M ) pri la (sufi c e) belaj paroj de modeloj de una O -plimalpova teorio. G … Resumo Ni montras propecon de el j eteco de la kvantoro (∃y ∈ M ) pri la (sufi c e) belaj paroj de modeloj de una O -plimalpova teorio. G i havas korolaron ke, se ni aldonas malkavajn unarajn predikatojn a la lingvo de kelka O -plimalpova strukturo, ni ricevas malforte O -plimalpovan strukturon. Tui c i rezultato estis en speciala kaso pruvita de [5], kaj la g ia g eneralize c o estis anoncita en [1].

Embedded o-minimal structures

2012-02-20

Assaf Hasson , Alf Onshuus

(Bull. London Math. Soc. 42 (2010) 64–74) There is a serious mistake in the proof of Theorem 1 in the above mentioned paper. Consequently, we must withdraw the claim of … (Bull. London Math. Soc. 42 (2010) 64–74) There is a serious mistake in the proof of Theorem 1 in the above mentioned paper. Consequently, we must withdraw the claim of having proved that theorem.

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DEFINABLY COMPACT ABELIAN GROUPS

2004-12-01

Mário J. Edmundo , Margarita Otero

Let M be an o-minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n-dimensional group definable in M. We show the following: the … Let M be an o-minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n-dimensional group definable in M. We show the following: the o-minimal fundamental group of G is isomorphic to ℤ n ; for each k>0, the k-torsion subgroup of G is isomorphic to (ℤ/kℤ) n , and the o-minimal cohomology algebra over ℚ of G is isomorphic to the exterior algebra over ℚ with n generators of degree one.

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TYPE-DEFINABILITY, COMPACT LIE GROUPS, AND o-MINIMALITY

2004-12-01

Anand Pillay

We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient … We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient structure is o-minimal. The gist is that in this o-minimal case, any definable group G should have a smallest type-definable subgroup of bounded index, and that the quotient, when equipped with the logic topology, should be a compact Lie group of the "right" dimension. I give positive answers to the conjectures in the special cases when G is 1-dimensional, and when G is definably simple.

A descending chain condition for groups definable in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>o</mml:mi></mml:math>-minimal structures

2005-03-22

Alessandro Berarducci , Margarita Otero , Ya’acov Peterzil +1 more

Weak generic types and coverings of groups I

2006-01-01

Ludomir Newelski , Marcin Petrykowski

We introduce the notion of a weak generic type in a group. We improve our earlier results on countable coverings of groups and types. We introduce the notion of a weak generic type in a group. We improve our earlier results on countable coverings of groups and types.

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Definable groups and 2-dependent theories

2007-03-01

Saharon Shelah

Let T be a (first order complete) dependent theory, C a kappa-saturated model of T and G a definable subgroup which is abelian. Among subgroups of bounded index which are … Let T be a (first order complete) dependent theory, C a kappa-saturated model of T and G a definable subgroup which is abelian. Among subgroups of bounded index which are the union of < kappa type definable subsets there is a minimal one, i.e. their intersection has bounded index. In fact, the bound is <= 2^{|T|} . We then deal with 2-dependent theories, a wider class of first order theories.

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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.

Uniform Laws of Large Numbers

1996-01-01

Luc Devroye , László Györfi , Gábor Lugosi

Valued fields, metastable groups

2019-07-23

Ehud Hrushovski , Silvain Rideau

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On central extensions and definably compact groups in o-minimal structures

2010-11-17

Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay

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G-Compactness and Groups II

2007-07-02

Jakub Gismatullin

We give a general exposition of model theoretic connected components of groups. We show that if a group G has NIP, then there exists the smallest invariant (over some small … We give a general exposition of model theoretic connected components of groups. We show that if a group G has NIP, then there exists the smallest invariant (over some small set) subgroup of G with bounded index (Theorem 5.3). This result extends theorem of Shelah. We consider also in this context the multiplicative and the additive groups of some rings (including infite 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.

Embedded o-minimal structures

2009-12-25

Assaf Hasson , Alf Onshuus

We prove the following two theorems on embedded o-minimal structures: Theorem 1. Let ℳ ≺ 풩 be o-minimal structures and let ℳ* be the expansion of ℳ by all traces … We prove the following two theorems on embedded o-minimal structures: Theorem 1. Let ℳ ≺ 풩 be o-minimal structures and let ℳ* be the expansion of ℳ by all traces in M of 1-variable formulas in 풩, that is all sets of the form φ(M, ā) for ā ⊆ N and φ(x, ȳ) ∈ ℒ(풩). Then, for any N-formula ψ(x1, …, xk), the set ψ(Mk) is ℳ*-definable. Theorem 2. Let 풩 be an ω1-saturated structure and let S be a sort in 풩eq. Let 풮 be the 풩-induced structure on S and assume that 풮 is o-minimal. Then 풮 is stably embedded.