Stable types in rosy theories

Authors

Assaf Hasson, Alf Onshuus

Type: Article

Publication Date: 2010-10-04

Citations: 12

DOI: https://doi.org/10.2178/jsl/1286198144

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Abstract

Abstract We study the behaviour of stable types in rosy theories. The main technical result is that a non-þ-forking extension of an unstable type is unstable. We apply this to show that a rosy group with a þ-generic stable type is stable. In the context of super-rosy theories of finite rank we conclude that non-trivial stable types of U þ -rank 1 must arise from definable stable sets.

Locations

Journal of Symbolic Logic
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Forking types and rank functions in stable theories

1982-03-01

T. G. Mustafin , T. A. Nurmagambetov

Geometry of forking in simple theories

2006-03-01

Assaf Peretz

Abstract We investigate the geometry of forking for SU-rank 2 elements in supersimple ω-categorical theories and prove stable forking and some structural properties for such elements. We extend this analysis … Abstract We investigate the geometry of forking for SU-rank 2 elements in supersimple ω-categorical theories and prove stable forking and some structural properties for such elements. We extend this analysis to the case of SU-rank 3 elements.

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Geometry of forking in simple theories

2004-01-01

Assaf Peretz

We investigate the geometry of forking for U-rank 2 elements in supersimple w-categorical theories and prove stable forking and some structural properties for such elements. We extend this analysis to … We investigate the geometry of forking for U-rank 2 elements in supersimple w-categorical theories and prove stable forking and some structural properties for such elements. We extend this analysis to the case of U-rank 3 elements.

On Lascar Rank in Non-Multidimensional ω-Stable Theories

1987-01-01

Andreas Baudisch

ω-Stable Theories

2009-10-01

ω-Stable Theories

2006-03-31

Stable theories

1969-09-01

Saharon Shelah

Characterizing rosy theories

2007-09-01

Clifton Ealy , Alf Onshuus

Abstract We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences … Abstract We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences of each of these conditions towards the rosiness of the theory. In particular we show that the existence of an ordinal valued equivalence relation rank is a (necessary and) sufficient condition for rosiness.

O-Stable Theories

2011-07-01

B. S. Baizhanov , V. V. Verbovskiĭ

Rank functions in stable theories

1981-01-01

T. G. Mustafin

Unstable structures definable in o-minimal theories

2010-03-09

Assaf Hasson , Alf Onshuus

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Stable theories without dense forking chains

1992-09-01

Bernhard Herwig , James Loveys , Anand Pillay +2 more

Minimal types in super-dependent theories

2007-01-01

Assaf Hasson , Alf Onshuus

We give necessary and sufficient geometric conditions for a theory definable in an o-minimal structure to interpret a real closed field. The proof goes through an analysis of thorn-minimal types … We give necessary and sufficient geometric conditions for a theory definable in an o-minimal structure to interpret a real closed field. The proof goes through an analysis of thorn-minimal types in super-rosy dependent theories of finite rank. We prove that such theories are coordinatised by thorn-minimal types and that such a type is unstable if an only if every non-algebraic extension thereof is. We conclude that a type is stable if and only if it admits a coordinatisation in thorn-minimal stable types. We also show that non-trivial thorn-minimal stable types extend stable sets.

On the n-uniqueness of types in rosy theories

2016-04-23

Byunghan Kim

Homology groups of types in stable theories and the Hurewicz correspondence

2014-12-12

John Goodrick , Byunghan Kim , Alexei Kolesnikov

We give an explicit description of the homomorphism group H_n(p) of a strong type p in any stable theory under the assumption that for every non-forking extension q of p … We give an explicit description of the homomorphism group H_n(p) of a strong type p in any stable theory under the assumption that for every non-forking extension q of p the groups H_i(q) are trivial for i at least 2 but less than n. The group H_n(p) turns out to be isomorphic to the automorphism group of a certain piece of the algebraic closure of n independent realizations of p; it was shown earlier by the authors that such a group must be abelian. We call this the correspondence in analogy with the Hurewicz Theorem in algebraic topology.

Invariants under stable equivalences of morita type

2012-03-01

Fang Li , Longgang Sun

Rank axioms and supersimplicity

2022-01-01

Santiago Cárdenas-Martín , Rafel Farré

Just as Lascar's notion of abstract rank axiomatizes the U-rank, we propose axioms for the ranks $SU^d$ and $SU^f$, the foundation ranks of dividing and forking. We study the relationships … Just as Lascar's notion of abstract rank axiomatizes the U-rank, we propose axioms for the ranks $SU^d$ and $SU^f$, the foundation ranks of dividing and forking. We study the relationships between these axioms. As with superstable, we characterize supersimple types and theories based on the existence of these ranks. We show that the U-rank is the foundation rank of the Lascar-splitting independence relationship. We also provide an alternative definition of SUd similar to the original definition of U. Finally, we check that if in the standard characterizations of simple and supersimple we change the non-forking independence for the non-lascar-splitting independence, we characterize stable and superstable.

Omega-stable groups

2009-03-13

Daniel Lascar

Tame Expansions of $\omega$-Stable Theories and Definable Groups

2017-10-22

Haydar Göral

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Stable Groups

1997-08-21

Frank Olaf Wagner

The study of stable groups connects model theory, algebraic geometry and group theory. It analyses groups which possess a certain very general dependence relation (Shelah's notion of 'forking'), and tries … The study of stable groups connects model theory, algebraic geometry and group theory. It analyses groups which possess a certain very general dependence relation (Shelah's notion of 'forking'), and tries to derive structural properties from this. These may be group-theoretic (nilpotency or solubility of a given group), algebro-geometric (identification of a group as an algebraic group), or model-theoretic (description of the definable sets). In this book, the general theory of stable groups is developed from the beginning (including a chapter on preliminaries in group theory and model theory), concentrating on the model- and group-theoretic aspects. It brings together the various extensions of the original finite rank theory under a unified perspective and provides a coherent exposition of the knowledge in the field.

Generic stability and stability

2012-10-22

Hans H. Adler , Enrique Casanovas , Anand Pillay

We prove two results about generically stable types $p$ in arbitrary theories. The first, on existence of strong germs, generalizes results from D. Haskell, E. Hrushovski and D. Macpherson on … We prove two results about generically stable types $p$ in arbitrary theories. The first, on existence of strong germs, generalizes results from D. Haskell, E. Hrushovski and D. Macpherson on stably dominated types. The second is an equivalence of forking and dividing, assuming generic stability of $p^{(m)}$ for all $m$. We use the latter result to answer in full generality a question posed by Hasson and Onshuus: If $p(x)\in S(B)$ is stable and does not fork over $A$ then $p\restriction A$ is stable. (They had solved some special cases.)

Definable structures in o-minimal theories: One dimensional types

2010-10-31

Assaf Hasson , Alf Onshuus , Ya’acov Peterzil

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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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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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Thorn orthogonality and domination in unstable theories

2011-01-01

Alf Onshuus , Alexander Usvyatsov

We study orthogonality, domination, weight, regular and minimal types in the contexts of rosy and super-rosy theories. We study orthogonality, domination, weight, regular and minimal types in the contexts of rosy and super-rosy theories.

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GENERIC STABILITY AND STABILITY

2014-03-01

Hans H. Adler , Enrique Casanovas , Anand Pillay

Abstract We prove two results about generically stable types p in arbitrary theories. The first, on existence of strong germs, generalizes results from [2] on stably dominated types. The second … Abstract We prove two results about generically stable types p in arbitrary theories. The first, on existence of strong germs, generalizes results from [2] on stably dominated types. The second is an equivalence of forking and dividing, assuming generic stability of p ( m ) for all m . We use the latter result to answer in full generality a question posed by Hasson and Onshuus: If P ( x ) ε S ( B ) is stable and does not fork over A then prestrictionA is stable. (They had solved some special cases.)

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Type decomposition in NIP theories

2016-05-02

Pierre Simon

We prove that any type in an NIP theory can be decomposed into a stable part (a generically stable partial type) and a distal-like quotient. We prove that any type in an NIP theory can be decomposed into a stable part (a generically stable partial type) and a distal-like quotient.

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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.

Non-forking and preservation of NIP and dp-rank

2019-09-10

Pedro Andrés Estevan , Itay Kaplan

We investigate the question of whether the restriction of a NIP type $p\in S(B)$ which does not fork over $A\subseteq B$ to $A$ is also NIP, and the analogous question … We investigate the question of whether the restriction of a NIP type $p\in S(B)$ which does not fork over $A\subseteq B$ to $A$ is also NIP, and the analogous question for dp-rank. We show that if $B$ contains a Morley sequence $I$ generated by $p$ over $A$, then $p\restriction AI$ is NIP and similarly preserves the dp-rank. This yields positive answers for generically stable NIP types and the analogous case of stable types. With similar techniques we also provide a new more direct proof for the latter. Moreover, we introduce a general construction of whose open cones are models of some and in particular an inp-minimal theory DTR of dense trees with random graphs on open cones, which exemplifies a negative answer to the question.

Unstable structures definable in o-minimal theories

2010-03-09

Assaf Hasson , Alf Onshuus

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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>.

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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Superrosy dependent groups having finitely satisfiable generics

2008-01-01

Clifton Ealy , Krzysztof Krupiński , Anand Pillay

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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.

Coordinatization in superstable theories. II

1988-01-01

Steven Buechler

In this paper we prove Theorem A. <italic>Suppose that</italic> <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> <italic>is superstable and</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper … In this paper we prove Theorem A. <italic>Suppose that</italic> <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> <italic>is superstable and</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U left-parenthesis a slash upper A right-parenthesis equals alpha plus 1"> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>α</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">U(a/A) = \alpha + 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <italic>for some</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. <italic>Then in</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Superscript eq"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>T</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>eq</mml:mtext> </mml:mrow> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{T^{{\text {eq}}}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <italic>there is a</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c element-of a c l left-parenthesis upper A a right-parenthesis minus a c l left-parenthesis upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>acl</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mi>a</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi class="MJX-variant" mathvariant="normal">∖</mml:mi> <mml:mi>acl</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">c \in \operatorname {acl} (Aa)\backslash \operatorname {acl} (A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <italic>such that one of the following holds</italic>. (i) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U left-parenthesis c slash upper A right-parenthesis equals 1"> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>c</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">U(c/A) = 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. (ii) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s t p left-parenthesis c slash upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>stp</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>c</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {stp} (c/A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <italic>has finite Morley rank. In fact, this strong type is semiminimal with respect to a strongly minimal set</italic>.

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Forking, normalization and canonical bases

1986-01-01

Anand Pillay

Measures and forking

1987-01-01

H. Jerome Keisler

A note on stable sets, groups, and theories with NIP

2007-05-18

Alf Onshuus , Ya’acov Peterzil

Abstract Let M be an arbitrary structure. Then we say that an M ‐formula φ ( x ) defines a stable set in M if every formula φ ( x … Abstract Let M be an arbitrary structure. Then we say that an M ‐formula φ ( x ) defines a stable set in M if every formula φ ( x ) ∧ α ( x , y ) is stable. We prove: If G is an M ‐definable group and every definable stable subset of G has U ‐rank at most n (the same n for all sets), then G has a maximal connected stable normal subgroup H such that G / H is purely unstable. The assumptions hold for example if M is interpretable in an o‐minimal structure. More generally, an M ‐definable set X is weakly stable if the M ‐induced structure on X is stable. We observe that, by results of Shelah, every weakly stable set in theories with NIP is stable. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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.

Characterizing rosy theories

2007-09-01

Clifton Ealy , Alf Onshuus

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.

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, 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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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.

On NIP and invariant measures

2007-01-01

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 math.LO/0607442. Among key results are: (i) if … We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of $NIP$ (not the independence property), continuing aspects of math.LO/0607442. Among key results are: (i) if $p = tp(b/A)$ 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 $bdd(A)$ (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 translation invariant Keisler measures on groups with finitely satisfiable generics (vi) A proof of the compact domination conjecture for definably compact commutative groups in $o$-minimal expansions of real closed fields.

Valued fields, metastable groups

2019-07-23

Ehud Hrushovski , Silvain Rideau

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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.

Orthogonality and domination in unstable theories

2009-01-01

Alf Onshuus , Alexander Usvyatsov

In the first part of the paper we study orthogonality, domination, weight, regular and minimal types in the contexts of rosy and super-rosy theories. Then we try to develop analogous … In the first part of the paper we study orthogonality, domination, weight, regular and minimal types in the contexts of rosy and super-rosy theories. Then we try to develop analogous theory for arbitrary dependent theories.