A GEOMETRIC INTRODUCTION TO FORKING AND THORN-FORKING

Authors

Hans H. Adler

Type: Article

Publication Date: 2009-06-01

Citations: 81

DOI: https://doi.org/10.1142/s0219061309000811

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Abstract

A ternary relation [Formula: see text] between subsets of the big model of a complete first-order theory T is called an independence relation if it satisfies a certain set of axioms. The primary example is forking in a simple theory, but o-minimal theories are also known to have an interesting independence relation. Our approach in this paper is to treat independence relations as mathematical objects worth studying. The main application is a better understanding of thorn-forking, which turns out to be closely related to modular pairs in the lattice of algebraically closed sets.

Locations

Journal of Mathematical Logic
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THORN-FORKING AS LOCAL FORKING

2009-06-01

Hans H. Adler

We introduce the notion of a preindependence relation between subsets of the big model of a complete first-order theory, an abstraction of the properties which numerous concrete notions such as … We introduce the notion of a preindependence relation between subsets of the big model of a complete first-order theory, an abstraction of the properties which numerous concrete notions such as forking, dividing, thorn-forking, thorn-dividing, splitting or finite satisfiability share in all complete theories. We examine the relation between four additional axioms (extension, local character, full existence and symmetry) that one expects of a good notion of independence. We show that thorn-forking can be described in terms of local forking if we localize the number k in Kim's notion of "dividing with respect to k" (using Ben-Yaacov's "k-inconsistency witnesses") rather than the forking formulas. It follows that every theory with an M-symmetric lattice of algebraically closed sets (in T eq ) is rosy, with a simple lattice theoretical interpretation of thorn-forking.

Explanation of Independence

2005-01-01

Hans H. Adler

An axiomatic treatment of `independence relations' (notions of independence) for complete first-order theories is presented, the principal examples being forking (due to Shelah) and thorn-forking (due to Onshuus). Thorn-forking is … An axiomatic treatment of `independence relations' (notions of independence) for complete first-order theories is presented, the principal examples being forking (due to Shelah) and thorn-forking (due to Onshuus). Thorn-forking is characterised in terms of modular pairs in the lattice of algebraically closed sets. Wherever possible, forking and thorn-forking are treated in a uniform way. They are dual in the sense that forking is the finest (most restrictive) and thorn-forking the coarsest independence relation worth examining. We finish by defining the kernel of a sequence of indiscernibles and studying its relation to canonical bases.

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.

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.

Axiomatic Theory of Independence Relations in Model Theory

2023-01-01

Christian d’Elbée

This course introduces the fruitful links between model theory and a combinatoric of sets given by independence relations. An independence relation on a set is a ternary relation between subsets. … This course introduces the fruitful links between model theory and a combinatoric of sets given by independence relations. An independence relation on a set is a ternary relation between subsets. Chapter 1 should be considered as an introductory chapter. It does not mention first-order theories or formulas. It introduces independence relations in a naive set theory framework. Its goal is to get the reader familiar with basic axioms of independence relations (which do not need an ambient theory to be stated) as well as introduce closure operators and pregeometries. Chapter 2 introduces the model-theoretic context. The two main examples (algebraically closed fields and the random graph) are described as well as independence relations in those examples. Chapter 3 gives the axioms of independence relations in a model-theoretic context. It introduces the general toolbox of the model-theorists (indiscernible sequences, Ramsey/Erdos-Rado and compactness) and the independence relations of heirs/coheirs with two main applications: Adler's theorem of symmetry (how symmetry emerges from a weaker set of axioms, which is rooted in the work of Kim and Pillay) and a criterion for NSOP4 using stationary independence relations in the style of Conant. Independence relations satisfying Adler's theorem of symmetry are here called 'Adler independence relations' or AIR. Chapter 4 treats forking and dividing. It is proved that dividing independence is always stronger than any AIR (even though it is not an AIR in general) a connection between the independence theorem and forking independence, which holds in all generality and is based on Kim-Pillay's approach. Then, simplicity is defined and the interesting direction of the Kim-Pillay theorem (namely that the existence of an Adler independence relation satisfying the independence theorem yields simplicity) is deduced from earlier results.

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Supersimple structures with a dense independent subset

2018-03-20

Alexander Berenstein , Juan Felipe Carmona , Evgueni Vassiliev

Based on the work done in \cite{BV-Tind,DMS} in the o-minimal and geometric settings, we study expansions of models of a supersimple theory with a new predicate distiguishing a set of … Based on the work done in \cite{BV-Tind,DMS} in the o-minimal and geometric settings, we study expansions of models of a supersimple theory with a new predicate distiguishing a set of forking-independent elements that is dense inside a partial type $\mathcal{G}(x)$, which we call $H$-structures. We show that any two such expansions have the same theory and that under some technical conditions, the saturated models of this common theory are again $H$-structures. We prove that under these assumptions the expansion is supersimple and characterize forking and canonical bases of types in the expansion. We also analyze the effect these expansions have on one-basedness and CM-triviality. In the one-based case, when $T$ has $SU$-rank $\omega^\alpha$ and the $SU$-rank is continuous, we take $\mathcal{G}(x)$ to be the type of elements of $SU$-rank $\omega^\alpha$ and we describe a natural geometry of generics modulo $H$ associated with such expansions and show it is modular.

Supersimple structures with a dense independent subset

2018-01-01

Alexander Berenstein , Juan Felipe Carmona , Evgueni Vassiliev

Based on the work done in \cite{BV-Tind,DMS} in the o-minimal and geometric settings, we study expansions of models of a supersimple theory with a new predicate distiguishing a set of … Based on the work done in \cite{BV-Tind,DMS} in the o-minimal and geometric settings, we study expansions of models of a supersimple theory with a new predicate distiguishing a set of forking-independent elements that is dense inside a partial type $\mathcal{G}(x)$, which we call $H$-structures. We show that any two such expansions have the same theory and that under some technical conditions, the saturated models of this common theory are again $H$-structures. We prove that under these assumptions the expansion is supersimple and characterize forking and canonical bases of types in the expansion. We also analyze the effect these expansions have on one-basedness and CM-triviality. In the one-based case, when $T$ has $SU$-rank $\omega^\alpha$ and the $SU$-rank is continuous, we take $\mathcal{G}(x)$ to be the type of elements of $SU$-rank $\omega^\alpha$ and we describe a natural "geometry of generics modulo $H$" associated with such expansions and show it is modular.

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THREE SURPRISING INSTANCES OF DIVIDING

2024-03-21

Gabriel Conant , Alex Kruckman

Abstract We give three counterexamples to the folklore claim that in an arbitrary theory, if a complete type p over a set B does not divide over $C\subseteq B$ , … Abstract We give three counterexamples to the folklore claim that in an arbitrary theory, if a complete type p over a set B does not divide over $C\subseteq B$ , then no extension of p to a complete type over $\operatorname {acl}(B)$ divides over C . Two of our examples are also the first known theories where all sets are extension bases for nonforking, but forking and dividing differ for complete types (answering a question of Adler). One example is an $\mathrm {NSOP}_1$ theory with a complete type that forks, but does not divide, over a model (answering a question of d’Elbée). Moreover, dividing independence fails to imply M-independence in this example (which refutes another folklore claim). In addition to these counterexamples, we summarize various related properties of dividing that are still true. We also address consequences for previous literature, including an earlier unpublished result about forking and dividing in free amalgamation theories, and some claims about dividing in the theory of generic $K_{m,n}$ -free incidence structures.

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Three surprising instances of dividing

2023-01-01

Gabriel Conant , Alex Kruckman

We give three counterexamples to the folklore claim that in an arbitrary theory, if a complete type $p$ over a set $B$ does not divide over $C\subseteq B$, then no … We give three counterexamples to the folklore claim that in an arbitrary theory, if a complete type $p$ over a set $B$ does not divide over $C\subseteq B$, then no extension of $p$ to a complete type over $\text{acl}(B)$ divides over $C$. Two of our examples are also the first known theories where all sets are extension bases for nonforking, but forking and dividing differ for complete types (answering a question of Adler). One example is an NSOP$_1$ theory with a complete type that forks, but does not divide, over a model (answering a question of d'Elbée). Moreover, dividing independence fails to imply M-independence in this example (which refutes another folklore claim). In addition to these counterexamples, we summarize various related properties of dividing that are still true. We also address consequences for previous literature, including an earlier unpublished result about forking and dividing in free amalgamation theories, and some claims about dividing in the theory of generic $K_{m,n}$-free incidence structures.

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Supersimple structures with a dense independent subset

2017-12-01

Alexander Berenstein , Juan Felipe Carmona , Evgueni Vassiliev

Abstract Based on the work done in [][] in the o‐minimal and geometric settings, we study expansions of models of a supersimple theory with a new predicate distiguishing a set … Abstract Based on the work done in [][] in the o‐minimal and geometric settings, we study expansions of models of a supersimple theory with a new predicate distiguishing a set of forking‐independent elements that is dense inside a partial type , which we call H ‐structures. We show that any two such expansions have the same theory and that under some technical conditions, the saturated models of this common theory are again H ‐structures. We prove that under these assumptions the expansion is supersimple and characterize forking and canonical bases of types in the expansion. We also analyze the effect these expansions have on one‐basedness and CM‐triviality. In the one‐based case, when T has SU‐rank and the SU‐rank is continuous, we take to be the type of elements of SU‐rank and we describe a natural “geometry of generics modulo H ” associated with such expansions and show it is modular.

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Properties and Consequences of th-Independence

2002-01-01

Alf Onshuus

We develop a new notion of independence suggested by Scanlon (th-independence). We prove that in a large class of theories (which includes all simple theories) this notion has many of … We develop a new notion of independence suggested by Scanlon (th-independence). We prove that in a large class of theories (which includes all simple theories) this notion has many of the properties needed for an adequate geometric structure in these models. We also prove that this definition agrees with the usual independence notions in stable, supersimple and o-minimal theories. Finally, many of the proofs and results we get for th-independence when restricted to simple theories seem to show there is some connection between th-independence and the stable forking conjecture. In particular, we prove that in any simple theory where this conjecture holds, our definition agrees with the classic definition.

Very simple theories without forking

2003-08-01

Ludomir Newelski

Forking

2000-01-01

Bruno Poizat

On geometry of quasi-minimal structures (New developments of independence notions in model theory)

2010-10-01

Hisatomo Maesono

On geometry of quasi-minimal structures (New developments of independence notions in model theory)

2010-10-01

久智前園

On Morley's Categoricity Theorem with an Eye Toward Forking

2011-01-01

Colin N. Craft

Lovely pairs for independence relations

2010-01-01

Antongiulio Fornasiero

In the literature there are two different notions of lovely pairs of a theory T, according to whether T is simple or geometric. We introduce a notion of lovely pairs … In the literature there are two different notions of lovely pairs of a theory T, according to whether T is simple or geometric. We introduce a notion of lovely pairs for an independence relation, which generalizes both the simple and the geometric case, and show how the main theorems for those two cases extend to our general notion.

Forking and fundamental order in simple theories

1999-09-01

Daniel Lascar , Anand Pillay

Abstract We give a characterisation of forking in the context of simple theories in terms of the fundamental order. Abstract We give a characterisation of forking in the context of simple theories in terms of the fundamental order.

th-Forking, Algebraic Independence and Examples of Rosy Theories

2003-01-01

Alf Onshuus

In a previous paper we developed the notions of th-independence and þ-ranks which define a geometric independence relation in a class of theories which we called ``rosy''. We proved that … In a previous paper we developed the notions of th-independence and þ-ranks which define a geometric independence relation in a class of theories which we called ``rosy''. We proved that rosy theories include simple and o-minimal theories and that for any theory for which the stable forking conjecture was true, þ-forking coincides with forking independence. In this article, we continue to study properties of th-forking and find more examples of rosy theories. Among the new properties we prove in this paper are some alternative characterizations of rosy theories and some tools to prove and analyze rosiness in particular cases. Finally, we use this to find two examples of rosy non simple theories: pseudo real closed fields (PRC-fields) and the uniform companion of a large differential field defined by Marcus Tressl.

A New Kim’s Lemma

2024-08-12

Alex Kruckman , Nicholas Ramsey

Kim's Lemma is a key ingredient in the theory of forking independence in simple theories.It asserts that if a formula divides, then it divides along every Morley sequence in type … Kim's Lemma is a key ingredient in the theory of forking independence in simple theories.It asserts that if a formula divides, then it divides along every Morley sequence in type of the parameters.Variants of Kim's Lemma have formed the core of the theories of independence in two orthogonal generalizations of simplicity -namely, the classes of NTP 2 and NSOP 1 theories.We introduce a new variant of Kim's Lemma that simultaneously generalizes the NTP 2 and NSOP 1 variants.We explore examples and nonexamples in which this lemma holds, discuss implications with syntactic properties of theories, and ask several questions. 1. Introduction 825 2. Preliminaries 828 3. A diversity of Kim's lemmas 838 4. Examples 844 5.

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þ-Forking and Stable Forking

2016-12-26

Clifton Ealy , Alf Onshuus

Usamos una contrucción particular de una relación de independencia para demostrar que en cualquier teoría þ-bifurcación es equivalente a bifurcación con una fórmula estable (en el sentido específico de st-bifurcación … Usamos una contrucción particular de una relación de independencia para demostrar que en cualquier teoría þ-bifurcación es equivalente a bifurcación con una fórmula estable (en el sentido específico de st-bifurcación dada en la Definición 1.3). También demostramos que si tenemos þ-división podemos lograr división fuerte sobre una base que pertenece a la clausura algebraica del conjunto parámetro.

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A note on some example of NSOP1 theories

2023-01-01

Yvon Bossut

We present here some known and some new examples of non-simple NSOP1 theories andsome behaviour that Kim-forking can exhibit in these theories, in particular that Kim-forking afterforcing base monotonicity can … We present here some known and some new examples of non-simple NSOP1 theories andsome behaviour that Kim-forking can exhibit in these theories, in particular that Kim-forking afterforcing base monotonicity can or can not satisfy extension (on arbitrary sets). This study is based onthe results of Chernikov, Ramsey, Dobrowolski and Granger.

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The Theory of Tracial Von Neumann Algebras Does Not Have A Model Companion

2013-09-01

Isaac Goldbring , Bradd Hart , Thomas Sinclair

Abstract In this note, we show that the theory of tracial von Neumann algebras does not have a model companion. This will follow from the fact that the theory of … Abstract In this note, we show that the theory of tracial von Neumann algebras does not have a model companion. This will follow from the fact that the theory of any locally universal, McDuff II 1 factor does not have quantifier elimination. We also show how a positive solution to the Connes Embedding Problem implies that there can be no model-complete theory of II 1 factors.

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Vector spaces with a dense-codense generic submodule

2024-03-20

Alexander Berenstein , Christian d’Elbée , Evgueni Vassiliev

We study expansions of a vector space V over a field F, possibly with extra structure, with a generic submodule over a subring of F. We construct a natural expansion … We study expansions of a vector space V over a field F, possibly with extra structure, with a generic submodule over a subring of F. We construct a natural expansion by existentially defined functions so that the expansion in the extended language satisfies quantifier elimination. We show that this expansion preserves tame model theoretic properties such as stability, NIP, NTP1, NTP2 and NSOP1. We also study induced independence relations in the expansion.

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Canonical forking in AECs

2016-04-11

Will Boney , Rami Grossberg , Alexei Kolesnikov +1 more

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Superrosy dependent groups having finitely satisfiable generics

2008-01-01

Clifton Ealy , Krzysztof Krupiński , Anand Pillay

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Expansions which introduce no new open sets

2012-01-20

Gareth Boxall , Philipp Hieronymi

Abstract We consider the question of when an expansion of a first-order topological structure has the property that every open set definable in the expansion is definable in the original … Abstract We consider the question of when an expansion of a first-order topological structure has the property that every open set definable in the expansion is definable in the original structure. This question has been investigated by Dolich, Miller and Steinhorn in the setting of ordered structures as part of their work on the property of having o-minimal open core. We answer the question in a fairly general setting and provide conditions which in practice are often easy to check. We give a further characterisation in the special case of an expansion by a generic predicate.

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Generic expansion and Skolemization in NSOP 1 theories

2018-05-29

Alex Kruckman , Nicholas Ramsey

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Neostability in countable homogeneous metric spaces

2017-04-25

Gabriel Conant

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Forking in short and tame abstract elementary classes

2017-03-02

Will Boney , Rami Grossberg

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Model theoretic properties of the Urysohn sphere

2015-10-20

Gabriel Conant , C. Terry

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Superrosy fields and valuations

2014-12-09

Krzysztof Krupiński

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A remark on strict independence relations

2016-03-14

Gabriel Conant

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Super/rosy<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msup></mml:math>-theories and classes of finite structures

2013-06-05

Cameron Donnay Hill

Using ultrapowers to compare continuous structures

2023-05-09

H. Jerome Keisler

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On the class of flat stable theories

2018-05-29

Daniel Palacín , Saharon Shelah

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On lovely pairs of geometric structures

2009-11-29

Alexander Berenstein , Evgueni Vassiliev

INDEPENDENCE IN GENERIC INCIDENCE STRUCTURES

2019-02-15

Gabriel Conant , Alex Kruckman

We study the theory $T_{m,n}$ of existentially closed incidence structures omitting the complete incidence structure $K_{m,n}$, which can also be viewed as existentially closed $K_{m,n}$-free bipartite graphs. In the case … We study the theory $T_{m,n}$ of existentially closed incidence structures omitting the complete incidence structure $K_{m,n}$, which can also be viewed as existentially closed $K_{m,n}$-free bipartite graphs. In the case $m = n = 2$, this is the theory of existentially closed projective planes. We give an $\forall\exists$-axiomatization of $T_{m,n}$, show that $T_{m,n}$ does not have a countable saturated model when $m,n\geq 2$, and show that the existence of a prime model for $T_{2,2}$ is equivalent to a longstanding open question about finite projective planes. Finally, we analyze model theoretic notions of complexity for $T_{m,n}$. We show that $T_{m,n}$ is NSOP$_1$, but not simple when $m,n\geq 2$, and we show that $T_{m,n}$ has weak elimination of imaginaries but not full elimination of imaginaries. These results rely on combinatorial characterizations of various notions of independence, including algebraic independence, Kim independence, and forking independence.

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On positive local combinatorial dividing-lines in model theory

2018-06-28

Vincent Guingona , Cameron Donnay Hill

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Independence in Model Theory

2016-01-01

Åsa Hirvonen

Strict independence

2014-06-23

Itay Kaplan , Alexander Usvyatsov

We investigate the notions of strict independence and strict non-forking, and establish basic properties and connections between the two. In particular, it follows from our investigation that in resilient theories … We investigate the notions of strict independence and strict non-forking, and establish basic properties and connections between the two. In particular, it follows from our investigation that in resilient theories strict non-forking is symmetric. Based on this study, we develop notions of weight which characterize NTP 2 , dependence and strong dependence. Many of our proofs rely on careful analysis of sequences that witness dividing. We prove simple characterizations of such sequences in resilient theories, as well as of Morley sequences which are witnesses. As a by-product we obtain information on types co-dominated by generically stable types in dependent theories. For example, we prove that every Morley sequence in such a type is a witness.

Stable embeddedness and <i>NIP</i>

2011-05-19

Anand Pillay

Abstract We give some sufficient conditions for a predicate P in a complete theory T to be “stably embedded”. Let be P with its “induced ∅-definable structure”. The conditions are … Abstract We give some sufficient conditions for a predicate P in a complete theory T to be “stably embedded”. Let be P with its “induced ∅-definable structure”. The conditions are that (or rather its theory) is “rosy”. P has NIP in T and that P is stably 1-embedded in T . This generalizes a recent result of Hasson and Onshuus [6] which deals with the case where P is o-minimal in T . Our proofs make use of the theory of strict nonforking and weight in NIP theories ([3], [10]).

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Superrosiness and dense pairs of geometric structures

2023-09-19

Gareth Boxall

Abstract Let T be a complete geometric theory and let $$T_P$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>P</mml:mi> </mml:msub> </mml:math> be the theory of dense pairs of models of T . We … Abstract Let T be a complete geometric theory and let $$T_P$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>P</mml:mi> </mml:msub> </mml:math> be the theory of dense pairs of models of T . We show that if T is superrosy with "Equation missing"-rank 1 then $$T_P$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>P</mml:mi> </mml:msub> </mml:math> is superrosy with "Equation missing"-rank at most $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> .

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Weak One-Basedness

2013-01-01

Gareth Boxall , David Bradley-Williams , Charlotte Kestner +2 more

We study the notion of weak one-basedness introduced in recent work of Berenstein and Vassiliev. Our main results are that this notion characterizes linearity in the setting of geometric þ-rank … We study the notion of weak one-basedness introduced in recent work of Berenstein and Vassiliev. Our main results are that this notion characterizes linearity in the setting of geometric þ-rank 1structures and that lovely pairs of weakly one-based geometric þ-rank 1 structures are weakly one-based with respect to þ-independence. We also study geometries arising from infinite-dimensional vector spaces over division rings.

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Superstability from categoricity in abstract elementary classes

2017-03-28

Will Boney , Rami Grossberg , Monica VanDieren +1 more

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Forking and Dividing in Henson Graphs

2017-01-01

Gabriel Conant

For n≥3, define Tn to be the theory of the generic Kn-free graph, where Kn is the complete graph on n vertices. We prove a graph-theoretic characterization of dividing in … For n≥3, define Tn to be the theory of the generic Kn-free graph, where Kn is the complete graph on n vertices. We prove a graph-theoretic characterization of dividing in Tn and use it to show that forking and dividing are the same for complete types. We then give an example of a forking and nondividing formula. Altogether, Tn provides a counterexample to a question of Chernikov and Kaplan.

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Model theory for real-valued structures

2023-05-09

H. Jerome Keisler

We consider general structures where formulas have truth values in the real unit interval as in continuous model theory, but whose predicates and functions need not be uniformly continuous with … We consider general structures where formulas have truth values in the real unit interval as in continuous model theory, but whose predicates and functions need not be uniformly continuous with respect to a distance predicate. Every general structure can be expanded to a pre-metric structure by adding a distance predicate that is a uniform limit of formulas. Moreover, that distance predicate is unique up to uniform equivalence. We use this to extend the central notions in the model theory of metric structures to general structures, and show that many model-theoretic results from the literature about metric structures have natural analogues for general structures.

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Three surprising instances of dividing

2023-01-01

Gabriel Conant , Alex Kruckman

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Weakly one-based geometric theories

2012-04-04

Alexander Berenstein , Evgueni Vassiliev

Abstract We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find … Abstract We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: “weak one-basedness”, absence of type definable “almost quasidesigns”, and “generic linearity”. Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based ω -categorical geometric theory interprets an infinite vector space over a finite field.

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Generic Expansions by a Reduct

2018-10-27

Christian d’Elbée

Consider the expansion $T_S$ of a theory $T$ by a predicate for a submodel of a reduct $T_0$ of $T$. We present a setup in which this expansion admits a … Consider the expansion $T_S$ of a theory $T$ by a predicate for a submodel of a reduct $T_0$ of $T$. We present a setup in which this expansion admits a model companion $TS$. We show that the nice features of the theory $T$ transfer to $TS$. In particular, we study conditions for which this expansion preserves the $\NSOP{1}$-ness, the simplicity or the stability of the starting theory $T$. We give concrete examples of new $\NSOP{1}$ not simple theories obtained by this process, among them the expansion of a perfect $\omega$-free $\PAC$ field of positive characteristic by generic additive subgroups, and the expansion of an algebraically closed field of \emph{any} characteristic by a generic multiplicative subgroup.

Forking, Imaginaries and other features of ACFG

2018-12-21

Christian d’Elbée

We study the generic theory of algebraically closed fields of fixed positive characteristic with a predicate for an additive subgroup, called $\mathrm{ACFG}$. This theory was introduced recently as a new … We study the generic theory of algebraically closed fields of fixed positive characteristic with a predicate for an additive subgroup, called $\mathrm{ACFG}$. This theory was introduced recently as a new example of $\mathrm{NSOP}_1$ non simple theory. In this paper we describe more features of $\mathrm{ACFG}$, such as imaginaries. We also study various independence relations in $\mathrm{ACFG}$, such as Kim-independence or forking independence, and describe interactions between them.

On dp-minimality, strong dependence and weight

2011-07-06

Alf Onshuus , Alexander Usvyatsov

Abstract We study dp-minimal and strongly dependent theories and investigate connections between these notions and weight. Abstract We study dp-minimal and strongly dependent theories and investigate connections between these notions and weight.

Dp-Minimality: Basic Facts and Examples

2011-07-01

Alfred Dolich , John Goodrick , David Lippel

We study the notion of dp-minimality, beginning by providing several essential facts about dp-minimality, establishing several equivalent definitions for dp-minimality, and comparing dp-minimality to other minimality notions. The majority of … We study the notion of dp-minimality, beginning by providing several essential facts about dp-minimality, establishing several equivalent definitions for dp-minimality, and comparing dp-minimality to other minimality notions. The majority of the rest of the paper is dedicated to examples. We establish via a simple proof that any weakly o-minimal theory is dp-minimal and then give an example of a weakly o-minimal group not obtained by adding traces of externally definable sets. Next we give an example of a divisible ordered Abelian group which is dp-minimal and not weakly o-minimal. Finally we establish that the field of p-adic numbers is dp-minimal.

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THORN-FORKING AS LOCAL FORKING

2009-06-01

Hans H. Adler

Forking in Short and Tame AECs

2013-06-27

Will Boney , Rami Grossberg

We develop a notion of forking for Galois-types in the context of Abstract Elementary Classes (AECs). Under the hypotheses that an AEC $K$ is tame, type-short, and failure of an … We develop a notion of forking for Galois-types in the context of Abstract Elementary Classes (AECs). Under the hypotheses that an AEC $K$ is tame, type-short, and failure of an order-property, we consider {\bf Definition.} Let $M_0 \prec N$ be models from $K$ and $A$ be a set. We say that the Galois-type of $A$ over $M$ \emph{does not fork over $M_0$} iff for all small $a \in A$ and all small $N^- \prec N$, we have that Galois-type of $a$ over $N^-$ is realized in $M_0$. Assuming property (E) (see Definition 3.3) we show that this non-forking is a well behaved notion of independence, in particular satisfies symmetry and uniqueness and has a corresponding U-rank. We find conditions for a universal local character, in particular derive superstability-like property from little more than categoricity in a \big cardinal. Finally, we show that under large cardinal axioms the proofs are simpler and the non-forking is more powerful. In [BGKV] it is established that this notion of non-forking is the only independence relation possible.

Forking in Short and Tame Abstract Elementary Classes

2013-06-27

Will Boney , Rami Grossberg

Advances in Classification Theory for Abstract Elementary Classes

2018-12-01

Will Boney

An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ … An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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THORN FORKING, WEAK NORMALITY, AND THEORIES WITH SELECTORS

2023-09-11

Daniel Max Hoffmann , Anand Pillay

Abstract We discuss the role of weakly normal formulas in the theory of thorn forking, as part of a commentary on the paper [5]. We also give a counterexample to … Abstract We discuss the role of weakly normal formulas in the theory of thorn forking, as part of a commentary on the paper [5]. We also give a counterexample to Corollary 4.2 from that paper, and in the process discuss “theories with selectors.”

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Generic expansions by a reduct

2021-01-06

Christian d’Elbée

Consider the expansion [Formula: see text] of a theory [Formula: see text] by a predicate for a submodel of a reduct [Formula: see text] of [Formula: see text]. We present … Consider the expansion [Formula: see text] of a theory [Formula: see text] by a predicate for a submodel of a reduct [Formula: see text] of [Formula: see text]. We present a setup in which this expansion admits a model companion [Formula: see text]. We show that some of the nice features of the theory [Formula: see text] transfer to [Formula: see text]. In particular, we study conditions for which this expansion preserves the [Formula: see text]-ness, the simplicity or the stability of the starting theory [Formula: see text]. We give concrete examples of new [Formula: see text] not simple theories obtained by this process, among them the expansion of a perfect [Formula: see text]-free [Formula: see text] field of positive characteristic by generic additive subgroups, and the expansion of an algebraically closed field of any characteristic by a generic multiplicative subgroup.

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Bounded ultraimaginary independence and its total Morley sequences

2024-04-30

James Hanson

We investigate the following model-theoretic independence relation: b | ⌣ bu A c if and only if bdd u (Ab) ∩ bdd u (Ac) = bdd u (A), where bdd … We investigate the following model-theoretic independence relation: b | ⌣ bu A c if and only if bdd u (Ab) ∩ bdd u (Ac) = bdd u (A), where bdd u (X ) is the class of all ultraimaginaries bounded over X .In particular, we sharpen a result of Wagner to show that b | ⌣ bu A c if and only if ⟨Autf(‫/ލ‬Ab) ∪ Autf(‫/ލ‬Ac)⟩ = Autf(‫/ލ‬A), and we establish full existence over hyperimaginary parameters (i.e., for any set of hyperimaginaries A and ultraimaginaries b and c, there is a b). Extension then follows as an immediate corollary.We also study total |

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Toward classifying unstable theories

1996-08-01

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.

Simplicity, and stability in there

2001-06-01

Byunghan Kim

Abstract Firstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover definability part of stability theory to simplicity theory. … Abstract Firstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover definability part of stability theory to simplicity theory. In particular, using elimination of hyperimaginaries we prove that for any supersimple T . canonical base of an amalgamation class is the union of names of ψ -definitions of , ψ ranging over stationary L -formulas in . Also, we prove that the same is true with stable formulas for an 1-based theory having elimination of hyperimaginaries. For such a theory, the stable forking property holds, too.

SIMPLICITY IN COMPACT ABSTRACT THEORIES

2003-11-01

Itay Ben-Yaacov

We continue [2], developing simplicity in the framework of compact abstract theories. Due to the generality of the context we need to introduce definitions which differ somewhat from the ones … We continue [2], developing simplicity in the framework of compact abstract theories. Due to the generality of the context we need to introduce definitions which differ somewhat from the ones use in first order theories. With these modified tools we obtain more or less classical behaviour: simplicity is characterized by the existence of a certain notion of independence, stability is characterized by simplicity and bounded multiplicity, and hyperimaginary canonical bases exist.

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Properties and consequences of Thorn-independence

2006-03-01

Alf Onshuus

Lattice of algebraically closed sets in one-based theories

1994-03-01

Lee Fong Low

Abstract Let T be a one-based theory. We define a notion of width, in the case of T having the finiteness property, for the lattice of finitely generated algebraically closed … Abstract Let T be a one-based theory. We define a notion of width, in the case of T having the finiteness property, for the lattice of finitely generated algebraically closed sets and prove Theorem. Let T be one-based with the finiteness property. If T is of bounded width, then every type in T is nonorthogonal to a weight one type. If T is countable, the converse is true .