An introduction to forking

Authors

Daniel Lascar, Bruno Poizat

Type: Article

Publication Date: 1979-09-01

Citations: 96

DOI: https://doi.org/10.2307/2273127

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Abstract

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 .

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Journal of Symbolic Logic
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Forking independence from the categorical point of view

2019-02-18

Michael Lieberman , Jir̆ı́ Rosický , Sebastien Vasey

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Non-forking independence in stable theories

2024-10-13

Amador Martin-Pizarro

We observe that a simple condition suffices to describes non-forking independence over models in a stable theory. Under mild assumptions, this description can be extended to non-forking independence over algebraically … We observe that a simple condition suffices to describes non-forking independence over models in a stable theory. Under mild assumptions, this description can be extended to non-forking independence over algebraically closed subsets, without having to use the full strength of the work of the seminal work of Kim and Pillay. The results in this note (which are surely well-known among most model theorists) essentially use that types over models in a stable theory are stationary.

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Independence relations in theories with the tree property

2013-01-01

Gwyneth Fae Harrison-Shermoen

Author(s): Harrison-Shermoen, Gwyneth Fae | Advisor(s): Scanlon, Thomas W | Abstract: This thesis investigates theories with the tree property and, in particular, notions of independence in such theories. We discuss … Author(s): Harrison-Shermoen, Gwyneth Fae | Advisor(s): Scanlon, Thomas W | Abstract: This thesis investigates theories with the tree property and, in particular, notions of independence in such theories. We discuss the example of the two-sorted theory of an infinite dimensional vector space over an algebraically closed field and with a bilinear form (which we refer to as T_∞), examined by Granger in his thesis. Granger notes that there is a well-behaved notion of independence, which he calls Γ-non-forking, in this theory, and that it can be viewed as the limit of the non-forking independence in the theories of its finite dimensional subspaces, which are ω-stable. He defines a notion of an ``approximating sequence'' of substructures, and shows that Γ-non-forking in a model of T_∞ corresponds to ``eventual non-forking in an approximating sequence. We generalize his notion of approximation by substructures, and in the case of a theory whose large models can be approximated in this way, define a notion of independence which is the ``limit'' of the non-forking independence in the theories of the approximating substructures. We show that if the approximating substructures have simple theories, the limit independence relation satisfies invariance, monotonicity, base monotonicity, transitivity, normality, extension, finite character, and symmetry. Under certain additional assumptions, limit independence also satisfies anti-reflexivity and the Independence Theorem over algebraically closed sets. We also consider the two-sorted theory of infinitely many cross-cutting equivalence relations, T^*_{feq}. We give a proof, explaining in detail the argument of Shelah and Usvyatsov for Theorem 2.1 in [Shelah-Usvyatsov'08], that T^*_{feq} does not have SOP_2 (equivalently, TP_1). The argument makes use of a theorem of Kim and Kim, from [Kim-Kim'11], along with several other lemmas involving tree indiscernibility.

Measures and forking

1987-01-01

H. Jerome Keisler

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.

NSOP-LIKE INDEPENDENCE IN AECATS

2022-12-12

Mark Kamsma

Abstract The classes stable, simple, and NSOP $_1$ in the stability hierarchy for first-order theories can be characterised by the existence of a certain independence relation. For each of them … Abstract The classes stable, simple, and NSOP $_1$ in the stability hierarchy for first-order theories can be characterised by the existence of a certain independence relation. For each of them there is a canonicity theorem: there can be at most one nice independence relation. Independence in stable and simple first-order theories must come from forking and dividing (which then coincide), and for NSOP $_1$ theories it must come from Kim-dividing. We generalise this work to the framework of Abstract Elementary Categories (AECats) with the amalgamation property. These are a certain kind of accessible category generalising the category of (subsets of) models of some theory. We prove canonicity theorems for stable, simple, and NSOP $_1$ -like independence relations. The stable and simple cases have been done before in slightly different setups, but we provide them here as well so that we can recover part of the original stability hierarchy. We also provide abstract definitions for each of these independence relations as what we call isi-dividing, isi-forking, and long Kim-dividing.

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NSOP$_1$-like independence in AECats

2021-01-01

Mark Kamsma

The classes stable, simple and NSOP$_1$ in the stability hierarchy for first-order theories can be characterised by the existence of a certain independence relation. For each of them there is … The classes stable, simple and NSOP$_1$ in the stability hierarchy for first-order theories can be characterised by the existence of a certain independence relation. For each of them there is a canonicity theorem: there can be at most one nice independence relation. Independence in stable and simple first-order theories must come from forking and dividing (which then coincide), and for NSOP$_1$ theories it must come from Kim-dividing. We generalise this work to the framework of AECats (Abstract Elementary Categories) with the amalgamation property. These are a certain kind of accessible category generalising the category of (subsets of) models of some theory. We prove canonicity theorems for stable, simple and NSOP$_1$-like independence relations. The stable and simple cases have been done before in slightly different setups, but we provide them here as well so that we can recover part of the original stability hierarchy. We also provide abstract definitions for each of these independence relations as what we call isi-dividing, isi-forking and long Kim-dividing.

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Stability Theory and its Variants

2000-01-01

Bradd Hart

Dimension theory plays a crucial technical role in stability theory and its relatives. The abstract dependence relations defined, although combinatorial in nature, often have surprising geometric meaning in particular cases. … Dimension theory plays a crucial technical role in stability theory and its relatives. The abstract dependence relations defined, although combinatorial in nature, often have surprising geometric meaning in particular cases. This article discusses several aspects of dimension theory, such as categoricity, strongly minimal sets, modularity and the Zil’ber principle, forking, simple theories, orthogonality and regular types and in the third, stability, definability of types, stable groups and 1-based groups. One of the achievements of the branch of model theory known as stability theory is the use of numerical invariants, dimensions, in a broad setting. In recent years, this dimension theory has been expanded to include the so-called simple theories. In this paper, I wish to give just a brief overview of the elements of this theory. In the first section, the special case of strongly minimal sets is considered. In the second section, the combinatorial definition of dividing is given and how it leads to a general independence relation is outlined. Only in the third section do stable theories appear and the theory surrounding them is developed there with an eye to other papers in this volume. 1. Strongly Minimal Sets Categorical Theories. One of the simplest questions one can ask about a first order theory is how many models it has of a given cardinality. If T is a countable theory with an infinite model then, by the Lowenheim–Skolem Theorem, it will have at least one model of every infinite power. The situation we will look at first is when a theory has exactly one model of some fixed power. Definition 1.1. A theory T is λ-categorical if T has exactly one model up to isomorphism of cardinality λ. T is said to be totally categorical if T is λ-categorical for every infinite cardinal λ. We will say that T is uncountably categorical if T is λ-categorical for all uncountable λ. Example 1.2. 1. The theory of a set in a language which has only equality is totally categorical.

On Kim-Independence

2017-01-01

Itay Kaplan , Nicholas Ramsey

We study NSOP$_{1}$ theories. We define Kim-independence, which generalizes non-forking independence in simple theories and corresponds to non-forking at a generic scale. We show that Kim-independence satisfies a version of … We study NSOP$_{1}$ theories. We define Kim-independence, which generalizes non-forking independence in simple theories and corresponds to non-forking at a generic scale. We show that Kim-independence satisfies a version of Kim's lemma, local character, symmetry, and an independence theorem and that, moreover, these properties individually characterize NSOP$_{1}$ theories. We describe Kim-independence in several concrete theories and observe that it corresponds to previously studied notions of independence in Frobenius fields and vector spaces with a generic bilinear form.

On Kim-Independence

2017-02-13

Itay Kaplan , Nicholas Ramsey

A GEOMETRIC INTRODUCTION TO FORKING AND THORN-FORKING

2009-06-01

Hans H. Adler

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 … 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.

A NOTE ON STABLE KIM-FORKING

2024-11-05

Yvon Bossut

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Fundamentals of Stability Theory

2017-03-02

John T. Baldwin

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been … Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the twelfth publication in the Perspectives in Logic series, John T. Baldwin presents an introduction to first order stability theory, organized around the spectrum problem: calculate the number of models a first order theory T has in each uncountable cardinal. The author first lays the groundwork and then moves on to three sections: independence, dependence and prime models, and local dimension theory. The final section returns to the spectrum problem, presenting complete proofs of the Vaught conjecture for ω-stable theories for the first time in book form. The book provides much-needed examples, and emphasizes the connections between abstract stability theory and module theory.

On Definability of Types in Dependent Theories

2011-01-01

Vincent Guingona

Title of dissertation: ON DEFINABILITY OF TYPES IN DEPENDENT THEORIES Vincent Guingona, Doctor of Philosophy, 2011 Dissertation directed by: Professor Michael Chris Laskowski Department of Mathematics Using definability of types … Title of dissertation: ON DEFINABILITY OF TYPES IN DEPENDENT THEORIES Vincent Guingona, Doctor of Philosophy, 2011 Dissertation directed by: Professor Michael Chris Laskowski Department of Mathematics Using definability of types for stable formulas, one develops the powerful tools of stability theory, such as canonical bases, a nice forking calculus, and stable embeddability. When one passes to the class of dependent formulas, this notion of definability of types is lost. However, as this dissertation shows, we can recover suitable alternatives to definability of types for some dependent theories. Using these alternatives, we can recover some of the power of stability theory. One alternative is uniform definability of types over finite sets (UDTFS). We show that all formulas in dp-minimal theories have UDTFS, as well as formulas with VC-density < 2. We also show that certain Henselian valued fields have UDTFS. Another alternative is isolated extensions. We show that dependent formulas are characterized by the existence of isolated extensions, and show how this gives a weak stable embeddability result. We also explore the idea of UDTFS rank and show how it relates to VC-density. Finally, we use the machinery developed in this dissertation to show that VCminimal theories satisfy the Kueker Conjecture. ON DEFINABILITY OF TYPES IN DEPENDENT THEORIES

Stability Theoretic Methods in Unstable Theories

2009-01-01

Bradd Hart , Alf Onshuus , Anand Pillay +2 more

The idea of applying methods and results from stability theory to unstable theories has been an important theme over the past 25 years, with o-minimality, smoothly approximable structures, and simple … The idea of applying methods and results from stability theory to unstable theories has been an important theme over the past 25 years, with o-minimality, smoothly approximable structures, and simple theories being key examples. But there have been some key recent developments which bring new ideas and techniques to the table. One of these is the investigation of abstract notions of independence, leading for example to the notions of thorn forking and rosiness. Another is the discovery that forking, weight, and related notions from stability are meaningful in dependent theories. Another is the formulation of notions of stable, compact, or more general domination, coming from the analysis of theories such as algebraically closed valued fields and o-minimal theories. The level of different approaches and techniques which end up overlapping was the reason we decided it would be a perfect time for a research meeting where the most prominent researchers would come together and discuss the ideas, results and goals that were showing up in different contexts. The dominant subjects of the meeting were the following.

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.

Independence of the axiomatization of fork

1998-08-01

Marcelo F. Frias

Forking

2000-01-01

Bruno Poizat

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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Fundamentals of forking

1984-06-01

Victor Harnik , Leo Harrington

Infinitary stability theory

2016-03-31

Sebastien Vasey

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Independence over arbitrary sets in NSOP1 theories

2021-10-25

Jan Dobrowolski , Byunghan Kim , Nicholas Ramsey

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Modules and stability theory

1987-02-01

Anand Pillay , Mike Prest

Modules are now widely recognized as important examples of stable structures. In fact, in the light of results and conjectures of Zilber [Zi] (${\aleph _1}$-categorical structures are âfield-likeâ, âmodule-likeâ or … Modules are now widely recognized as important examples of stable structures. In fact, in the light of results and conjectures of Zilber [Zi] (${\aleph _1}$-categorical structures are âfield-likeâ, âmodule-likeâ or âtrivialâ), we may consider modules as one of the typical examples of stable structures. Our aim here is both to prove some new results in the model theory of modules and to highlight the particularly clear form of, and the algebraic content of, the concepts of stability theory when applied to modules. One of the main themes of this paper is the connection between stability-theoretic notions, such as ranks, and algebraic decomposition of models. We will usually work with $T$, a complete theory of $R$-modules, for some ring $R$. In $\S 2$ we show that the various stability-theoretic ranks, when defined, are the same. In $\S 3$ we show that $T$ (not necessarily superstable) is nonmultidimensional (in the sence of Shelah [Sh1]). In $\S 4$ we consider the algebraic content of saturation and we show, for example, that if $M$ is a superstable module then $M$ is $F_{{\aleph _0}}^a$-saturated just if $M$ is pure-injective and realizes all types in finitely many free variables over $\phi$. In $\S 5$ we use our methods to reprove Zieglerâs theorem on the possible spectrum functions. In $\S 6$ we show the profusion (in a variety of senses) of regular types. In $\S 7$ we give a structure theorem for the models of $T$ in the case where $T$ has $U$-rank 1.

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Model theory of modules

1984-04-01

Martin Ziegler

Les modèles dénombrables d’une théorie ayant des fonctions de Skolem

1981-01-01

Daniel Lascar

Let <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> be a countable complete theory having Skolem functions. We prove that if all the types over … Let <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> be a countable complete theory having Skolem functions. We prove that if all the types over finitely generated models are definable (this is the case for example if <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> is stable), then either <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> has <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript normal alef 0"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{2^{{\aleph _0}}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> countable models or all its models are homogeneous. The proof makes heavy use of stability techniques.

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Coordinatization in superstable theories. I. Stationary types

1985-01-01

Steven Buechler

Suppose <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> is superstable and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> … Suppose <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> is superstable and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a complete type over some finite set with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U left-parenthesis p right-parenthesis equals alpha plus 1"> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</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(p) = \alpha + 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some <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>. We show how to associate with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> an incidence geometry which measures the complexity of the family of extensions of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of rank <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>. When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is stationary we give a characterization of the possible incidence geometries. As an application we prove Theorem. <italic>Suppose</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <italic>is superstable and has only one</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p element-of upper S left-parenthesis normal empty-set right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">∅</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p \in S(\emptyset )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. <italic>Further suppose</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <italic>is stationary with</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U left-parenthesis p right-parenthesis equals alpha plus 1"> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</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(p) = \alpha + 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some <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 one of the following holds</italic>: (i) <italic>There is an equivalence relation</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E subset-of upper M squared"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>⊂</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">E \subset {M^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <italic>with infinitely many infinite classes definable over</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal empty-set"> <mml:semantics> <mml:mi mathvariant="normal">∅</mml:mi> <mml:annotation encoding="application/x-tex">\emptyset</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="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <italic>is the algebraic closure of a set of Morley rank</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. <italic>In particular</italic>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</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="normal alef 0"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\aleph _0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-<italic>stable of finite rank</italic>.

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Paires de structures stables

1983-06-01

Bruno Poizat

Le paradigme de théorie stable est la théorie T d'un corps algébriquement clos; une autre théorie stable T ′ est celle de la structure formée d'un corps algébriquement clos, avec … Le paradigme de théorie stable est la théorie T d'un corps algébriquement clos; une autre théorie stable T ′ est celle de la structure formée d'un corps algébriquement clos, avec en outre un symbole relationnel unaire interprétant un de ses sous-corps propres algébriquement clos. C'est à l'éclaircissement des rapports de T et de T ′ qu'est consacré cet article. J'y considère une théorie complète T stable, et les structures formées d'un modèle N de T, avec en outre un symbole relationnel unaire ( x ) interprétant une restriction élémentaire M de N ; j'appelle ces structures paires de modèles de T . Et je dis que la paire ( N, M ) est belle si d'une part M est ∣ T ∣ + -saturé, et d'autre part pour tout n -uplet ā d'éléments de N , tout type, au sens de T , sur M ⋃ {α} est réalisé dans N . Le premier résultat (Théorème 4) est que deux belles paires sont élémentairement équivalentes. Plus précisément, si ( N 1 , M 1 ) et ( N 2 , M 2 ) sont deux belles paires, et si ā est dans la première, b¯ dans la seconde, le fait que le type de ā sur M 1 et celui de b¯ sur M 2 soient équivalents dans l'ordre fondamental au sens de T suffit (et est bien sûr nécessaire) pour que ā at b¯ aient même type (sur ⊘) au sens de la théorie T ′ des belles paires.

Definable structures in o-minimal theories: One dimensional types

2010-10-31

Assaf Hasson , Alf Onshuus , Ya’acov Peterzil

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F

1997-01-01

Michiel Hazewinkel

SATURATED FREE ALGEBRAS REVISITED

2015-09-01

Anand Pillay , Rizos Sklinos

Abstract We give an exposition of results of Baldwin–Shelah [2] on saturated free algebras, at the level of generality of complete first order theories T with a saturated model M … Abstract We give an exposition of results of Baldwin–Shelah [2] on saturated free algebras, at the level of generality of complete first order theories T with a saturated model M which is in the algebraic closure of an indiscernible set. We then make some new observations when M is a saturated free algebra, analogous to (more difficult) results for the free group, such as a description of forking.

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The number of uncountable models of ω-stable theories

1983-09-01

Jürgen Saffe

Measures and forking

1987-01-01

H. Jerome Keisler

The geometry of weakly minimal types

1985-12-01

Steven Buechler

Abstract Let T be superstable. We say a type p is weakly minimal if R ( p, L , ∞) = 1. Let M ⊨ T be uncountable and saturated, … Abstract Let T be superstable. We say a type p is weakly minimal if R ( p, L , ∞) = 1. Let M ⊨ T be uncountable and saturated, H = p ( M ). We say D ⊂ H is locally modular if for all X, Y ⊂ D with X = acl( X ) ∩ D, Y = acl( Y ) ∩ D and X ∩ Y ≠ ∅, Theorem 1. Let p ∈ S(A) be weakly minimal and D the realizations of stp( a/A) for some a realizing p. Then D is locally modular or p has Morley rank 1. Theorem 2. Let H, G be definable over some finite A, weakly minimal, locally modular and nonorthogonal. Then for all a ∈ H ∖acl( A ), b ∈ G ∖acl( A ) there are a ′ ∈ H , b ′ ∈ G such that a′ ∈ acl( abb ′ A )∖acl( aA). Similarly when H and G are the realizations of complete types or strong types over A .

Decidable models of small theories

2015-10-01

Alex Gavryushkin

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The notion of independence in categories of algebraic structures, part I: Basic properties

1988-05-01

Gabriel Srour

Kueker's conjecture for superstable theories

1984-09-01

Steven Buechler

Abstract We prove that if every uncountable model of a first-order theory T is ω -saturated and T is superstable then T is categorical in some infinite power. Abstract We prove that if every uncountable model of a first-order theory T is ω -saturated and T is superstable then T is categorical in some infinite power.

Forking, normalization and canonical bases

1986-01-01

Anand Pillay

Countable models of stable theories

1983-01-01

Anand Pillay

The notion of a normal theory such a theory <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T comma upper I left-parenthesis normal alef 0 comma upper T right-parenthesis equals 1 or greater-than-or-slanted-equals … The notion of a normal theory such a theory <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T comma upper I left-parenthesis normal alef 0 comma upper T right-parenthesis equals 1 or greater-than-or-slanted-equals normal alef 0"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thickmathspace" /> <mml:mi>I</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext> or </mml:mtext> </mml:mrow> <mml:mo>⩾</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">T,\;I({\aleph _0},T) = 1{\text { or }} \geqslant {\aleph _0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. theorem that for superstable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T comma upper I left-parenthesis normal alef 0 comma upper T right-parenthesis equals 1 or greater-than-or-slanted-equals normal alef 0"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thickmathspace" /> <mml:mi>I</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext> or </mml:mtext> </mml:mrow> <mml:mo>⩾</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">T,\;I({\aleph _0},T) = 1{\text { or }} \geqslant {\aleph _0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> stronger than stability but incomparable is introduced, and it is proved that for We also include a short proof of Lachlan’s (The property of normality is to superstability.)

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A Class of ℵ<sub>0</sub>‐Categorical Theories

1981-01-01

Anand Pillay

Saturated free algebras revisited

2014-09-30

Anand Pillay , Rizos Sklinos

We give an exposition of results of Baldwin-Shelah on saturated free algebras, at the level of generality of complete first order theories $T$ with a saturated model $M$ which is … We give an exposition of results of Baldwin-Shelah on saturated free algebras, at the level of generality of complete first order theories $T$ with a saturated model $M$ which is in the algebraic closure of an indiscernible set. We then make some new observations when $M$ is a saturated free algebra, analogous to (more difficult) results for the free group, such as a description of forking.

Maximal chains in the fundamental order

1986-06-01

Steven Buechler

Abstract Suppose T is superstable. Let ≤ denote the fundamental order on complete types, [ p ] the class of the bound of p , and U (—) Lascar's foundation … Abstract Suppose T is superstable. Let ≤ denote the fundamental order on complete types, [ p ] the class of the bound of p , and U (—) Lascar's foundation rank (see [LP]). We prove Theorem 1. If q < p and there is no r such that q < r < p, then U(q) + 1 = U(p) . Theorem 2. Suppose U(p) < ω and ξ 1 < … < ξ k is a maximal descending chain in the fundamental order with ξ k = [p]. Then k = U(p) . That the finiteness of U(p) in Theorem 2 is necessary follows from Theorem 3. There is an ω-stable theory with a type p ϵ S 1 ( ϕ ) such that (1) U(p) = ω + 1, and (2) there is a maximal descending chain of proper extensions of [p] which has order type ω .

On two hierarchies of dimensions

1987-12-01

Andreas Baudisch

Abstract Let T be a countable, complete, ω -stable, nonmultidimensional theory. By Lascar [7], in T eq there is in every dimension of T a type with Lascar rank ω … Abstract Let T be a countable, complete, ω -stable, nonmultidimensional theory. By Lascar [7], in T eq there is in every dimension of T a type with Lascar rank ω α for some α . We give sufficient conditions for α to coincide with the level of that dimension in Pillay's [10] RK-hierarchy of dimensions computed in T eq . In particular, this is fulfilled for modules.

Classification of superstable theories by rank functions

1985-01-01

T. G. Mustafin

Minimal stable types in Banach spaces

2019-08-19

Saharon Shelah , Alexander Usvyatsov

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Building independence relations in abstract elementary classes

2016-05-01

Sebastien Vasey

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The theory of ordered abelian groups does not have the independence property

1984-01-01

Yuri Gurevich , Peter H. Schmitt

We prove that no complete theory of ordered abelian groups has the independence property, thus answering a question by B. Poizat. The main tool is a result contained in the … We prove that no complete theory of ordered abelian groups has the independence property, thus answering a question by B. Poizat. The main tool is a result contained in the doctoral dissertation of Yuri Gurevich and also in P. H. Schmitt’s <italic>Elementary properties of ordered abelian groups</italic>, which basically transforms statements on ordered abelian groups into statements on coloured chains. We also prove that every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-type in the theory of coloured chains has at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{2^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> coheirs, thereby strengthening a result by B. Poizat.

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From Stability to Simplicity

1998-03-01

Byunghan Kim , Anand Pillay

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

Ordre de Rudin‐Keisler et Poids Dans les Theories Stables

1982-01-01

Daniel Lascar

Flat Morley sequences

1999-09-01

Ludomir Newelski

Abstract Assume T is a small superstable theory. We introduce the notion of a flat Morley sequence, which is a counterpart of the notion of an infinite Morley sequence in … Abstract Assume T is a small superstable theory. We introduce the notion of a flat Morley sequence, which is a counterpart of the notion of an infinite Morley sequence in a type p , in case when p is a complete type over a finite set of parameters. We show that for any flat Morley sequence Q there is a model M of T which is τ -atomic over { Q }. When additionally T has few countable models and is 1-based, we prove that within M there is an infinite Morley sequence I , with I ⊂ dcl( Q ), such that M is prime over I .

On generically stable types in dependent theories

2009-01-04

Alexander Usvyatsov

Abstract We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent theories. Abstract We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent theories.

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Some remarks on indiscernible sequences

2003-07-03

Enrique Casanovas

We prove a property of generic homogeneity of tuples starting an infinite indiscernible sequence in a simple theory and we use it to give a shorter proof of the Independence … We prove a property of generic homogeneity of tuples starting an infinite indiscernible sequence in a simple theory and we use it to give a shorter proof of the Independence Theorem for Lascar strong types. We also characterize the relation of starting an infinite indiscernible sequence in terms of coheirs.

Heirs of box types in polynomially bounded structures

2009-10-05

Marcus Tressl

Abstract A box type is an n -type of an o-minimal structure which is uniquely determined by the projections to the coordinate axes. We characterize heirs of box types of … Abstract A box type is an n -type of an o-minimal structure which is uniquely determined by the projections to the coordinate axes. We characterize heirs of box types of a polynomially bounded o-minimal structure M . From this, we deduce various structure theorems for subsets of M k , definable in the expansion of M by all convex subsets of the line. We show that after naming constants, is model complete provided M is model complete.

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Regular types in nonmultidimensional ω-stable theories

1984-09-01

Anand Pillay

Abstract We define a hierarchy on the regular types of an ω -stable nonmultidimensional theory, using generalised notions of algebraic and strongly minimal formulae. As an application we show that … Abstract We define a hierarchy on the regular types of an ω -stable nonmultidimensional theory, using generalised notions of algebraic and strongly minimal formulae. As an application we show that any resplendent model of an ω -stable finite-dimensional theory is saturated.

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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Independence in Model Theory

2016-01-01

Åsa Hirvonen

Countable models of nonmultidimensional ℵ<sub>0</sub>-stable theories

1983-03-01

Élisabeth Bouscaren , Daniel Lascar

In this paper T will always be a countable ℵ 0 -stable theory, and in this introduction a model of T will mean a countable model. One of the main … In this paper T will always be a countable ℵ 0 -stable theory, and in this introduction a model of T will mean a countable model. One of the main notions we introduce is that of almost homogeneous model: we say that a model M of T is almost homogeneous if for all ā and finite sequences of elements in M , if the strong type of ā is the same as the strong type of (i.e. for all equivalence relations E , definable over the empty set and with a finite number of equivalence classes, ā and are in the same equivalence class), then there is an automorphism of M taking ā to . Although this is a weaker notion than homogeneity, these models have strong properties, and it can be seen easily that the Scott formula of any almost homogeneous model is in L 1 . In fact, Pillay [Pi.] has shown that almost homogeneous models are characterized by the set of types they realize. The motivation of this research is to distinguish two classes of ℵ 0 -Stable theories: (1) theories such that all models are almost homogeneous; (2) theories with 2 ℵ 0 nonalmost homogeneous models. The example of theories with Skolem functions [L. 1] (almost homogeneous is then equivalent to homogeneous) seems to indicate a link between these properties and the notion of multidimensionality, and that nonmultidimensional theories are in the first case.

Indépendance et liberté

2016-12-19

Bruno Poizat

Fundamentals of Stability Theory

2017-03-02

John T. Baldwin

Lascar rank and the finite cover property for complete theories of unars

1993-11-01

A. N. Ryaskin

A. D. Taimanov and model theory in Kazakhstan

2020-02-19

B. S. Baizhanov , B. Sh. Kulpeshov , T. S. Zambarnaya

The article gives an overview of the development of model theory in Kazakhstan over the past 60 years from the standpoint of the tasks that stood at that time.We consider … The article gives an overview of the development of model theory in Kazakhstan over the past 60 years from the standpoint of the tasks that stood at that time.We consider directions that naturally arose from the definition of truth in a model and under the influence of Doctor of Physical and Mathematical Sciences, Academician A. Taimanov, initiator of the growth of model theory in Kazakhstan.

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