Lascar strong types in some simple theories

Buechler, Steven

doi:10.2307/2586503

Cited by 25 publications

(46 citation statements)

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“…Let a realize ϕ, and b realize ψ. Since a and b are in the same class C , they realize the same strong type over ∅, so by Theorem 3.1 of [5], they realize the same Lascar strong type over ∅. Hence, by the Independence Theorem for Lascar strong types (Theorem 5.8 of [25]), there is an element c ∈ acl(A ∪ B) , a i )), as required.…”

Section: Proposition 46 Let γ = (V γ E) Be a Countably Infinite Grmentioning

confidence: 81%

One-dimensional asymptotic classes of finite structures

Macpherson

Steinhorn

2008

Trans. Amer. Math. Soc.

107

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Abstract. A collection C of finite L-structures is a 1-dimensional asymptotic class if for every m ∈ N and every formula ϕ(x,ȳ), whereȳ = (y 1 , . . . , y m ):(i) There is a positive constant C and a finite set E ⊂ R >0 such that forOne-dimensional asymptotic classes are introduced and studied here. These classes come equipped with a notion of dimension that is intended to provide for the study of classes of finite structures a concept that is central in the development of model theory for infinite structures. Connections with the model theory of infinite structures are also drawn.

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Section: Proposition 46 Let γ = (V γ E) Be a Countably Infinite Grmentioning

confidence: 81%

One-dimensional asymptotic classes of finite structures

Macpherson

Steinhorn

2008

Trans. Amer. Math. Soc.

107

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“…Dividing in type denable, so in dependent theories all these notions are type-denable over A (i.e. dependent theories are low, see [Bue99]) Proof. (Due to Itai Ben Yaacov) First we shall see that for any set B, if ϕ (x, a) divides over B then it k := alt (ϕ) divides over B. if a i |i < ω is an indiscernible sequence witnessing m > k dividing but not k dividing, it means that ∃x i<k ϕ (x, a i ), and by indiscerniblity, ∃x i<k ϕ (x, a mi ).…”

Section: Proof Suppose ϕ (X A)mentioning

confidence: 99%

Forking and Dividing in NTP₂ theories

Chernikov¹,

Kaplan

2012

J. symb. log.

120

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“…For simple theories, (2) is equivalent to Buechler's original definition[1] which asked that for every ϕ, D(x = x, {ϕ}, ℵ 0 ) < ω.…”

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confidence: 99%

A dividing line within simple unstable theories

Malliaris

Shelah

2013

Advances in Mathematics

106

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Abstract. We give the first (ZFC) dividing line in Keisler's order among the unstable theories, specifically among the simple unstable theories. That is, for any infinite cardinal λ for which there is µ < λ ≤ 2 µ , we construct a regular ultrafilter D on λ so that (i) for any model M of a stable theory or of the random graph,The non-saturation result relies on the notion of flexible ultrafilters. To prove the saturation result we develop a property of a class of simple theories, called Qr 1 , generalizing the fact that whenever B is a set of parameters in some sufficiently saturated model of the random graph, |B| = λ and µ < λ ≤ 2 µ , then there is a set A with |A| = µ so that any nonalgebraic p ∈ S(B) is finitely realized in A. In addition to giving information about simple unstable theories, our proof reframes the problem of saturation of ultrapowers in several key ways. We give a new characterization of good filters in terms of "excellence," a measure of the accuracy of the quotient Boolean algebra. We introduce and develop the notion of moral ultrafilters on Boolean algebras. We prove a so-called "separation of variables" result which shows how the problem of constructing ultrafilters to have a precise degree of saturation may be profitably separated into a more set-theoretic stage, building an excellent filter, followed by a more model-theoretic stage: building so-called moral ultrafilters on the quotient Boolean algebra, a process which highlights the complexity of certain patterns, arising from first-order formulas, in certain Boolean algebras.

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Lascar strong types in some simple theories

Abstract: In this paper a class of simple theories, called the low theories is developed, and the following is proved.Theorem. Let T be a low theory, A a set and a, b elements realizing the same strong type over A. Then, a and b realize the same Lasear strong type over A.

Cited by 25 publications

References 1 publication

One-dimensional asymptotic classes of finite structures

One-dimensional asymptotic classes of finite structures

Forking and Dividing in NTP₂ theories

A dividing line within simple unstable theories

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Lascar strong types in some simple theories

Abstract: In this paper a class of simple theories, called the low theories is developed, and the following is proved.Theorem. Let T be a low theory, A a set and a, b elements realizing the same strong type over A. Then, a and b realize the same Lasear strong type over A.

Cited by 25 publications

References 1 publication

One-dimensional asymptotic classes of finite structures

One-dimensional asymptotic classes of finite structures

Forking and Dividing in NTP2 theories

A dividing line within simple unstable theories

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Product

Resources

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Forking and Dividing in NTP₂ theories