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34
A Survey on the Model Theory of Difference Fields
, 2000
"... We survey the model theory of difference fields, that is, fields with a distinguished automorphism σ. After introducing the theory ACFA and stating elementary results, we discuss independence and the various concepts of rank, the dichotomy theorems, and, as an application, the Manin–Mumford conject ..."
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Cited by 67 (9 self)
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We survey the model theory of difference fields, that is, fields with a distinguished automorphism σ. After introducing the theory ACFA and stating elementary results, we discuss independence and the various concepts of rank, the dichotomy theorems, and, as an application, the Manin–Mumford conjecture over a number field. We conclude with some other applications.
Nonexistence of Universal Orders in Many Cardinals
 Journal of Symbolic Logic
, 1992
"... Our theme is that not every interesting question in set theory is independent of ZF C. We give an example of a first order theory T with countable D(T) which cannot have a universal model at ℵ1 without CH; we prove in ZF C a covering theorem from the hypothesis of the existence of a universal model ..."
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Cited by 19 (15 self)
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Our theme is that not every interesting question in set theory is independent of ZF C. We give an example of a first order theory T with countable D(T) which cannot have a universal model at ℵ1 without CH; we prove in ZF C a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove — again in ZFC — that for a large class of cardinals there is no universal linear order (e.g. in every ℵ1 < λ < 2 ℵ0). In fact, what we show is that if there is a universal linear order at a regular λ and its existence is not a result of a trivial cardinal arithmetical reason, then λ “resembles ” ℵ1 — a cardinal for which the consistency of having a universal order is known. As for singular cardinals, we show that for many singular cardinals, if they are not strong limits then they have no universal linear order. As a result of the non existence of a universal linear order, we show the nonexistence of universal models for all theories possessing the strict order property (for example, ordered fields and groups, Boolean algebras, padic rings and fields, partial orders, models of PA and so on).
Introduction to theories without the independence property
"... We present an updated exposition of the classical theory of complete first order theories ..."
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Cited by 17 (1 self)
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We present an updated exposition of the classical theory of complete first order theories
SIMPLE HOMOGENEOUS MODELS
, 2002
"... Geometrical stability theory is a powerful set of modeltheoretic tools that can lead to structural results on models of a simple firstorder theory. Typical results offer a characterization of the groups definable in a model of the theory. The work is carried out in a universal domain of the theor ..."
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Cited by 16 (2 self)
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Geometrical stability theory is a powerful set of modeltheoretic tools that can lead to structural results on models of a simple firstorder theory. Typical results offer a characterization of the groups definable in a model of the theory. The work is carried out in a universal domain of the theory (a saturated model) in which the Stone space topology on ultrafilters of definable relations is compact. Here we operate in the more general setting of homogeneous models, which typically have noncompact Stone topologies. A structure M equipped with a class of finitary relations R is strongly λ−homogeneous if orbits under automorphisms of (M, R) have finite character in the following sense: Given α an ordinal < λ ≤ M  and sequences ā = { ai: i < α}, ¯ b = { bi: i < α} from M, if (ai1,..., ain) and (bi1,..., bin) have the same orbit, for all n and i1 < · · · < in < α, then f(ā) = ¯ b for some automorphism f of (M, R). In this paper strongly λ−homogeneous models (M, R) in which the elements of R induce a symmetric and transitive notion of independence with bounded character are studied. This notion of independence, defined using a combinatorial condition called “dividing”, agrees with forking independence when (M, R) is saturated. The concept central to the development of geometrical stability theory for saturated structures, namely the canonical base, is also shown to exist in this setting. These results broaden the scope of the
Simplicity and the Lascar group
, 1998
"... This paper contains a series of easy constructions and observations relating to the Lascar group and to simple theories. ..."
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Cited by 14 (0 self)
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This paper contains a series of easy constructions and observations relating to the Lascar group and to simple theories.
Toward classifying unstable theories
 Annals of Pure and Applied Logic
, 1995
"... (500) revision:19960622 modified:19960622 Abstract. We prove a consistency results saying, that for a simple (first order) theory, it is easier to have a universal model in some cardinalities, than for the theory of linear order. We define additional properties of first order theories, the nstr ..."
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Cited by 12 (4 self)
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(500) revision:19960622 modified:19960622 Abstract. We prove a consistency results saying, that for a simple (first order) theory, it is easier to have a universal model in some cardinalities, than for the theory of linear order. We define additional properties of first order theories, the nstrong order property (SOPn in short). The main result is that a first order theory with the 4strong order property behaves like linear orders concerning existence of universal models. Key words and phrases. Model theory, classification thoery, stability theory, unstable theories, universal models, simple theories, Keisler’s order. Done: §1: with 457; section 2: 8/92: 2.2 + sufficiency for non existence of universal; 12/92 rest
Lascar Strong Types in Some Simple Theories
, 1997
"... 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 Lascar strong type over A . The reader is expected to be famil ..."
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Cited by 11 (2 self)
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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 Lascar strong type over A . The reader is expected to be familiar with forking in simple theories, as developed in Kim's thesis [Kim]. The Lascar strong type of a over A is denoted lstp(a=A) . Unless stated otherwise, we work in the context of a simple theory in this paper. 1 Amalgamation properties Type amalgamation (the Independence Theorem) is perhaps the most useful property of forking dependence in a simple theory. First, we stress an important fact from [Kim]. Lemma 1.1 Let A be a set, a; b elements such that lstp(a=A) = lstp(b=A) and a j
A geometrical introduction to forking and thornforking
, 2007
"... A ternary relation  ⌣ between subsets of the big model of a complete firstorder theory ..."
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Cited by 11 (1 self)
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A ternary relation  ⌣ between subsets of the big model of a complete firstorder theory
Stable group theory and approximate subgroups
 J. Amer. Math. Soc
"... Abstract. We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sumproduct phenomenon. For a simple linear group G, we show that a finite subset X with XX −1 X/X  bounded is close to a finite subgroup, or else to a subset of a pro ..."
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Cited by 9 (0 self)
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Abstract. We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sumproduct phenomenon. For a simple linear group G, we show that a finite subset X with XX −1 X/X  bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of G. We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Modeltheoretically we prove the independence theorem and the stabilizer theorem in a general firstorder setting. 1.