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32
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.
Simple theories
, 2000
"... Abstract. Groups definable in simple theories retain the chain conditions and decomposition properties known from stable groups, up to commensurability. In the small case, if a generic type of G is not foreign to some type q, there is a qinternal quotient. In the supersimple case, the BerlineLasca ..."
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Cited by 45 (0 self)
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Abstract. Groups definable in simple theories retain the chain conditions and decomposition properties known from stable groups, up to commensurability. In the small case, if a generic type of G is not foreign to some type q, there is a qinternal quotient. In the supersimple case, the BerlineLascar decomposition works. Onebased simple groups are finitebyabelianbyfinite.
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.
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
A primer of simple theories
 Archive Math. Logic
"... Abstract. We present a selfcontained exposition of the basic aspects of simple theories while developing the fundamentals of forking calculus. We expound also the deeper aspects of S. Shelah’s 1980 paper Simple unstable theories. The concept of weak dividing has been replaced with that of forking. ..."
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Cited by 9 (2 self)
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Abstract. We present a selfcontained exposition of the basic aspects of simple theories while developing the fundamentals of forking calculus. We expound also the deeper aspects of S. Shelah’s 1980 paper Simple unstable theories. The concept of weak dividing has been replaced with that of forking. The exposition is from a contemporary perspective and takes into account contributions due to S. Buechler, E. Hrushovski, B. Kim, O. Lessmann, S. Shelah and A. Pillay.