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35
Definable sets in ordered structures
 Bull. Amer. Math. Soc. (N.S
, 1984
"... Abstract. This paper introduces and begins the study of a wellbehaved class of linearly ordered structures, the ^minimal structures. The definition of this class and the corresponding class of theories, the strongly ©minimal theories, is made in analogy with the notions from stability theory of m ..."
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Cited by 96 (7 self)
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Abstract. This paper introduces and begins the study of a wellbehaved class of linearly ordered structures, the ^minimal structures. The definition of this class and the corresponding class of theories, the strongly ©minimal theories, is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories. Theorems 2.1 and 2.3, respectively, provide characterizations of Cminimal ordered groups and rings. Several other simple results are collected in §3. The primary tool in the analysis of ¿¡minimal structures is a strong analogue of "forking symmetry, " given by Theorem 4.2. This result states that any (parametrically) definable unary function in an (5minimal structure is piecewise either constant or an orderpreserving or reversing bijection of intervals. The results that follow include the existence and uniqueness of prime models over sets (Theorem 5.1) and a characterization of all N0categorical ¿¡¡minimal structures (Theorem 6.1). 1. Introduction. The
Shelah’s stability spectrum and homogeneity spectrum in finite diagrams
 Arch. Math. Logic
"... Spectrum theorems, as well as the equivalence between the order property and instability in the framework of Finite Diagrams. Finite Diagrams is a context which generalizes the first order case. Localized versions of these theorems are presented. Our presentation is based on several papers; the poin ..."
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Cited by 20 (16 self)
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Spectrum theorems, as well as the equivalence between the order property and instability in the framework of Finite Diagrams. Finite Diagrams is a context which generalizes the first order case. Localized versions of these theorems are presented. Our presentation is based on several papers; the point of view is contemporary and some of the proofs are new. The treatment of local stability in Finite Diagrams is new.
A new uncountably categorical group
 Trans. Amer. Math. Soc
, 1996
"... Abstract. We construct an uncountably categorical group with a geometry that is not locally modular. It is not possible to interpret a field in this group. We show the group is CMtrivial. 1. ..."
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Cited by 17 (3 self)
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Abstract. We construct an uncountably categorical group with a geometry that is not locally modular. It is not possible to interpret a field in this group. We show the group is CMtrivial. 1.
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.
Ranks and pregeometries in finite diagrams
 Ann. Pure Appl. Logic
, 2000
"... ABSTRACT. The study of classes of models of a finite diagram was initiated by S. Shelah in 1969. A diagram D is a set of types over the empty set, and the class of models of the diagram D consists of the models of T which omit all the types not in D. In this work, we introduce a natural dependence r ..."
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Cited by 8 (1 self)
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ABSTRACT. The study of classes of models of a finite diagram was initiated by S. Shelah in 1969. A diagram D is a set of types over the empty set, and the class of models of the diagram D consists of the models of T which omit all the types not in D. In this work, we introduce a natural dependence relation on the subsets of the models for the ℵ0stable case which share many of the formal properties of forking. This is achieved by considering a rank for this framework which is bounded when the diagram D is ℵ0stable. We can also obtain pregeometries with respect to this dependence relation. The dependence relation is the natural one induced by the rank, and the pregeometries exist on the set of realizations of types of minimal rank. Finally, these concepts are used to generalize many of the classical results for models of a totally transcendental firstorder theory. In fact, strong analogies arise: models are determined by their pregeometries or their relationship with their pregeometries; however the proofs are different, as we do not have compactness. This is illustrated with positive results (categoricity) as well as negative results (construction of nonisomorphic models).
Unimodular minimal structures
 J. London Math. Soc
, 1992
"... A strongly minimal structure D is called unimodular if any two finitetoone maps with the same domain and range have the same degree; that is if/4: (/» • V is everywhere fc4tol, then kx = kc,. THEOREM. Unimodular strongly minimal structures are locally modular. This extends Zil'ber's theorem on ..."
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Cited by 8 (2 self)
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A strongly minimal structure D is called unimodular if any two finitetoone maps with the same domain and range have the same degree; that is if/4: (/» • V is everywhere fc4tol, then kx = kc,. THEOREM. Unimodular strongly minimal structures are locally modular. This extends Zil'ber's theorem on locally finite strongly minimal sets, Urbanik's theorem on free algebras with the Steinitz property, and applies also to minimal types in N0categorical stable theories. Strongly minimal sets A strongly minimal set is a structure D such that every definable subset of D is finite or cofinite, uniformly in the parameters. For the importance of these in model theory, see [1] and [4]; relations to combinatorial geometry are discussed in [5] and [3]. We will use the existence of a theory of rank and multiplicity (Morley rank and
A New Spectrum of Recursive Models
"... . We describe a strongly minimal theory S in an effective language such that, in the chain of countable models of S, only the second model has a computable presentation. Thus there is a spectrum of an ! 1 categorical theory which is neither upward nor downward closed. We also give an upper bou ..."
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Cited by 6 (0 self)
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. We describe a strongly minimal theory S in an effective language such that, in the chain of countable models of S, only the second model has a computable presentation. Thus there is a spectrum of an ! 1 categorical theory which is neither upward nor downward closed. We also give an upper bound on the complexity of spectra. 1. Introduction Our main purpose is to find a strongly minimal theory in an effective language whose spectrum of recursive models is the set f1g. We rely on some concepts in Khoussainov, Nies, and Shore [2], reviewed here briefly. Baldwin and Lachlan [1] showed that the countable models of an ! 1 categorical theory T form an ! + 1 chain M 0 (T ) OE M 1 (T ) OE : : : OE M! (T ) under elementary embeddings. In [2], we defined the spectrum of computable models of T , SRM(T ) = fi ! : M i (T ) has a computable presentationg: We gave an example of an ! 1 categorical (in fact, strongly minimal) theory T such that SRM(T ) = (! \Gamma f0g) [ f!g. Kudeiber...
Integration in valued fields
 in Algebraic Geometry and Number Theory, Progr. Math. 253, Birkhäuser
, 2006
"... Abstract. We develop a theory of integration over valued fields of residue characteristic zero. In particular we obtain new and basefield independent foundations for integration over local fields of large residue characteristic, extending results of Denef,Loeser, Cluckers. The method depends on an ..."
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Cited by 6 (2 self)
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Abstract. We develop a theory of integration over valued fields of residue characteristic zero. In particular we obtain new and basefield independent foundations for integration over local fields of large residue characteristic, extending results of Denef,Loeser, Cluckers. The method depends on an analysis of definable sets up to definable bijections. We obtain a precise description of the Grothendieck semigroup of such sets in terms of related groups over the residue field and value group. This yields new invariants of all definable bijections, as well as invariants of measure preserving bijections.
Notes on quasiminimality and excellence
 Bulletin of Symbolic Logic
"... Zilber’s proposes [60] to prove ‘canonicity results for pseudoanalytic ’ structures. Informally, ‘canonical means the theory of the structure in a suitable possibly infinitary language (see Section 2) has one model in each uncountable power ’ while ‘pseudoanalytic means the model of power 2 ℵ0 can ..."
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Cited by 5 (4 self)
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Zilber’s proposes [60] to prove ‘canonicity results for pseudoanalytic ’ structures. Informally, ‘canonical means the theory of the structure in a suitable possibly infinitary language (see Section 2) has one model in each uncountable power ’ while ‘pseudoanalytic means the model of power 2 ℵ0 can be taken as a reduct of an expansion of the complex numbers by analytic functions’. This program interacts with two other lines of research. First is the general study of categoricity theorems in infinitary languages. After initial results by Keisler, reported in [31], this line was taken up in a long series of works by Shelah. We place Zilber’s work in this context. The second direction stems from Hrushovski’s construction of a counterexample to Zilber’s conjecture that every strongly minimal set is ‘trivial’, ‘vector spacelike’, or ‘fieldlike’. This construction turns out to be very concrete example of the Abstract Elementary Classes which arose in Shelah’s analysis. This paper examines the intertwining of these three themes. The study of (C, +, ·, exp) leads one immediately to some extension of first order logic; the integers with all their arithmetic are first order definable in (C, +, ·, exp). Thus, the first order theory of complex exponentiation is horribly complicated; it is certainly unstable and so can’t be first order categorical. One solution is to use infinitary logic to pin down the pathology. Insist that the kernel of the exponential map is fixed as a single copy of the integers while allowing the rest of the structure to grow. We describe in Section 5 Zilber’s program to