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54
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 127 (8 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 23 (15 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 21 (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.
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 20 (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.
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 15 (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.
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 t ..."
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Cited by 13 (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
Classification from a computable viewpoint
 The Bulletin of Symbolic Logic
"... Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism ..."
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Cited by 9 (0 self)
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Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism
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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. 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...
Trivial, strongly minimal theories are model complete after naming constants
 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
, 2002
"... We prove that if M is any model of a trivial, strongly minimal theory, then the elementary diagram Th(MM) is a model complete LMtheory. We conclude that all countable models of a trivial, strongly minimal theory with at least one computable model are 0 ′ ′decidable, and that the spectrum of compu ..."
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We prove that if M is any model of a trivial, strongly minimal theory, then the elementary diagram Th(MM) is a model complete LMtheory. We conclude that all countable models of a trivial, strongly minimal theory with at least one computable model are 0 ′ ′decidable, and that the spectrum of computable models of any trivial, strongly minimal theory is Σ 0 5.