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28
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 19 (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
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 14 (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
SUPERSIMPLE FIELDS AND DIVISION RINGS
 MATHEMATICAL RESEARCH LETTERS
, 1998
"... It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types in groups and fields whose theory is simple. ..."
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Cited by 13 (2 self)
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It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types in groups and fields whose theory is simple.
Simplicity in compact abstract theories
 J. Math. Log
, 2003
"... Abstract. We continue [Ben03], developing simplicity in the framework of compact abstract theories. Due to the generality of the context we need to introduce definitions which differ somewhat from the ones use in first order theories. With these modified tools we obtain more or less classical behavi ..."
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Cited by 11 (2 self)
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Abstract. We continue [Ben03], developing simplicity in the framework of compact abstract theories. Due to the generality of the context we need to introduce definitions which differ somewhat from the ones use in first order theories. With these modified tools we obtain more or less classical behaviour: simplicity is characterised by the existence of a certain notion of independence, stability is characterised by simplicity and bounded multiplicity, and hyperimaginary canonical bases exist.
From Stability To Simplicity
 Bulletin of Symbolic Logic 4
, 1998
"... this report we wish to describe recent work on a class of first order theories first introduced by Shelah in [32], the simple theories. Major progress was made in the first author's doctoral thesis [17]. We will give a survey of this, as well as further works by the authors and others. ..."
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Cited by 10 (2 self)
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this report we wish to describe recent work on a class of first order theories first introduced by Shelah in [32], the simple theories. Major progress was made in the first author's doctoral thesis [17]. We will give a survey of this, as well as further works by the authors and others.
Prospects for mathematical logic in the twentyfirst century
 BULLETIN OF SYMBOLIC LOGIC
, 2002
"... The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently. ..."
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Cited by 9 (0 self)
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The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
Simplicity, And Stability In There
 JOURNAL OF SYMBOLIC LOGIC
, 1999
"... Firstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover definability part of stability theory to simplicity theory. In particular, using elimination of hyperimaginaries we prove that for any supersimple T , canonical base of ..."
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Firstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover definability part of stability theory to simplicity theory. In particular, using elimination of hyperimaginaries we prove that for any supersimple T , canonical base of an amalgamation class P is the union of names of /definitions of P , / ranging over stationary Lformulas in P . Also, we prove that the same is true with stable formulas for an 1based theory having elimination of hyperimaginaries. For such a theory, the stable forking property holds, too.
Uncountable dense categoricity in cats
 Journal of Symbolic Logic
"... theory) is metric, and develop some of the theory of metric cats. We generalise Morley’s theorem: if a countable Hausdorff cat T has a unique complete model of density character λ ≥ ω1, then it has a unique complete model of density character λ for every λ ≥ ω1. ..."
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Cited by 7 (2 self)
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theory) is metric, and develop some of the theory of metric cats. We generalise Morley’s theorem: if a countable Hausdorff cat T has a unique complete model of density character λ ≥ ω1, then it has a unique complete model of density character λ for every λ ≥ ω1.
Some modeltheoretic and geometric properties of fields with jet operators
, 2001
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Dimension and measure in finite first order structures
, 2005
"... The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the works of others. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without pro ..."
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Cited by 2 (2 self)
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The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the works of others. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement. 2Acknowledgments The research undertaken in Chapter 6 was performed jointly by myself and Mark Ryten. My contribution to the work was throughout: in the formation of the initial strategy, in the development of the methods, in the exposition of the proofs, and in their correction and finetuning. Many thanks to my PhDsupervisor Dugald Macpherson for his guidance, assistance, and support. Thanks also to the University of Leeds and especially the staff of the School of Mathematics for providing me with the facilities and support to undertake this research. Thanks to Agatha WalczakTypke for her help with LaTeX.