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  1. A geometric introduction to forking and thorn-forking.Hans Adler - 2009 - Journal of Mathematical Logic 9 (1):1-20.
    A ternary relation [Formula: see text] between subsets of the big model of a complete first-order theory T is called an independence relation if it satisfies a certain set of axioms. The primary example is forking in a simple theory, but o-minimal theories are also known to have an interesting independence relation. Our approach in this paper is to treat independence relations as mathematical objects worth studying. The main application is a better understanding of thorn-forking, which turns out to be (...)
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  • Real closed rings II. model theory.Gregory Cherlin & Max A. Dickmann - 1983 - Annals of Pure and Applied Logic 25 (3):213-231.
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  • Characterizing Rosy Theories.Clifton Ealy & Alf Onshuus - 2007 - Journal of Symbolic Logic 72 (3):919 - 940.
    We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences of each of these conditions towards the rosiness of the theory. In particular we show that the existence of an ordinal valued equivalence relation rank is a (necessary and) sufficient condition for rosiness.
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  • Properties and Consequences of Thorn-Independence.Alf Onshuus - 2006 - Journal of Symbolic Logic 71 (1):1 - 21.
    We develop a new notion of independence (þ-independence, read "thorn"-independence) that arises from a family of ranks suggested by Scanlon (þ-ranks). We prove that in a large class of theories (including simple theories and o-minimal theories) this notion has many of the properties needed for an adequate geometric structure. We prove that þ-independence agrees with the usual independence notions in stable, supersimple and o-minimal theories. Furthermore, we give some evidence that the equivalence between forking and þ-forking in simple theories might (...)
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  • T-Convexity and Tame Extensions.Dries Lou Van Den & H. Lewenberg Adam - 1995 - Journal of Symbolic Logic 60 (1):74 - 102.
    Let T be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of T and show that the residue field of such a convex hull has a natural expansion to a model of T. We give a quantifier elimination relative to T for the theory of pairs (R, V) where $\mathscr{R} \models T$ and V ≠ R is the convex hull of an elementary substructure of R. We deduce (...)
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  • T-Convexity and Tame Extensions II.Lou Van Den Dries - 1997 - Journal of Symbolic Logic 62 (1):14 - 34.
    I solve here some problems left open in “T-convexity and Tame Extensions” [9]. Familiarity with [9] is assumed, and I will freely use its notations. In particular,Twill denote a completeo-minimal theory extending RCF, the theory of real closed fields. Let (,V) ⊨Tconvex, let=V/m(V)be the residue field, with residue class mapx↦:V↦, and let υ:→ Γ be the associated valuation. “Definable” will mean “definable with parameters”.The main goal of this article is to determine the structure induced by(,V)on its residue fieldand on its (...)
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  • T-convexity and tame extensions II.Lou van den Dries - 1997 - Journal of Symbolic Logic 62 (1):14-34.
    I solve here some problems left open in “T-convexity and Tame Extensions” [9]. Familiarity with [9] is assumed, and I will freely use its notations. In particular,Twill denote a completeo-minimal theory extending RCF, the theory of real closed fields. Let (,V) ⊨Tconvex, let=V/m(V)be the residue field, with residue class mapx↦:V↦, and let υ:→ Γ be the associated valuation. “Definable” will mean “definable with parameters”.The main goal of this article is to determine the structure induced by(,V)on its residue fieldand on its (...)
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  • Forking in VC-minimal theories.Sarah Cotter & Sergei Starchenko - 2012 - Journal of Symbolic Logic 77 (4):1257-1271.
    We consider VC-minimal theories admitting unpackable generating families, and show that in such theories, forking of formulae over a model M is equivalent to containment in global types definable over M, generalizing a result of Dolich on o-minimal theories in [4].
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  • Imaginaries in real closed valued fields.Timothy Mellor - 2006 - Annals of Pure and Applied Logic 139 (1):230-279.
    The paper shows elimination of imaginaries for real closed valued fields to suitable sorts. We also show that this result is in some sense optimal. The paper includes a quantifier elimination theorem for real closed valued fields in a language with sorts for the field, value group and residue field.
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  • Simple unstable theories.Saharon Shelah - 1980 - Annals of Mathematical Logic 19 (3):177.
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  • Unexpected imaginaries in valued fields with analytic structure.Deirdre Haskell, Ehud Hrushovski & Dugald Macpherson - 2013 - Journal of Symbolic Logic 78 (2):523-542.
    We give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the ‘geometric' sorts which suffice to code all imaginaries in the corresponding algebraic setting.
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  • $t$-convexity And Tame Extensions.Lou van den Dries & Adam H. Lewenberg - 1995 - Journal of Symbolic Logic 60 (1):74-102.
    Let $T$ be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of $T$ and show that the residue field of such a convex hull has a natural expansion to a model of $T$. We give a quantifier elimination relative to $T$ for the theory of pairs $$ where $\mathscr{R} \models T$ and $V \neq \mathscr{R}$ is the convex hull of an elementary substructure of $\mathscr{R}$. We deduce that (...)
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