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  1.  3
    Some Model-Theoretic Results in the Algebraic Theory of Quadratic Forms.Vincent Astier - 2001 - Annals of Pure and Applied Logic 112 (2-3):189-223.
    This paper studies some model-theoretic properties of special groups of finite type. Special groups are a first-order axiomatization of the algebraic theory of quadratic forms, introduced by Dickmann and Miraglia, which is essentially equivalent to abstract Witt rings. More precisely, we consider elementary equivalence, saturation, elementary embeddings, quantifier elimination, stability and Morley rank.
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  2.  67
    Axiomatization of Local-Global Principles for Pp-Formulas in Spaces of Orderings.Vincent Astier & Marcus Tressl - 2004 - Archive for Mathematical Logic 44 (1):77-95.
    .We use a model theoretic approach to investigate properties of local-global principles for positive primitive formulas in spaces of orderings, such as the existence of bounds and the axiomatizability of local-global principles. As a consequence we obtain various classes of special groups satisfying local-global principles for all positive primitive formulas, and we show that local-global principles are preserved by some natural constructions in special groups.
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  3.  11
    Special Groups Whose Isometry Relation Is a Finite Union of Cosets.Vincent Astier - 2008 - Journal of Symbolic Logic 73 (2):448 - 473.
    N₀-stable N₀-categorical linked quaternionic mappings are studied and are shown to correspond (in some sense) to special groups which are N₀-stable. N₀-categorical, satisfy AP(3) and have finite 2-symbol length. They are also related to special groups whose isometry relation is a finite union of cosets, which are then considered on their own, as well as their links with pseudofinite, profinite and weakly normal special groups.
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  4.  39
    Elementary Equivalence of Some Rings of Definable Functions.Vincent Astier - 2008 - Archive for Mathematical Logic 47 (4):327-340.
    We characterize elementary equivalences and inclusions between von Neumann regular real closed rings in terms of their boolean algebras of idempotents, and prove that their theories are always decidable. We then show that, under some hypotheses, the map sending an L-structure R to the L-structure of definable functions from R n to R preserves elementary inclusions and equivalences and gives a structure with a decidable theory whenever R is decidable. We briefly consider structures of definable functions satisfying an extra condition (...)
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  5.  36
    On Some Sheaves of Special Groups.Vincent Astier - 2007 - Archive for Mathematical Logic 46 (5-6):481-488.
    Using sheaves of special groups, we show that a general local-global principle holds for every reduced special group whose associated space of orderings only has a finite number of accumulation points. We also compute the behaviour of the Boolean hull functor applied to sheaves of special groups.
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