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On the fine structure of the polygroup blow-up

Itay Ben-Yaacov

Archive for Mathematical Logic 42 (7):649-663 (2003)

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The group configuration in simple theories and its applications.Itay Ben-Yaacov, Ivan Tomašić & Frank O. Wagner - 2002 - Bulletin of Symbolic Logic 8 (2):283-298.details In recent work, the authors have established the group configuration theorem for simple theories, as well as some of its main applications from geometric stability theory, such as the binding group theorem, or in the $\omega$-categorical case, the characterization of the forking geometry of a finitely based non-trivial locally modular regular type as projective geometry over a finite field and the equivalence of pseudolinearity and local modularity. The proof necessitated an extension of the model-theoretic framework to include almost hyperimaginaries, and (...) the study of polygroups. (shrink) Download Export citation Bookmark 3 citations
Simple almost hyperdefinable groups.Itaï Ben-Yaacov - 2006 - Journal of Mathematical Logic 6 (01):69-88.details We lay down the groundwork for the treatment of almost hyperdefinable groups: notions from [5] are put into a natural hierarchy, and new notions, essential to the study to such groups, fit elegantly into this hierarchy. We show that "classical" properties of definable and hyperdefinable groups in simple theories can be generalised to this context. In particular, we prove the existence of stabilisers of Lascar strong types and of the connected and locally connected components of subgroups, and that in a (...) Download Export citation Bookmark
On almost orthogonality in simple theories.Itay Ben-Yaacov & Frank O. Wagner - 2004 - Journal of Symbolic Logic 69 (2):398-408.details Abstract.1. We show that ifpis a real type which is internal in a set Σ of partial types in a simple theory, then there is a typep′ interbounded withp, which is finitely generated over Σ, and possesses a fundamental system of solutions relative to Σ.2. Ifpis a possibly hyperimaginary Lascar strong type, almost Σ-internal, but almost orthogonal to Σω, then there is a canonical non-trivial almost hyperdefinable polygroup which multi-acts onpwhile fixing Σ generically In casepis Σ-internal andTis stable, this is (...) the binding group ofpover Σ. (shrink) Download Export citation Bookmark 1 citation
Compactness and independence in non first order frameworks.Itay Ben-Yaacov - 2005 - Bulletin of Symbolic Logic 11 (1):28-50.details This communication deals with positive model theory, a non first order model theoretic setting which preserves compactness at the cost of giving up negation. Positive model theory deals transparently with hyperimaginaries, and accommodates various analytic structures which defy direct first order treatment. We describe the development of simplicity theory in this setting, and an application to the lovely pairs of models of simple theories without the weak non finite cover property. Download Export citation Bookmark 2 citations
Constructing an almost hyperdefinable group.Itay Ben-Yaacov, Ivan Tomašić & Frank O. Wagner - 2004 - Journal of Mathematical Logic 4 (02):181-212.details This paper completes the proof of the group configuration theorem for simple theories started in [1]. We introduce the notion of an almost hyperdefinable structure, and show that it has a reasonable model theory. We then construct an almost hyperdefinable group from a polygroup chunk. Download Export citation Bookmark 6 citations
On Almost Orthogonality in Simple Theories.Itay Ben-Yaacov & Frank O. Wagner - 2004 - Journal of Symbolic Logic 69 (2):398 - 408.details 1. We show that if p is a real type which is internal in a set $\sigma$ of partial types in a simple theory, then there is a type p' interbounded with p, which is finitely generated over $\sigma$ , and possesses a fundamental system of solutions relative to $\sigma$ . 2. If p is a possibly hyperimaginary Lascar strong type, almost \sigma-internal$ , but almost orthogonal to $\sigma^{\omega}$ , then there is a canonical non-trivial almost hyperdefinable polygroup which multi-acts (...) Download Export citation Bookmark 2 citations

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