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Preprint Number 406

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406. Cédric Milliet
Groupes fins
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Submission date: 29 December 2011.

Abstract:

We investigate some common points between stable and weakly small structures and define a structure M to be “fine” if the topological space S_\phi(dcl^{eq}(A)) has an ordinal Cantor-Bendixson rank for every formula phi and finite subset A of M. By definition, a theory is "fine" if every of its models is so. Weakly minimal, small, and stable structures are all examples of fine structures. For any of its finite subset A, a fine structure has local descending chain conditions on the algebraic closure acl(A) of A for subgroups uniformly definable over acl(A). An infinite field with fine theory has no additive or multiplicative proper subgroup of finite index, and no Artin-Schreier extension.

Résumé : Pour réunir à la fois les structures stables et menues, nous introduisons les structures fines et mettons en évidence plusieurs caractéristiques communes aux structures (faiblement) minimales, stables ou menues : une condition de chaîne uniforme et locale et une notion de presque phi-stabilisateur local. Nous en déduissons qu'un corps infini dont la théorie est fine n'a pas de sous-groupes additifs ni multiplicatifs d'indice fini, ni d'extension d'Artin-Schreier.

Mathematics Subject Classification: 03C45, 03C60

Keywords and phrases: model theory - Cantor-Bendixson rank - local descending chain conditions - Artin-Schreier extensions

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