Groupes fins

Cédric Milliet (1, 2)

1 ICJ - Institut Camille Jordan
2 Département de Mathématiques

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.

Mots clés: théorie des modèles; rang de Cantor-Bendixson; condition de chaîne locale; extension d'Artin-Schreier

Domaines: Logique [math.LO]

Cédric Milliet. Groupes fins. The Journal of Symbolic Logic, 2014, 79 (4), pp.1120-1132. ⟨10.1017/jsl.2014.12⟩. ⟨hal-00655411⟩