GROUPES FINS


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Milliet C.

JOURNAL OF SYMBOLIC LOGIC, vol.79, no.4, pp.1120-1132, 2014 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 79 Issue: 4
  • Publication Date: 2014
  • Doi Number: 10.1017/jsl.2014.12
  • Title of Journal : JOURNAL OF SYMBOLIC LOGIC
  • Page Numbers: pp.1120-1132

Abstract

We investigate some common points between stable structures and weakly small structures and define a structure M to be fine if the Cantor-Bendixson rank of the topological space S-phi(dcl(eq) (A)) is an ordinal for every finite subset A of M and every formula phi(x, y) where x is of arity 1. By definition, a theory is fine if all its models are so. Stable theories and small theories are fine, and weakly minimal structures are fine. For any finite subset A of a fine group G, the traces on the algebraic closure acl (A) of A of definable subgroups of G over acl (A) which are boolean combinations of instances of an arbitrary fixed formula can decrease only finitely many times. An infinite field with a fine theory has no additive nor multiplicative proper definable subgroups of finite index, nor Artin-Schreier extensions.