Research Information System
JOURNAL OF SYMBOLIC LOGIC, vol.79, no.4, pp.1120-1132, 2014 (SCI-Expanded)
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.