Generalized stability of torsion-free Abelian groups

Abstract

Stable theories were introduced in [1] for constructing classification theory and are a generalization of the concept of a totally transcendental theory as defined in [2]. In [3], it was proved that the property of being stable for a theory is equivalent to being definable for every complete type of the theory. This property plays a fundamental part in research on stable theories. In [4], the notion of E∗-stability (generalized stability) was introduced, and it was proved that types for E∗-stable theories are definable. A consequence of that result was stating, along with definability of types for stable theories (see [3]), that types over any P -sets in P -stable theories likewise are definable (which had been established in [5] for types over P -models). The notion of E∗-stability is a new stability scale, whose basic parameter is a mapping of types of a complete theory into types of another theory. An interesting example of E∗-stability is (P, a)-stability, defined by adding to a language a unary predicate symbol and adding to types the condition of being algebraically closed for that predicate. In this paper, we work to prove a (P, a)-stability theorem for theories of torsion-free Abelian groups. In so doing, use is made of quantifier elimination down to positive primitive formulas in Abelian groups with a predicate distinguishing a subgroup. By virtue of this fact, the question of being (P, a)-stable for a theory reduces to asking if a system of linear equations with integer coefficients has a solution in the algebraic closure of constants involved in the system. To ∗Supported by RFBR (project No. 09-01-00336-a).