The number of P-expansions of Abelian groups

Abstract

In [1], there is a theorem which says that under the assumption of the generalized continuum hypothesis, a theory of any Abelian group is P -superstable if P defines an elementary subsystem. There, also, for this type of subgroups, a list of possible P -spectra of complete theories T of Abelian groups is given on the assumption that T is superstable. In [2], it was proved that a theory of every torsion-free Abelian group is P -stable if P defines an algebraically closed subgroup. A complete description of Abelian groups whose theories are P -stable if P defines an algebraically closed subgroup is contained in [3]. Below P -spectra of Abelian groups will be completely described for subgroups P of the following types: pure subgroups, elementary subsystems, algebraically closed subgroups, and arbitrary subgroups. In particular, we generalize the above-mentioned results in [1, 2]. A substructure B of a structure A is said to be algebraically closed if B contains each finite set X ⊆ A definable in A by a formula Φ(x) with parameters in B. By L(X) we denote a language obtained from L by adding X as a set of new constants. For a complete theory T in a language L, T (X) denotes an arbitrary completion of T in a language L(X). We say that X is a set in a theory T , bearing in mind that some completion T (X) has been fixed. Let LP be a language obtained from L by adding a new unary predicate symbol P . Definition. Let T be a complete L-theory, Δ a set of LP -sentences, and X a set in T . Denote by CTΔ(X) the cardinality of a set of completions in a language (L(X))P of the set