Approximations for Theories of Abelian Groups

Approximations of syntactic and semantic objects play an important role in various elds of mathematics. They can create theories and structures in one given class by means of others, usually simpler. For instance, in certain situations, innite objects can be approximated by nite or strongly minimal ones. Thus, complicated objects can be collected using simplied ones. Among these objects, Abelian groups, their rst order theories, connections and dynamics are of interests. Theories of Abelian groups are characterized by Szmielew invariants leading to the study and descriptions of approximations in terms of these invariants. In the paper we apply a general approach for approximating theories to the class of theories of Abelian groups which characterizes the approximability of a theory of Abelian groups by a given family of theories of Abelian groups in terms of Szmielew invariants and their limits. We describe some forms of approximations for theories of Abelian groups. In particular, approximations of theories of Abelian groups by theories of nite ones are characterized. In addition, we describe approximations by quasi-cyclic and torsion-free Abelian groups and their combinations with respect to given families of prime numbers. Approximations and closures of families of theories with respect to standard Abelian groups for various sets of prime numbers are also described.