Ranks for families of theories of abelian groups

The rank for families of theories is similar to Morley rank and can be considered as a measure for complexity or richness of these families. Increasing the rank by extensions of families we produce more rich families and obtaining families with the infinite rank that can be considered as “rich enough”. In the paper, we realize ranks for families of theories of abelian groups. In particular, we study ranks and closures for families of theories of finite abelian groups observing that the set of theories of finite abelian groups in not totally transcendental, i.e., its rank equals infinity. We characterize pseudofinite abelian groups in terms of Szmielew invariants. Besides we characterize $e$-minimal families of theories of abelian groups both in terms of dimension, i.e., the number of independent limits for Szmielew invariants, and in terms of inequalities for Szmielew invariants. These characterizations are obtained both for finite abelian groups and in general case. Furthermore we give characterizations for approximability of theories of abelian groups and show the possibility to count Szmielew invariants via these parameters for approximations. We describe possibilities to form $d$-definable families of theories of abelian groups having given countable rank and degree.