Families of theories of abelian groups and their closures

Abstract

In studying the structural properties of elementary theories, a relationship between theories with respect to a series of natural operators plays an important role. This relationship can be determined by placing models of given theories in various formula definable sets. Such sets include, for example, sets defined by unary predicates or equivalence relations. In this way, P-operators and E-operators arise, as well as their closures, and e-spectra, i.e. the numbers of new theories that may be generated by these operators. For E-operators, applicable to the families of theories of abelian groups, closures and generating sets, as well as their e-spectra are described. Szmielew invariants are used as a tool for the established characterization of a theory belonging to the E-closure of a family of theories of abelian groups. A series of families of theories corresponding to the sets of Szmielew invariants, properties of these families, and values of e-spectra are also described.