On algebraic and definable closures for theories of abelian groups

Classifying abelian groups and their elementary theories, a series of characteristics arises that describe certain features of the objects under consideration. Among these characteristics, an important role is played by Szmielew invariants, which define the possibilities of divisibility of elements, orders of elements, dimension of subgroups, and allow describing given abelian groups up to elementary equivalence. Thus, in terms of Szmielew invariants, the syntactic properties of Abelian groups are represented, i.e. properties that depend only on their elementary theories. The work, based on Szmielew invariants, provides a description of the behavior of algebraic and definable closure operators based on two characteristics: degrees of algebraization and the difference between algebraic and definable closures. Thus, possibilities for algebraic and definable closures, adapted to theories of Abelian groups, are studied and described. A theorem on trichotomy for degrees of algebraization is proved: either this degree is minimal, if in the standard models, except for the only two-element group, there are no positively finitely many cyclic and quasi-cyclic parts, or the degree is positive and natural, if in a standard model there are no positively finitely many cyclic and quasi-cyclic parts, except a unique copy of a two-element group and some finite direct sum of finite cyclic parts, and the degree is infinite if the standard model contains unboundedly many non-isomorphic finite cyclic parts or positively finitely many of copies of quasi-finite parts. In addition, a dichotomy of the values of the difference between algebraic closures and definable closures for abelian groups defined by Szmielew invariants for cyclic parts is established. In particular, it is shown that torsion-free abelian groups are quasi-Urbanik.