Strong subgroup chains and the Baer-Specker group

TL;DR

This paper investigates the structural properties of the Baer-Specker group, focusing on subgroup chains, cotorsion-free groups, and their model-theoretic classification, revealing new insights into their algebraic and logical characteristics.

Contribution

It introduces new results on the non-elementary properties preserved under C-filtrations and characterizes the Baer-Specker group within an abstract elementary class framework.

Findings

The Baer-Specker group is not a union of proper subgroups with cotorsionfree quotients.

Cotorsion-free groups form an abstract elementary class (AEC).

Kaplansky invariants help determine the stability spectrum of related quotient groups.

Abstract

Examples are given of non-elementary properties that are preserved under C-filtrations for various classes C of Abelian groups. The Baer-Specker group is never the union of a chain of proper subgroups with cotorsionfree quotients. Cotorsion-free groups form an abstract elementary class (AEC). The Kaplansky invariants of the Baer-Specker group are used to determine the AECs defined by the perps of the Baer-Specker quotient groups that are obtained by factoring the Baer-Specker group B of a ZFC extension by the Baer-Specker group A of the ground model, under various hypotheses, yielding information about its stability spectrum.