Completely decomposable pure subgroups of completely decomposable Abelian groups (Q1891518)

The authors investigate conditions under which pure subgroups of completely decomposable groups are again completely decomposable. For a pure subgroup \(A\) of a completely decomposable group \(B\), the following are equivalent: (a) \(A\) is completely decomposable; (b) there is an \(\aleph_0\)-prebalanced chain \(\{A_\sigma\}\) from \(A\) to \(B\) with countable factors such that, for each member \(A_\sigma\) of the chain, every countable subgroup of \(A_\sigma\) can be embedded in a countable completely decomposable pure subgroup of \(A_\sigma\); (c) there is a continuous well-ordered ascending chain of pure subgroups \(\{A_\sigma\}\) from \(A\) to \(B\) with countable factors such that each member \(A_\sigma\) is completely decomposable; (d) there is a \(G(\aleph_0)\)-family of completely decomposable pure subgroups of \(B\) over \(A\) (Th. 6).