Direct decompositions of Abelian groups without isomorphic almost refinements (Q1109146)

Using a well known construction of \textit{A. L. S. Corner}'s [Proc. Camb. Philos. Soc. 57, 230-233 (1961; Zbl 0100.029)] an example of two direct decompositions of an (abelian) torsion-free group, \(K=\oplus G_ n=\oplus H_ n\) is given, which does not admit isomorphic almost refinements (i.e. any two direct decompositions \(K=\oplus X_ k=\oplus Y_ k\), where \(X_ k\) (resp. \(Y_ k)\) are finite direct sums of \(G_ n's\) (resp. \(H_ n's)\) admit no isomorphic refinements). All \(G_ n\), \(H_ n\) are indecomposable, \(G_ 0\), \(H_ 0\) have countable rank and all remaining \(G_ n\), \(H_ n\) have rank 2.