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We conclude this paper with an example of two mixed abelian gruops of torsion free rank one with the same torsion parts and the same height matrices which are not lattice isomorphic.
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Mahdavi, K., Poland, J. On lattice isomorphic of mixed abelian groups. Arch. Math 60, 327–329 (1993). https://doi.org/10.1007/BF01207186
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DOI: https://doi.org/10.1007/BF01207186