The structure of 𝜔₁-separable groups

Eklof, Paul C.

doi:10.1090/S0002-9947-1983-0709565-8

The structure of $\omega _{1}$-separable groups
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by Paul C. Eklof

Trans. Amer. Math. Soc. 279 (1983), 497-523

DOI: https://doi.org/10.1090/S0002-9947-1983-0709565-8

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Abstract:

A classification theorem is proved for ${\omega _1}$-separable ${\omega _1}$-free abelian groups of cardinality ${\omega _1}$ assuming Martin’s Axiom $(\text {MA})$ and ${2^{\aleph _0}} > {\aleph _1}$. As a consequence, several structural results about direct sum decompositions of ${\omega _1}$-separable groups are proved. These results are proved independent of $\text {ZFC}$, and, in addition, another structural property is proved undecidable in ${\text {ZFC}} + {\text {MA}} + {2^{\aleph _0}} > {\aleph _1}$. The problem of classifying these groups in a model of ${2^{\aleph _0}} = {\aleph _1}$ is also investigated.

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