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arXiv · 2312.04162

Torsion-free abelian groups are faithfully Borel complete and pure embeddability is a complete analytic quasi-order

Gianluca Paolini·Saharon Shelah
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Abstract

In [9] we proved that the space of countable torsion-free abelian groups is Borel complete. In this paper we show that our construction from [9] satisfies several additional properties of interest. We deduce from this that countable torsion-free abelian groups are faithfully Borel complete, in fact, more strongly, we can $\mathfrak{L}_{ω_1, ω}$-interpret countable graphs in them. Secondly, we show that the relation of pure embeddability (equiv., elementary embeddability) among countable models of $\mathrm{Th}(\mathbb{Z}^{(ω)})$ is a complete analytic quasi-order.

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Gianluca Paolini, Saharon Shelah. 2026-01-26. Torsion-free abelian groups are faithfully Borel complete and pure embeddability is a complete analytic quasi-order. https://arxiv.org/abs/2312.04162

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