不可数无挠的阿贝尔群之间可嵌入关系的复杂性
The complexity of the embeddability relation between torsion-free abelian groups of uncountable size
摘要：
证明了对每个不可数基数$κ$，使得$κ^{<κ}=κ$，$κ$ -大小的图Borel在$κ$ -空间上的可嵌入性的拟阶约化为$κ$ -大小的无挠的阿贝尔群在$κ$ -空间上的可嵌入性.然后我们用同样的技巧证明了对于基数小于连续统的$\mathbb{S}$ -无余挠环$R$，前Borel约化为$κ$-模在$κ$ -空间上的可嵌入性.由此我们得到所有的前序都是完备的$\boldsymbolΣ^1_1$拟序。
英文摘要：
We prove that for every uncountable cardinal $κ$ such that $κ^{<κ}=κ$, the quasi-order of embeddability on the $κ$-space of $κ$-sized graphs Borel reduces to the embeddability on the $κ$-space of $κ$-sized torsion-free abelian groups. Then we use the same techniques to prove that the former Borel reduces to the embeddability on the $κ$-space of $κ$-sized $R$-modules, for every $\mathbb{S}$-cotorsion-free ring $R$ of cardinality less than the continuum. As a consequence we get that all the previous are complete $\boldsymbolΣ^1_1$ quasi-orders.
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