The complexity of the embeddability relation between torsion-free abelian groups of uncountable size

Abstract

We prove that for every uncountable cardinal $\kappa$ such that ${\kappa }^{<\kappa }=\kappa$, the quasi-order of embeddability on the $\kappa$-space of $\kappa$-sized graphs Borel reduces to the embeddability on the $\kappa$-space of $\kappa$-sized torsion-free abelian groups. Then we use the same techniques to prove that the former Borel reduces to the embeddability on the $\kappa$-space of $\kappa$-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 ${\mathbf{\Sigma }}_1^1$ quasi-orders.

14 pages, final version

References in corpus (4)

• Generalized Descriptive Set Theory and Classification Theory
• An "almost" full embedding of the category of graphs into the category of groups
• An almost full embedding of the category of graphs into the category of abelian groups
• An axiomatic construction of an almost full embedding of the category of graphs into the category of R-objects

Cited by in corpus (1)

• Classification problems from the descriptive set theoretical perspective