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

Gianluca Paolini; Saharon Shelah

doi:10.1007/s11425-024-2401-0

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

Gianluca Paolini^*
, Saharon Shelah

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

In Paolini and Shelah (2024), we proved that the space of countable torsion-free abelian groups is Borel complete. In this paper, we show that our construction from Paolini and Shelah (2024) 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 -interpret countable graphs in them. Secondly, we show that the relation of pure embeddability (i.e., elementary embeddability) among countable models of Th(ℤ^(ω)) is a complete analytic quasi-order.

Original language	English
Pages (from-to)	2809-2814
Number of pages	6
Journal	Science China Mathematics
Volume	68
Issue number	12
DOIs	https://doi.org/10.1007/s11425-024-2401-0
State	Published - Dec 2025

Bibliographical note

Publisher Copyright:
© Science China Press 2025.

Keywords

03E15
20K20
complete analytic quasi-order
faithful Borel completeness
torsion-free abelian groups

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10.1007/s11425-024-2401-0

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abstract = "In Paolini and Shelah (2024), we proved that the space of countable torsion-free abelian groups is Borel complete. In this paper, we show that our construction from Paolini and Shelah (2024) 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 -interpret countable graphs in them. Secondly, we show that the relation of pure embeddability (i.e., elementary embeddability) among countable models of Th(ℤ(ω)) is a complete analytic quasi-order.",

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T1 - Torsion-free abelian groups are faithfully Borel complete and pure embeddability is a complete analytic quasi-order

AU - Paolini, Gianluca

AU - Shelah, Saharon

N1 - Publisher Copyright: © Science China Press 2025.

PY - 2025/12

Y1 - 2025/12

N2 - In Paolini and Shelah (2024), we proved that the space of countable torsion-free abelian groups is Borel complete. In this paper, we show that our construction from Paolini and Shelah (2024) 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 -interpret countable graphs in them. Secondly, we show that the relation of pure embeddability (i.e., elementary embeddability) among countable models of Th(ℤ(ω)) is a complete analytic quasi-order.

AB - In Paolini and Shelah (2024), we proved that the space of countable torsion-free abelian groups is Borel complete. In this paper, we show that our construction from Paolini and Shelah (2024) 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 -interpret countable graphs in them. Secondly, we show that the relation of pure embeddability (i.e., elementary embeddability) among countable models of Th(ℤ(ω)) is a complete analytic quasi-order.

KW - 03E15

KW - 20K20

KW - complete analytic quasi-order

KW - faithful Borel completeness

KW - torsion-free abelian groups

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SN - 1674-7283

VL - 68

SP - 2809

EP - 2814

JO - Science China Mathematics

JF - Science China Mathematics

IS - 12

ER -

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

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Keywords

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