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arXiv · 2608.17861

A note on definable endomorphisms of ordered abelian groups

Haitao Ji·Fengshuo Xu·Dong Qiu
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Abstract

We answer Kourovka Notebook Problem~18.16 affirmatively in the pure language of ordered abelian groups, with parameters allowed. The piecewise-affine description of definable functions reduces the question to an algebraic rigidity theorem: an additive endomorphism of a torsion-free abelian group that is covered by finitely many rational-affine laws has one global rational slope. The parameter-free pure-language case is included, whereas the unrestricted expansion-language variant admits a simple counterexample.

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Haitao Ji, Fengshuo Xu, Dong Qiu. 2026-08-18. A note on definable endomorphisms of ordered abelian groups. https://arxiv.org/abs/2608.17861

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