The Grothendieck ring of a non-divisible ordered abelian group is trivial
Blaise Boissonneau, Mathias Stout, Floris Vermeulen
https://arxiv.org/abs/2603.28483 https://arxiv.org/pdf/2603.28483 https://arxiv.org/html/2603.28483
arXiv:2603.28483v1 Announce Type: new
Abstract: We consider the model-theoretic Grothendieck ring of definable sets in ordered abelian groups. It is well-known that $\mathrm{K} \mathbb{Q} \cong \mathbb{Z}[T]/(T^2 T)$ and $\mathrm{K} \mathbb{Z} =0$, but surprisingly little is known about other cases. We present a short computation which shows that they all collapse: $\mathrm{K} G = 0$, unless $G$ is divisible.
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