The Grothendieck ring of a non-divisible ordered abelian group is trivial

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.

Theorems & Definitions (5)

• Theorem 1
• Lemma 2
• proof
• proof : Proof of Theorem \ref{['th:KG_triv']}
• Remark 1