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

TL;DR

This paper proves that the Grothendieck ring of definable sets in non-divisible ordered abelian groups is trivial, extending known results and showing all such rings collapse to zero unless the group is divisible.

Contribution

It provides a short computation demonstrating that the Grothendieck ring is trivial for all non-divisible ordered abelian groups, filling a gap in the understanding of these structures.

Findings

Grothendieck ring of non-divisible ordered abelian groups is zero

Known cases: $ ext{K} bQ eq 0$, $ ext{K} bZ=0

All non-divisible cases collapse to zero

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.