Grothendieck ring of the pairing function without cycles

Abstract

A bijection $(l,r)$ between $M^2$ and $M$ is said to be a pairing function with no cycles, if any composition of its coordinate functions has no fixed point. We compute here the Grothendieck ring of the pairing function without cycles to be isomorphic to $\mathbb{Z}^2\simeq \mathbb{Z}[X]/(X-X^2)$. More generally, for any $n\in \mathbb{N}^*$ and any bijetion without cycles betwen $M$ and $M^n$, the exact same method proves that $K_0(M)=\mathbb{Z}[X]/(X-X^n)$.