Grothendieck ring of the pairing function without cycles

TL;DR

This paper computes the Grothendieck ring of a special class of cycle-free pairing functions, revealing a specific algebraic structure that generalizes to higher dimensions.

Contribution

It determines the Grothendieck ring for cycle-free pairing functions and extends the result to bijections between M and M^n.

Findings

Grothendieck ring is isomorphic to bZ^2 for the pairing function without cycles.

Generalization to bijections between M and M^n yields bZ[X]/(X - X^n).

Provides algebraic characterization of cycle-free pairing functions.

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 $K_0(M)=\mathbb{Z}[X]/(X-X^n)$.