Grothendieck Ring of the Pairing Function without Cycles

Abstract

A bijection $(f_1,f_2)$ 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 $\mathbb{Z}[X]∕(X-X^2)\simeq {\mathbb{Z}}^2$.