Right adjoints for the forgetful functors on $\lambda$-rings and bi-rings are applied to motivic measures and their zeta functions on the Grothendieck ring of $\mathbb{ F}_1$-varieties in the sense of Lorscheid and Lopez-Pena (torified schemes). This leads us to a specific subring of $\mathbb{ W}(\mathbb{ Z})$, properly containing Almkvist's ring $\mathbb{ W}_0(\mathbb{ Z})$, which might be a natural receptacle for all local factors of completed zeta functions.
Categories: math.AG math.QA math.RA math.RT