  We define an infinite graded graph of ordered pairs and a~canonical action of
the group $\mathbb{Z}$ (the adic action) and of the infinite sum of groups of
order two~$\mathcal{D}=\sum_1^{\infty} \mathbb{Z}/2\mathbb{Z}$ on the path
space of the graph. It is proved that these actions are universal for both
groups in the following sense: every ergodic action of these groups with
invariant measure and binomial generator, multiplied by a~special action (the
`odometer'), is metrically isomorphic to the canonical adic action on the path
space of the graph with a~central measure. We consider a~series of related
problems.
