  We study the structure algebra $\mathcal{Z}$ of the stable moment graph for
the case of the affine root system $A_{1}$. The structure algebra $\mathcal{Z}$
is an algebra over a symmetric algebra and in particular, it is a module over a
symmetric algebra. We study this module structure on $\mathcal{Z}$ and we
construct a basis. By "setting $c$ equal to zero" in $\mathcal{Z}$, we obtain
the module $\mathcal{Z}_{c=0}$. This module can be described in terms of the
finite root system $A_{1}$ and we show that it is determined by a set of
certain divisibility relations. These relations can be regarded as a
generalization of ordinary moment graph relations that define sections of
sheaves on moment graphs, and because of this we call them higher-order
congruence relations.
