  A discrete multidimensional system is the set of solutions to a system of
linear partial difference equations defined on the lattice $\Z^n$. This paper
shows that it is determined by a unique coarsest sublattice, in the sense that
the solutions of the system on this sublattice determine the solutions on
$\Z^n$; it is therefore the correct domain of definition of the discrete
system. In turn, the defining sublattice is determined by a Galois group of
symmetries that leave invariant the equations defining the system. These
results find application in understanding properties of the system such as
controllability and autonomy, and in its order reduction.
