The coarsest lattice that determines a discrete multidimensional system

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