Consistency on Cubic Lattices for Determinants of Arbitrary Orders

We consider a special class of two-dimensional discrete equations defined by relations on elementary $N\times N$ squares, $N>2$, of the square lattice $\mathbb Z^2$, and propose a new type of consistency conditions on cubic lattices for such discrete equations that is connected to bending elementary $N\times N$ squares, $N>2$, in the cubic lattice $\mathbb Z^3$. For an arbitrary $N$ we prove such consistency on cubic lattices for two-dimensional discrete equations defined by the condition that the determinants of values of the field at the points of the square lattice $\mathbb Z^2$ that are contained in elementary $N\times N$ squares vanish.