On consistency of determinants on cubic lattices

Consider the square lattice Z with vertices at points with integer-valued coordinates in R = {(x1, x2)| xk ∈ R, k = 1, 2} and complex (or real) scalar fields u on the lattice Z, u : Z → C, that are defined by their values ui1i2 , ui1i2 ∈ C, at each vertex of the lattice with the coordinates (i1, i2), ik ∈ Z, k = 1, 2. Consider a class of two-dimensional discrete equations on the lattice Z for the field u that are defined by functions Q(x1, x2, x3, x4) of four variables with the help of the relations