Abstract: We define discrete total differential forms on lattice space by changing coefficients of discrete differential forms from functions only of n to functions also of dependent variables un and their partial differences. And the discrete exterior derivative extends to be discrete total differential map which is also nilpotent. Then a discrete horizontal complex can be derived and be proved to be exact by constructing homotopy operators.

Key words: discrete horizontal complex, noncommutative differential calculus, discrete higher Euler operator, homotopy operator