Integrable non-commutative equations on quad-graphs. The consistency approach

We extend integrable systems on quad-graphs, such as the Hirota equation and the cross-ratio equation, to the non-commutative context, when the fields take values in an arbitrary associative algebra. We demonstrate that the three-dimensional consistency property remains valid in this case. We derive the non-commutative zero curvature representations for these systems, based on the latter property. Quantum systems with their quantum zero curvature representations are particular cases of the general non-commutative ones.