Higher Hirota difference equations and their reductions

We previously proposed an approach for constructing integrable equations based on the dynamics in associative algebras given by commutator relations. In the framework of this approach, evolution equations determined by commutators of (or similarity transformations with) functions of the same operator are compatible by construction. Linear equations consequently arise, giving a base for constructing nonlinear integrable equations together with the corresponding Lax pairs using a special dressing procedure. We propose an extension of this approach based on introducing higher analogues of the famous Hirota difference equation. We also consider some $(1+1)$-dimensional discrete integrable equations that arise as reductions of either the Hirota difference equation itself or a higher equation in its hierarchy.