  The purpose of this article is to develop an algebraic approach to the
problem of integrable classification of differential-difference equations with
one continuous and two discrete variables. As a classification criterion, we
put forward the following hypothesis. Any integrable equation of the type under
consideration admits an infinite sequence of finite-field Darboux-integrable
reductions. The property of Darboux integrability of a finite-field system is
formalized as finite-dimensionality condition of its characteristic
Lie-Rinehart algebras. That allows one to derive effective integrability
conditions in the form of differential equations on the right hand side of the
equation under study. To test the hypothesis, we use known integrable equations
from the class under consideration. In this article, we show that all known
examples do have this property.
