Factorization of Darboux–Laplace transformations for discrete hyperbolic operators

We classify elementary Darboux–Laplace transformations for semidiscrete and discrete second-order hyperbolic operators. We prove that there are two types of elementary Darboux–Laplace transformations in the $($semi$)$discrete case as in the continuous case: Darboux transformations constructed from a particular element in the kernel of the initial hyperbolic operator and classical Laplace transformations that are defined by the operator itself and are independent of the choice of an element in the kernel. We prove that on the level of equivalence classes in the discrete case, any Darboux–Laplace transformation is a composition of elementary transformations.