Recursion operators, conservation laws and integrability conditions for difference equations

In this paper we make an attempt to give a consistent background and definitions suitable for the theory of integrable difference equations. We adapt a concept of recursion operator to difference equations and show that it generates an infinite sequence of symmetries and canonical conservation laws for a difference equation. Similar to the case of partial differential equations these canonical densities can serve as integrability conditions for difference equations. We have found the recursion operators for the Viallet and all ABS equations.