  A new integrable discrete system is constructed and studied, based on the
algebraization of the difference operator. The model is named the discrete
generalized nonlinear Schrodinger (GNLS) equation for which can be reduced to
classical discrete nonlinear Schrodinger (NLS) equation. To show the complete
integrability of the discrete GNLS equation, the recursion operator, symmetries
and conservation quantities are obtained. Furthermore, all of reductions for
the discrete GNLS equation are given and the discrete NLS equation is obtained
by one of the reductions. At the same time, the recursion operator and
symmetries of continuous GNLS equation are successfully recovered by its
corresponding discrete ones.
