  In this paper we work, first, with forced discrete-time mechanical systems on
the configuration space $Q$ and construct closed $2$-forms $\omega^+$ and
$\omega^-$ on $Q \times Q$, that are symplectic if and only if the system is
regular. For a special type of discrete force, we prove that $\omega^+$ and
$\omega^-$ are invariant by the flow of the system. We also consider the
Lagrangian reduction of a discrete mechanical system by a symmetry group (using
an affine discrete connection derived from the discrete momentum) and prove
that, under some conditions on the action, the trajectories of the reduced
system (with constant discrete momentum $\mu$) can be seen as trajectories of a
forced discrete mechanical system, where the discrete force is of the type
analyzed before. Therefore, we prove that there is a symplectic structure that
is invariant by the flow of the forced reduced system; the symplectic structure
can be seen as a pullback of a canonical cotangent structure plus a magnetic
term. This discrete reduction process is the (discrete) Routh reduction and the
behavior obtained runs parallel to the well known case for (continuous) Routh
reduction.
