  It is well known that from two-dimensional lattice equations one can derive
one-dimensional lattice equations by imposing periodicity in some direction. In
this paper we generalize the periodicity condition by adding a symmetry
transformation and apply this idea to autonomous and non-autonomous lattice
equations. As results of this approach, we obtain new reductions of the
discrete potential Korteweg-de Vries equation, discrete modified Korteweg-de
Vries equation and the discrete Schwarzian Korteweg-de Vries equation. We will
also describe a direct method for obtaining Lax representations for the reduced
equations.
