  A higher order difference equation may be generally defined in an arbitrary
nonempty set S as: \[
f_{n}(x_{n},x_{n-1},...,x_{n-k})=g_{n}(x_{n},x_{n-1},...,x_{n-k}) \] where
$f_{n},g_{n} :S^{k+1}\rightarrow S$ are given functions for $n=1,2,... $ and
$k$ is a positive integer. We present conditions that imply the above equation
can be factored into an equivalent pair of lower order difference equations
using possible form symmetries (order-reducing changes of variables). These
results extend and generalize semiconjugate factorizations of recursive
difference equations on groups. We apply some of this theory to obtain new
factorization results for the important class of quadratic difference equations
on algebraic fields. We also discuss the nontrivial issue of the existence of
solutions for quadratic equations.
