  We consider $k$-dimensional discrete-time systems of the form
$x_{n+1}=F(x_n,\ldots,x_{n-k+1})$ in which the map $F$ is continuous and
monotonic in each one of its arguments. We define a partial order on
$\mathbb{R}^{2k}_+$, compatible with the monotonicity of $F$, and then use it
to embed the $k$-dimensional system into a $2k$-dimensional system that is
monotonic with respect to this poset structure. An analogous construction is
given for periodic systems. Using the characteristics of the higher-dimensional
monotonic system, global stability results are obtained for the original
system. Our results apply to a large class of difference equations that are
pertinent in a variety of contexts. As an application of the developed theory,
we provide two examples that cover a wide class of difference equations, and in
a subsequent paper, we provide additional applications of general interest.
