We present a global attractivity result
for maps generated by systems of autonomous difference equations.
It is assumed that the map of the system leaves invariant
a box, is monotone in a coordinate-wise sense
(but not necessarily monotone with respect to a standard cone),
and satisfies certain algebraic condition. It is shown that there exists
a unique equilibrium, and that it is a global attractor.
As an application, it is shown that a discretized version of the
Lotka-Volterra system of differential equations of order $k$
has a global attractor in the positive orthant
for certain range of parameters.