Sufficient conditions are given under which the higher order difference equation x _n+1= f(x _n,x_n-1,...,x_n-k ), n=0,1,2,... generates an order preserving discrete dynamical system with respect to the discrete exponential ordering. It is shown that under the above monotonicity assumption the boundedness of all solutions as well as the local and global stability of an equilibrium hold if and only if they hold for the much simpler first order equation x _n+1=h(x _n ), where h(x)=f(x,x,…,x). As an application, a second order nonlinear difference equation from macroeconomics and a discrete analogue of a model of haematopoiesis are discussed.