  Discrete dynamical systems defined on the state space {0,1,...,p-1}^n have
been used in multiple applications, most recently for the modeling of gene and
protein networks. In this paper we study to what extent well-known theorems by
Smale and Hirsch, which form part of the theory of (continuous) monotone
dynamical systems, generalize or fail to do so in the discrete case.
  We show that that arbitrary m-dimensional systems cannot necessarily be
embedded into n-dimensional cooperative systems for n=m+1, as in the Smale
theorem for the continuous case, but we show that this is possible for n=m+2 as
long as p is sufficiently large.
  We also prove that a natural discrete analogue of strong cooperativity
implies nontrivial bounds on the lengths of periodic orbits and imposes a
condition akin to Lyapunov stability on all attractors. Finally, we explore
several natural candidates for definitions of irreducibility of a discrete
system. While some of these notions imply the strong cooperativity of a given
cooperative system and impose even tighter bounds on the lengths of periodic
orbits than strong cooperativity alone, other plausible definitions allow the
existence of exponentially long periodic orbits.
