  In this paper we study homomorphisms of Probabilistic Regulatory Gene
Networks(PRN) introduced in arXiv:math.DS/0603289 v1 13 Mar 2006. The model PRN
is a natural generalization of the Probabilistic Boolean Networks (PBN),
introduced by I. Shmulevich, E. Dougherty, and W. Zhang in 2001, that has been
using to describe genetic networks and has therapeutic applications. In this
paper, our main objectives are to apply the concept of homomorphism and
$\epsilon$-homomorphism of probabilistic regulatory networks to the dynamic of
the networks. The meaning of $\epsilon$ is that these homomorphic networks have
similar distributions and the distance between the distributions is upper
bounded by $\epsilon$. Additionally, we prove that the class of PRN together
with the homomorphisms form a category with products and coproducts.
Projections are special homomorphisms, and they always induce invariant
subnetworks that contain all the cycles and steady states in the network. Here,
it is proved that the $\epsilon$-homomorphism for $0<\epsilon<1$ produce
simultaneous Markov Chains in both networks, that permit to introduce the
concept of $\epsilon$-isomorphism of Markov Chains, and similar networks.
