  In this paper we study finite dynamical systems with $n$ functions acting on
the same set $X$, and probabilities assigned to these functions, that it is
called Probabilistic Regulatory Gene Networks (PRN. his concept is the same or
a natural generalization of the concept Probabilistic Boolean Networks (PBN),
introduced by I. Shmulevich, E. Dougherty, and W. Zhang, particularly the model
PBN has been using to describe genetic networks and has therapeutic
applications. In PRNs the most important question is to describe the steady
states of the systems, so in this paper we pay attention to the idea of
transforming a network to another without lost all the properties, in
particular the probability distribution. Following this objective we develop
the concepts of homomorphism and $\epsilon$-homomorphism of probabilistic
regulatory networks, since these concepts bring the properties from one
networks to another. Projections are special homomorphisms, and they always
induce invariant subnetworks that contain all cycles and steady states in the
network.
