  Given a finite state space E, we build a universal dilation for all possible
discrete time Markov chains on E, homogeneous or not: we introduce a second
system (an ``environment'') and a deterministic invertible time-homogeneous
global evolution of the system E with this environment such that any Markov
evolution of E can be realized by a proper choice of the initial (random) state
of the environment, which therefore determines the transition probabilities of
the system. We also compare this dilation with the quantum dilations of a
Quantum Dynamical Semigroup: given a Classical Markov Semigroup, we show that
it can be extended to a Quantum Dynamical Semigroup for which we can find a
quantum dilation to a group of *-automorphisms admitting an invariant abelian
subalgebra where this quantum dilation gives just our classical dilation.
