Modelling processes on the Z^d-lattice

Suppose that we are given a stationary stochastic process
{X_n}_{n\in Z}. Can we model it by another stationary stochastic
process {Y_n}_{n\in Z} where Y_n can take only two values? In 1971,
Krieger answered with an affirmative under certain natural
assumptions. It is now well-known that the analogous result holds true
for modelling stationary random fields {X_n}_{n\in Z^d} as well. What
if we now constrain the stationary stochastic process {Y_n}_{n\in Z^d}
to take only three values such that adjacent values are distinct?
Along with Tom Meyerovitch, we find that this is true thereby
answering a question of Şahin and Robinson. No background in
stochastic processes or ergodic theory will be assumed.