Let $Y$ be a discrete stationary stochastic process. Suppose that $Y$ is not $n$-step Markov for any positive integer $n$. Let $Z$ be a 1-block factor of $Y$. For what condition on $Y$ or the factor map, will $Z$ be Markov?

The problem has been well-studied when $Y$ is assumed to be Markov. Then $Z$ is a hidden Markov chain. However, I am not familiar with any literature addressing the question when $Y$ is not Markov. Any references would be greatly appreciated. Thank you.