Let $X$ be a finite partially ordered set, let $f\colon X\to X$ be an order-preserving map [edit: meaning $x\le y\implies f(x)\le f(y)$], and let $x_0$ be an initial point. Define $x_n = f(x_{n-1})$ for all $n$; then the sequence $(x_n)$ is ultimately periodic. What is its worst-case period? I.e. what are the minimal $k<\ell$ such that $x_k=x_\ell$?

With no order assumption, one could have $\ell=\#X$ since the map $f$ could cycle through all of $X$. I'm particularly interested in the case when $X$ is the family of subsets of a set $Y$, and $x_0$ is a small subset. Does there then exist a non-trivial bound, say polynomial in $\#Y$ and exponential in $\#x_0$?

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