Let $P$ be a poset with a least element $\bot$ ($\forall x \in P.\ \bot \le x$). Let $M$ be a set of monotone maps $P \to P$. Call $x \in P$ reachable if $x = f_1(f_2(...f_n(\bot)...))$ for some sequence $f_i \in M$. Call $x$ fixed if $x = f(x)$ for all $f \in M$.
If $x$ reachable, $y$ fixed, then $x \le y$. For then $x = F(\bot)$ for $F$ some composition of functions in $M$. Since $F$ is monotone and $\bot \le y$, we have $x = F(\bot) \le F(y) = y$. Thus there is at most one reachable fixed point.
Question #1: If there is a reachable fixed point $x$, will applying functions in $M$ uniformly at random (starting from $\bot$) eventually find $x$ with probability 1?
This requires a uniform probability distribution over $M$, so is ill-defined if $M$ is infinite. So:
Question #2: If there is a reachable fixed point $x$, will any schedule which eventually includes every sequence of $M$s infinitely often eventually find $x$ (starting from $\bot$)?
For our purposes, $M$ is at most countable. I'd even be interested in a solution to the case where $M$ is finite.