Monotonicity equivalence and synchronizability for a system of probability distributions

A system $(P_\alpha: \alpha\in\mathcal{A})$ of probability distributions on a partially ordered set (poset) $\mathcal{S}$ indexed by another poset $\mathcal{A}$ can be realized by a system of $\mathcal{S}$-valued random variables $X_\alpha$'s marginally distributed as $P_\alpha$. It is called realizably monotone if $X_\alpha\le X_\beta$ in $\mathcal{S}$ whenever $\alpha\le\beta$ in $\mathcal{A}$. Such a system necessarily is stochastically monotone, that is, it satisfies $P_\alpha\preceq P_\beta$ in stochastic ordering whenever $\alpha \le \beta$. It has been known exactly when these notions of monotonicity are equivalent except for a certain subclass of acyclic posets, called Class W. In this paper we introduce inverse probability transforms and synchronizing bijections recursively when $\mathcal{S}$ is a poset of Class W and $\mathcal{A}$ is synchronizable, and validate monotonicity equivalence by constructing $(X_\alpha: \alpha\in\mathcal{A})$ explicitly. We also show that synchronizability is necessary for monotonicity equivalence when $\mathcal{S}$ is in Class W.