The problem comes from an article: Order Structure on the algebra of permutations and of planar bianry trees. I will describe the problem independently here.
Question
Suppose $(S, \leq)$ is a partially ordered set and Y is a set. There is a surjective map $\Psi: S \longrightarrow Y$ such that $\forall t\in Y$, $\Psi^{-1}(t)$ is an interval, that is, $\exists Min(t),Max(t)\in S$,
$\Psi^{-1}(t) = \{ \sigma\in S | Min(t) \leq \sigma \leq Max(t) \}$.
Then how can the partial order of S induce a partial order of Y such that $\forall \sigma,\tau\in S$, $\sigma \leq \tau \Longrightarrow \Psi(\sigma) \leq \Psi(\tau)$?
My Idea
I try to define the partial order of Y as the transitive closure of
$\{ w \leq_B t \big| Min(w) \leq Max(t) \}$
But this is not right. I can not verify the asymmetric axiom.