Definition:Dual Order Embedding

Definition
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets.

A dual order embedding is a mapping $\phi: S \to T$ such that:


 * $\forall x, y \in S: x \preceq_1 y \iff \phi \left({y}\right) \preceq_2 \phi \left({x}\right)$

That is:
 * if $\phi$ is an order embedding of $\left({S, \preceq_1}\right)$ into $\left({T, \succeq_2}\right)$

where $\succeq_2$ is the dual of $\preceq_2$.