\begin{definition}[Definition:Dual Order Embedding]
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 $\preceq_2$.
Category:Definitions/Order Embeddings
\end{definition}