Definition:Dual Order Embedding

Definition

Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.

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

$\forall x, y \in S: x \preceq_1 y \iff \map \phi y \preceq_2 \map \phi x$

That is:

if $\phi$ is an order embedding of $\struct {S, \preceq_1}$ into $\struct {T, \succeq_2}$

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