Definition:Order Embedding/Definition 1

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

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


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

That is, an order embedding is an order-preserving, order-reflecting mapping.

Also see

 * Equivalence of Definitions of Order Embedding