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 defined as
It is usual to state in the definition for an order embedding that it be injective.

As can be seen in Order Embedding is Injection, that condition is redundant.

Also see

 * Equivalence of Definitions of Order Embedding