Definition:Order Embedding/Definition 2

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

Let $\phi: S \to T$ be a mapping.

$\phi$ is an order embedding of $S$ into $T$ both of the following conditions hold:


 * $(1): \quad \phi$ is an injection


 * $(2): \quad \forall x, y \in S: x \preceq_1 y \iff \map \phi x \preceq_2 \map \phi y$

Also defined as
As seen in Order Embedding is Injection, the specification of injectivity is redundant, so some sources omit it.

Also see

 * Equivalence of Definitions of Order Embedding