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 $T$ if and only if 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 known as

An order embedding is also known as an order monomorphism.

Some sources call it an order-preserving mapping, but this term is also used (in particular on $\mathsf{Pr} \infty \mathsf{fWiki}$ to be the same thing as an increasing mapping: that is, a mapping which preserves an ordering in perhaps only one direction.