Definition:Order Embedding/Definition 4

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Definition

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

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

Let $T' = \Img S$ be the image of $S$ under $\phi$.

$\phi$ is an order embedding of $S$ into $T$ if and only if:

the restriction of $\phi$ to $S \times T'$ is an order isomorphism between $\struct {S, \preceq_1}$ and $\struct {T', \preceq_2 \restriction_{T' \times T'} }$.


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.


Also see