# Definition:Order Embedding

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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.

### Definition 1

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

$\forall x, y \in S: x \preceq_1 y \iff \map \phi x \preceq_2 \map \phi y$

### Definition 2

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

$(1): \quad \phi$ is an injection
$(2): \quad \forall x, y \in S: x \preceq_1 y \iff \map \phi x \preceq_2 \map \phi y$

### Definition 3

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

$(1): \quad \phi$ is an injection
$(2): \quad \forall x, y \in S: x \prec_1 y \iff \map \phi x \prec_2 \map \phi y$

### Definition 4

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.