# Definition:Order Embedding

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## Definition

### Definition 1

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:

$\forall x, y \in S: x \preceq_1 y \iff \phi \left({x}\right) \preceq_2 \phi \left({y}\right)$

### Definition 2

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets.

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

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

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

### Definition 3

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets.

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

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

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

### Definition 4

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets.

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

Let $T' = \operatorname{Im} \left({S}\right)$ be the image of $S$ under $\phi$.

Then $\phi$ is an order embedding if and only if:

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

## Also known as

An order embedding is also known as an order monomorphism.