# Definition:Order Embedding

## 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**.

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.

## Examples

### Finite Subsets of Natural Numbers in Divisibility Structure

Consider the relational structures:

- $\struct {\Z_{>0}, \divides}$, where $\Z_{>0}$ denotes the strictly positive integers and $\divides$ denotes the divisor relation
- $\struct {\FF, \subseteq}$, where $\FF$ denotes the finite subsets of the natural numbers without zero $\N_{\ne 0}$ and $\subseteq$ denotes the subset relation.

Let $\pi: \FF \to \Z_{>0}$ be the mapping defined as:

- $\forall S \in \FF: \map \pi S = \ds \prod_{n \mathop \in S} \map p n$

where $\map p n$ denotes the $n$th prime number:

- $\map p 1 = 2, \map p 2 = 3, \map p 3 = 5, \ldots$

Then $\pi$ is an order embedding of $\FF$ into $\Z_{>0}$.

## Also see

- Results about
**order embeddings**can be found**here**.