# Definition:Order Embedding/Definition 2

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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.

$\phi$ 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.

## Also see

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings