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.