Order embedding

Formally, given two partially ordered sets (posets) ${\displaystyle (S,\leq )}$ and ${\displaystyle (T,\preceq )}$, a function ${\displaystyle f:S\to T}$ is an order embedding if ${\displaystyle f}$ is both order-preserving and order-reflecting, i.e. for all ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle S}$, one has

${\displaystyle x\leq y{\text{ if and only if }}f(x)\preceq f(y).}$[1]

Such a function is necessarily injective, since ${\displaystyle f(x)=f(y)}$ implies ${\displaystyle x\leq y}$ and ${\displaystyle y\leq x}$.[1] If an order embedding between two posets ${\displaystyle S}$ and ${\displaystyle T}$ exists, one says that ${\displaystyle S}$ can be embedded into ${\displaystyle T}$.

Mutual order embedding of ${\displaystyle (0,1)}$ and ${\displaystyle [0,1]}$, using ${\displaystyle f(x)=(94x+3)/100}$ in both directions.

The set ${\displaystyle S}$ of divisors of 6, partially ordered by x divides y. The embedding ${\displaystyle id:\{1,2,3\}\to S}$ cannot be a coretraction.

An order isomorphism can be characterized as a surjective order embedding. As a consequence, any order embedding f restricts to an isomorphism between its domain S and its image f(S), which justifies the term "embedding".[1] On the other hand, it might well be that two (necessarily infinite) posets are mutually order-embeddable into each other without being order-isomorphic.

An example is provided by the open interval ${\displaystyle (0,1)}$ of real numbers and the corresponding closed interval ${\displaystyle [0,1]}$. The function ${\displaystyle f(x)=(94x+3)/100}$ maps the former to the subset ${\displaystyle (0.03,0.97)}$ of the latter and the latter to the subset ${\displaystyle [0.03,0.97]}$of the former, see picture. Ordering both sets in the natural way, ${\displaystyle f}$ is both order-preserving and order-reflecting (because it is an affine function). Yet, no isomorphism between the two posets can exist, since e.g. ${\displaystyle [0,1]}$ has a least element while ${\displaystyle (0,1)}$ does not. For a similar example using arctan to order-embed the real numbers into an interval, and the identity map for the reverse direction, see e.g. Just and Weese (1996).[2]

A retract is a pair ${\displaystyle (f,g)}$ of order-preserving maps whose composition ${\displaystyle g\circ f}$ is the identity. In this case, ${\displaystyle f}$ is called a coretraction, and must be an order embedding.[3] However, not every order embedding is a coretraction. As a trivial example, the unique order embedding ${\displaystyle f:\emptyset \to \{1\}}$ from the empty poset to a nonempty poset has no retract, because there is no order-preserving map ${\displaystyle g:\{1\}\to \emptyset }$. More illustratively, consider the set ${\displaystyle S}$ of divisors of 6, partially ordered by x divides y, see picture. Consider the embedded sub-poset ${\displaystyle \{1,2,3\}}$. A retract of the embedding ${\displaystyle id:\{1,2,3\}\to S}$ would need to send ${\displaystyle 6}$ to somewhere in ${\displaystyle \{1,2,3\}}$ above both ${\displaystyle 2}$ and ${\displaystyle 3}$, but there is no such place.

Posets can straightforwardly be viewed from many perspectives, and order embeddings are basic enough that they tend to be visible from everywhere. For example:

• (Model theoretically) A poset is a set equipped with a (reflexive, antisymmetric and transitive) binary relation. An order embedding A → B is an isomorphism from A to an elementary substructure of B.
• (Graph theoretically) A poset is a (transitive, acyclic, directed, reflexive) graph. An order embedding A → B is a graph isomorphism from A to an induced subgraph of B.
• (Category theoretically) A poset is a (small, thin, and skeletal) category such that each homset has at most one element. An order embedding A → B is a full and faithful functor from A to B which is injective on objects, or equivalently an isomorphism from A to a full subcategory of B.

• Dushnik–Miller theorem
• Laver's theorem