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 ${\displaystyle T}$ .

 

 

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