In order theory, a branch of mathematics, an order embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another. Like Galois connections, order embeddings constitute a notion which is strictly weaker than the concept of an order isomorphism. Both of these weakenings may be understood in terms of category theory.
Formally, given two partially ordered sets (posets)
(S,\leq)
(T,\preceq)
f:S\toT
f
x
y
S
x\leqyifandonlyiff(x)\preceqf(y).
Such a function is necessarily injective, since
f(x)=f(y)
x\leqy
y\leqx
S
T
S
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.
(0,1)
[0,1]
f(x)=(94x+3)/100
(0.03,0.97)
[0.03,0.97]
f
[0,1]
(0,1)
A retract is a pair
(f,g)
g\circf
f
f:\emptyset\to\{1\}
g:\{1\}\to\emptyset
S
\{1,2,3\}
id:\{1,2,3\}\toS
6
\{1,2,3\}
2
3
Posets can straightforwardly be viewed from many perspectives, and order embeddings are basic enough that they tend to be visible from everywhere. For example: