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) and , a function is an order embedding if is both order-preserving and order-reflecting, i.e. for all and in , one has
Such a function is necessarily injective, since implies and .^{[1]} If an order embedding between two posets and exists, one says that can be embedded into .
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 of real numbers and the corresponding closed interval . The function maps the former to the subset of the latter and the latter to the subset of the former, see picture. Ordering both sets in the natural way, 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. has a least element while 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 of order-preserving maps whose composition is the identity. In this case, 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 from the empty poset to a nonempty poset has no retract, because there is no order-preserving map . More illustratively, consider the set of divisors of 6, partially ordered by x divides y, see picture. Consider the embedded sub-poset . A retract of the embedding would need to send to somewhere in above both and , 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: