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Order embedding

Type of monotone function

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated April 18, 2026
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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.

01Formal definition

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

x\leq y{\text{ if and only if }}f(x)\preceq f(y).

Such a function is necessarily injective, since f(x)=f(y) implies x\leq y and y\leq x. If an order embedding exists from a poset S to a poset T, one says that S can be embedded into T.

Mutual order embedding of and , using in both directions.
Mutual order embedding of and , using in both directions.

02Properties

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". 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 (0,1) of real numbers and the corresponding closed interval [0,1]. The function f(x)=(94x+3)/100 maps the former to the subset (0.03,0.97) of the latter and the latter to the subset [0.03,0.97] of the former, see picture. Ordering both sets in the natural way, 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. [0,1] has a least element while (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).

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

03Additional perspectives

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

Watch videos about Order embeddingExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Order embedding”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Dushnik-Miller theorem

In mathematics, the Dushnik-Miller theorem is a result in order theory stating that every countably infinite linear order has a non-identity order embedding into itself. It is named for Ben Dushnik and E. W. Miller, who proved this result in a paper of 1940; in the same paper, they showed that the statement does not always hold for uncountable linear orders, using the axiom of choice to build a suborder of the real line of cardinality continuum with no non-identity order embeddings into itself.

Laver's theorem

Laver's theorem, in order theory, states that order embeddability of countable total orders is a well-quasi-ordering. That is, for every infinite sequence of totally-ordered countable sets, there exists an order embedding from an earlier member of the sequence to a later member.