Order embedding

In mathematical order theory, 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 (S, ≤) and (T, ≤), a function f: S → 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

Note that such a function is necessarily injective, since f(x) = f(y) implies x ≤ y and y ≤ x. If an order embedding between two posets S and T exists, one says that S can be embedded into 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 range f(S), which justifies the term "embedding". On the other hand, it might well be that two (necessarily infinite) posets are mutually embeddable into each other without being isomorphic. An example is provided by the set of real numbers and its interval [−1,1]. Ordering both sets in the natural way, one clearly finds that [−1,1] can be embedded into the reals. On the other hand, one can define a function e from the real numbers to [−1,1] as

This is the projection of the real number line to (half of) the circle with circumference 4 (see trigonometric functions for details) and embeds the reals into [−1,1]. Yet, the two posets are not isomorphic: [−1,1] has both a least and a greatest element, which are not present in the case of the real numbers. This shows that an isomorphism cannot exist.

...
Wikipedia