Orderembedding

In mathematical order theory, an orderembedding is a special kind of monotone function, which provides a way to include one partially ordered set into another. Like Galois connections, orderembeddings constitute a notion which is strictly weaker than the concept of an order isomorphism. A way to understand the different nature of both of these weakenings by exploiting some category theory is discussed at the end of this article.
Formal definition
Formally, given two partially ordered sets (S, ≤) and (T, <=), a function f: S → T is an orderembedding if f is both orderpreserving and orderreflecting, i.e. for all x and y in S, one has
 x ≤ y iff f(x) <= f(y)
Note that such a function is necessarily injective, since f(x) = f(y) implies x ≤ y and y ≤ x. If an orderembedding between two posets S and T exists, one says that S can be embedded into T.
Properties
An order isomorphism can be characterized as a surjective orderembedding. As a consequence, any orderembedding 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
 <math>e(x) = \frac{2}{\pi}\arctan r<math>
This is in fact just the projection of the real number line to (half of) the circle with circumference 4 (see trigonometric functions for details). Now it is easy to see that e 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.
Categorically speaking
The basic category for the study of partial orders is the category of posets and monotone functions. Since a categorical perspective usually has an enlightening effect in the study of mathematical subjects, one may well wonder what role orderembeddings play in this category. While the categorical significance of orderisomorphisms is quite obvious (being just isomorphisms in the categorical sense), embeddings may deserve some discussion. It turns out that the orderembeddings with nonempty domain correspond exactly to the sections in the category of posets. Remember that a section s is just a morphism that has a leftinverse r (called retraction), where r o s = id. Although all mappings from the empty poset to some other poset are orderembeddings, these can in general not be inverted (since there is no mapping from a nonempty set to the empty set). Note that in addition to injectivity (implying that a mapping is monic), one needs reflection of the order to find a monotone leftinverse. Thus orderembeddings specialize the notion of a subposet in the same way that sections specify monomorphisms.
This correspondence also helps to understand the difference of Galois connections and orderembeddings, which in a sense are both weakenings of the concept of an orderisomorphism. It is helpful to consider the abovementioned category as a category of categories: this can be done by noting that any poset is in fact a smallcategory with at most one arrow between two objects (pointing "upwards" in the order) and that functors between those categories correspond exactly to monotone mappings. As just demonstrated orderembeddings basically agree with sections in this category. On the other hand, Galois connections yield an adjunction between the related posetcategories. Hence orderembeddings (being sections) are a weaker form of isomorphisms within this category (i.e. of isomorphisms of categories, i.e. of orderisomorphisms), while Galois connections weaken the concept of an equivalence of categories between posetcategories  it just happens that both these isomorphisms and the categorical equivalences agree in our special setting.