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High Quality Content by WIKIPEDIA articles! 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. 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.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].