Inclusion of one mathematical structure in another, preserving properties of interest / From Wikipedia, the free encyclopedia

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In mathematics, an embedding (or imbedding[1]) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

When some object is said to be embedded in another object , the embedding is given by some injective and structure-preserving map . The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which and are instances. In the terminology of category theory, a structure-preserving map is called a morphism.

The fact that a map is an embedding is often indicated by the use of a "hooked arrow" (U+21AA RIGHTWARDS ARROW WITH HOOK);[2] thus: (On the other hand, this notation is sometimes reserved for inclusion maps.)

Given and , several different embeddings of in may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the natural numbers in the integers, the integers in the rational numbers, the rational numbers in the real numbers, and the real numbers in the complex numbers. In such cases it is common to identify the domain with its image contained in , so that .