Inclusion map

In mathematics, if is a subset of , then the inclusion map (also inclusion function, insertion, or canonical injection) is the function that sends each element, of to , treated as an element of :

A "hooked arrow" is sometimes used in place of the function arrow above to denote an inclusion map.

This and other analogous injective functions from substructures are sometimes called natural injections.

Given any morphism f between objects X and Y, if there is an inclusion map into the domain , then one can form the restriction fi of f. In many instances, one can also construct a canonical inclusion into the codomain R→Y known as the range of f.

Applications of inclusion maps

Inclusion maps tend to be homomorphisms of algebraic structures; thus, such inclusion maps are embeddings. More precisely, given a sub-structure closed under some operations, the inclusion map will be an embedding for tautological reasons. For example, for a binary operation , to require that

is simply to say that is consistently computed in the sub-structure and the large structure. The case of a unary operation is similar; but one should also look at nullary operations, which pick out a constant element. Here the point is that closure means such constants must already be given in the substructure.