A is a subset of B, and B is a superset of A.

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

i : A → B, i(x) = x. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function

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. In Universal algebra, an (induced substructure or (induced subalgebra is a structure whose domain is a subset of that of a bigger structure and

Given any morphism between objects X and Y, if there is an inclusion map into the domain i : A → X, then one can form the restriction fi of f. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In many instances, one can also construct a canonical inclusion into the codomain R→Y known as the range of f. In Mathematics, the codomain, or target, of a function f: X → Y is the set In Mathematics, the range of a function is the set of all "output" values produced by that function

Inclusion maps

Inclusion maps tend to be homomorphisms of algebraic structures; more precisely, given a sub-structure closed under some operations, the inclusion map will be a homomorphism for tautological reasons, given the very definition by restriction of what one checks. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, For example, for a binary operation @, to require that

i(x@y) = i(x)@i(y)

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. In Mathematics, a unary operation is an operation with only one Operand, i In Logic, Mathematics, and Computer science, the arity (synonyms include type, adicity, and rank) of a function Here the point is that closure means such constants must already be given in the substructure. In Mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set

Inclusion maps in geometry come in different kinds: for example embeddings of submanifolds. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group In Mathematics, a submanifold of a Manifold M is a Subset S which itself has the structure of a manifold and for which the Inclusion Contravariant objects such as differential forms restrict to submanifolds, giving a mapping in the other direction. In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is Another example, more sophisticated, is that of affine schemes, for which the inclusions

Spec(R/I) → Spec(R)

and

Spec(R/I²) → Spec(R)

may be different morphisms, where R is a commutative ring and I an ideal. In Abstract algebra and Algebraic geometry, the spectrum of a Commutative ring R, denoted by Spec( R) is defined to be the set of In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring.

See also

• Identity function