Inclusion (mathematics)

In mathematics, inclusion is a partial order on sets. Under this order, A ≤ B if A is a subset of B.

When A is a subset of B, the inclusion function (inclusion map, or canonical injection) is the function i that sends each element of A to the same element in B:

i:A → B, i(x) = x.

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

Inclusion as partial order

The order on ordinal numbers is given by inclusion.

For the power set of a set X, the inclusion partial order is (up to isomorphism) the direct product of |X| copies of the partial order on {0,1} for which 0 < 1.

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. 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. Here the point is that closure means such constants must already be given in the substructure.

Inclusion maps in geometry come in different kinds: for example embeddings of submanifolds. Contravariant objects such as differential forms restrict to submanifolds, giving a mapping in the other direction. 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 a ideal.