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 or inclusion map is the function i that sends each element of A to the same element in B:

i:A → B, i(x) = x

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.