In mathematics, a direct product of objects already known can often be defined by giving a new one.
That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects.
More abstractly, the product in category theory is mentioned, which formalizes those notions.
Examples are the product of sets, groups (described below), rings, and other algebraic structures.
There is also the direct sum, which in some areas used interchangeably but in others is a different concept.
In a similar manner, the direct product of finitely many algebraic structures can be talked about; for example,
That relies on the direct product being associative up to isomorphism.
The construction gives a new group, which has a normal subgroup that is isomorphic to
A relaxation of those conditions by requiring only one subgroup to be normal gives the semidirect product.
are taken as two copies of the unique (up to isomorphisms) group of order 2,
With a direct product, some natural group homomorphisms are obtained for free: the projection maps defined by
to the direct product is totally determined by its component functions
repeated application of the direct product gives the group of all
Only sequences with a finite number of non-zero elements are in
some index set, once again makes use of the Cartesian product
For finitely many factors, it is the obvious and natural thing to do: simply take as a basis of open sets to be the collection of all Cartesian products of open subsets from each factor:
(disjoint unions of open intervals), the basis for that topology would consist of all disjoint unions of open rectangles in the plane (as it turns out, it coincides with the usual metric topology).
However, it is not too difficult to find an example of bunch of continuous component functions whose product function is not continuous (see the separate entry box topology for an example and more).
The problem that makes the twist necessary is ultimately rooted in the fact that the intersection of open sets is guaranteed to be open only for finitely many sets in the definition of topology.
That last one, called Tychonoff's theorem, is yet another equivalence to the axiom of choice.
For more properties and equivalent formulations, see product topology.
On the Cartesian product of two sets with binary relations
are both reflexive, irreflexive, transitive, symmetric, or antisymmetric, then
If the properties are combined, that also applies for being a preorder and being an equivalence relation.
is an arbitrary (possibly infinite) index set, and
the above definition of the direct product of groups is obtained by using the notation
Similarly, the definition of the direct product of modules is subsumed here.
The direct product can be abstracted to an arbitrary category.
In the special case of the category of groups, a product always exists.
, the group operation is componentwise multiplication, and the (homo)morphism