In mathematics, one can often define a direct product of objects already known, giving a new one.
This induces a structure on the Cartesian product of the underlying sets from that of the contributing objects.
More abstractly, one talks about the product in category theory, which formalizes these notions.
Examples are the product of sets, groups (described below), rings, and other algebraic structures.
There is also the direct sum – in some areas this is used interchangeably, while in others it is a different concept.
In a similar manner, we can talk about the direct product of finitely many algebraic structures, for example,
This relies on the direct product being associative up to isomorphism.
A relaxation of these conditions, requiring only one subgroup to be normal, gives the semidirect product.
two copies of the unique (up to isomorphisms) group of order 2,
With a direct product, we get some natural group homomorphisms 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
[1][2] The direct product for a collection of topological spaces
some index set, once again makes use of the Cartesian product
For finitely many factors, this 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 this topology would consist of all disjoint unions of open rectangles in the plane (as it turns out, it coincides with the usual metric topology).
The product topology for infinite products has a twist, and this has to do with being able to make all the projection maps continuous and to make all functions into the product continuous if and only if all its component functions are continuous (that is, to satisfy the categorical definition of product: the morphisms here are continuous functions): we take as a basis of open sets to be the collection of all Cartesian products of open subsets from each factor, as before, with the proviso that all but finitely many of the open subsets are the entire factor:
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 only guaranteed to be open 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 the separate entry product topology.
On the Cartesian product of two sets with binary relations
are both reflexive, irreflexive, transitive, symmetric, or antisymmetric, then
Combining properties it follows that this 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, using the notation
Similarly, the definition of the direct product of modules is subsumed here.
In the special case of the category of groups, a product always exists: the underlying set of
, the group operation is componentwise multiplication, and the (homo)morphism