Direct product

In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. 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. The product of topological spaces is another instance.
There is also the direct sum – in some areas this is used interchangeably, while in others it is a different concept.
Examples.
In a similar manner, we can talk about the direct product of finitely many algebraic structures, e.g. formula_17. This relies on the fact that the direct product is associative up to isomorphism. That is, formula_18 for any algebraic structures formula_19, formula_20, and formula_21 of the same kind. The direct product is also commutative up to isomorphism, i.e. formula_22 for any algebraic structures formula_19 and formula_20 of the same kind. We can even talk about the direct product of infinitely many algebraic structures; for example we can take the direct product of countably many copies of formula_25, which we write as formula_26.
Group direct product.
In group theory one can define the direct product of two groups ("G", ∘) and ("H", ∙), denoted by "G" × "H". For abelian groups which are written additively, it may also be called the direct sum of two groups, denoted by formula_27.
It is defined as follows:
This construction gives a new group. It has a normal subgroup isomorphic to "G" (given by the elements of the form ("g", 1)), and one isomorphic to "H" (comprising the elements (1, "h")).
The reverse also holds, there is the following recognition theorem: If a group "K" contains two normal subgroups "G" and "H", such that "K"= "GH" and the intersection of "G" and "H" contains only the identity, then "K" is isomorphic to "G" × "H". A relaxation of these conditions, requiring only one subgroup to be normal, gives the semidirect product.
As an example, take as "G" and "H" two copies of the unique (up to isomorphisms) group of order 2, "C"2: say {1, "a"} and {1, "b"}. Then "C"2×"C"2 = {(1,1), (1,"b"), ("a",1), ("a","b")}, with the operation element by element. For instance, (1,"b")*("a",1) = (1*"a", "b"*1) = ("a","b"), and (1,"b")*(1,"b") = (1,"b"2) = (1,1).
With a direct product, we get some natural group homomorphisms for free: the projection maps define by
called the coordinate functions.
Also, every homomorphism "f" to the direct product is totally determined by its component functions 
formula_29.
For any group ("G", ∘) and any integer "n" ≥ 0, repeated application of the direct product gives the group of all "n"-tuples "G""n" (for "n" = 0 we get the trivial group), for example Z"n" and R"n".
Direct product of modules.
The direct product for modules (not to be confused with the tensor product) is very similar to the one defined for groups above, using the Cartesian product with the operation of addition being componentwise, and the scalar multiplication just distributing over all the components. Starting from R we get Euclidean space R"n", the prototypical example of a real "n"-dimensional vector space. The direct product of R"m" and R"n" is R"m"+"n".
Note that a direct product for a finite index formula_30 is identical to the direct sum formula_31. The direct sum and direct product differ only for infinite indices, where the elements of a direct sum are zero for all but for a finite number of entries. They are dual in the sense of category theory: the direct sum is the coproduct, while the direct product is the product.
For example, consider formula_32 and formula_33, the infinite direct product and direct sum of the real numbers. Only sequences with a finite number of non-zero elements are in "Y". For example, (1,0,0,0...) is in "Y" but (1,1,1,1...) is not. Both of these sequences are in the direct product "X"; in fact, "Y" is a proper subset of "X" (that is, "Y" ⊂ "X").
Topological space direct product.
The direct product for a collection of topological spaces "Xi" for "i" in "I", some index set, once again makes use of the Cartesian product
Defining the topology is a little tricky. 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:
This topology is called the product topology. For example, directly defining the product topology on R2 by the open sets of R (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 (i.e. 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:
The more natural-sounding topology would be, in this case, to take products of infinitely many open subsets as before, and this does yield a somewhat interesting topology, the box 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 which 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.
Products (with the product topology) are nice with respect to preserving properties of their factors; for example, the product of Hausdorff spaces is Hausdorff; the product of connected spaces is connected, and the product of compact spaces is compact. 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.
Direct product of binary relations.
On the Cartesian product of two sets with binary relations R and S, define ("a", "b")T("c", "d") as "a"R"c" and "b"S"d". If R and S are both reflexive, irreflexive, transitive, symmetric, or antisymmetric, then T will be also. Similarly, seriality of "T" is inherited from "R" and "S". Combining properties it follows that this also applies for being a preorder and being an equivalence relation. However if R and S are connected relations, T need not be connected; for example, the direct product of ≤ on ℕ with itself does not relate (1,2) and (2,1).
Direct product in universal algebra.
If Σ is a fixed signature, "I" is an arbitrary (possibly infinite) index set, and (A"i")"i"∈"I" is an indexed family of Σ algebras, the direct product A = ∏"i"∈"I" A"i" is a Σ algebra defined as follows:
For each , the "i"th projection is defined by . It is a surjective homomorphism between the Σ algebras A and A"i".
As a special case, if the index set the direct product of two Σ algebras A1 and A2 is obtained, written as . If Σ just contains one binary operation "f", the above definition of the direct product of groups is obtained, using the notation , , , , and . Similarly, the definition of the direct product of modules is subsumed here.
Categorical product.
The direct product can be abstracted to an arbitrary category. In a general category, given a collection of objects "Ai" "and" a collection of morphisms "pi" from "A" to "Ai" with "i" ranging in some index set "I", an object "A" is said to be a categorical product in the category if, for any object "B" and any collection of morphisms "fi" from "B" to "Ai", there exists a unique morphism "f" from "B" to "A" such that "fi = pi f" and this object "A" is unique. This not only works for two factors, but arbitrarily (even infinitely) many.
For groups we similarly define the direct product of a more general, arbitrary collection of groups "Gi" for "i" in "I", "I" an index set. Denoting the Cartesian product of the groups by "G" we define multiplication on "G" with the operation of componentwise multiplication; and corresponding to the "pi" in the definition above are the projection maps
the functions that take formula_38 to its "i"th component "gi".
Internal and external direct product.
Some authors draw a distinction between an internal direct product and an external direct product. If formula_39 and formula_40, then we say that "X" is an "internal" direct product of "A" and "B", while if "A" and "B" are not subobjects then we say that this is an "external" direct product.