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.
There is also the direct sum – in some areas this is used interchangeably, while in others it is a different concept.
If we think of as the set of real numbers, then the direct product is just the Cartesian product
If we think of as the group of real numbers under addition, then the direct product still has as its underlying set. The difference between this and the preceding example is that is now a group, and so we have to also say how to add their elements. This is done by defining
If we think of as the ring of real numbers, then the direct product again has as its underlying set. The ring structure consists of addition defined by and multiplication defined by
However, if we think of as the field of real numbers, then the direct product does not exist – naively defining addition and multiplication componentwise as in the above example would not result in a field since the element does not have a multiplicative inverse.
In a similar manner, we can talk about the direct product of finitely many algebraic structures, for example, This relies on the fact that the direct product is associativeup toisomorphism. That is, for any algebraic structures and of the same kind. The direct product is also commutative up to isomorphism, that is, for any algebraic structures and 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 which we write as
the set of the elements of the new group is the Cartesian product of the sets of elements of that is
on these elements put an operation, defined element-wise:
Note that may be the same as
This construction gives a new group. It has a normal subgroup isomorphic to (given by the elements of the form ), and one isomorphic to (comprising the elements ).
The reverse also holds. There is the following recognition theorem: If a group contains two normal subgroups such that and the intersection of contains only the identity, then is isomorphic to A relaxation of these conditions, requiring only one subgroup to be normal, gives the semidirect product.
As an example, take as two copies of the unique (up to isomorphisms) group of order 2, say Then with the operation element by element. For instance, and
With a direct product, we get some natural group homomorphisms for free: the projection maps defined by
are called the coordinate functions.
Also, every homomorphism to the direct product is totally determined by its component functions
For any group and any integer repeated application of the direct product gives the group of all -tuples (for this is the trivial group), for example and
Direct product of modulesEdit
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 we get Euclidean space the prototypical example of a real -dimensional vector space. The direct product of and is
Note that a direct product for a finite index is canonically isomorphic to the direct sum The direct sum and direct product are not isomorphic 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 and the infinite direct product and direct sum of the real numbers. Only sequences with a finite number of non-zero elements are in For example, is in but is not. Both of these sequences are in the direct product in fact, is a proper subset of (that is, ).
Topological space direct productEdit
The direct product for a collection of topological spaces for in 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 by the open sets of (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:
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.
As a special case, if the index set the direct product of two algebras is obtained, written as If just contains one binary operation 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.
The direct product can be abstracted to an arbitrary category. In a category, given a collection of objects indexed by a set , a product of these objects is an object together with morphisms for all , such that if is any other object with morphisms for all , there exists a unique morphism whose composition with equals for every .
Such and do not always exist. If they do exist, then is unique up to isomorphism, and is denoted .
In the special case of the category of groups, a product always exists: the underlying set of is the Cartesian product of the underlying sets of the , the group operation is componentwise multiplication, and the (homo)morphism is the projection sending each tuple to its th coordinate.
Internal and external direct productEdit
Some authors draw a distinction between an internal direct product and an external direct product. If and then we say that is an internal direct product of while if are not subobjects then we say that this is an external direct product.
Direct sum – Operation in abstract algebra composing objects into "more complicated" objects