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.[dubious ]

There is also the direct sum – in some areas this is used interchangeably, while in others it is a different concept.

ExamplesEdit

  • 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 associative up to isomorphism. 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  

Group direct productEdit

In group theory one can define the direct product of two groups   and   denoted by   For abelian groups which are written additively, it may also be called the direct sum of two groups, denoted by  

It is defined as follows:

  • 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,  ).[1][2]

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.

Direct product of binary relationsEdit

On the Cartesian product of two sets with binary relations   define   as   If   are both reflexive, irreflexive, transitive, symmetric, or antisymmetric, then   will be also.[3] Similarly, totality of   is inherited from   Combining properties it follows that this also applies for being a preorder and being an equivalence relation. However if   are connected relations,   need not be connected; for example, the direct product of   on   with itself does not relate  

Direct product in universal algebraEdit

If   is a fixed signature,   is an arbitrary (possibly infinite) index set, and   is an indexed family of   algebras, the direct product   is a   algebra defined as follows:

  • The universe set   of   is the Cartesian product of the universe sets   of   formally:  
  • For each   and each  -ary operation symbol   its interpretation   in   is defined componentwise, formally: for all   and each   the  th component of   is defined as  

For each   the  th projection   is defined by   It is a surjective homomorphism between the   algebras  [4]

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.

Categorical productEdit

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.

See alsoEdit

NotesEdit

  1. ^ Weisstein, Eric W. "Direct Product". mathworld.wolfram.com. Retrieved 2018-02-10.
  2. ^ Weisstein, Eric W. "Group Direct Product". mathworld.wolfram.com. Retrieved 2018-02-10.
  3. ^ "Equivalence and Order" (PDF).
  4. ^ Stanley N. Burris and H.P. Sankappanavar, 1981. A Course in Universal Algebra. Springer-Verlag. ISBN 3-540-90578-2. Here: Def.7.8, p.53 (=p. 67 in pdf file)

ReferencesEdit