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In
mathematics, one can often define a '''direct product''' of objects
already known, giving a new one. Examples are the product of groups (described below), the
product of rings and of other
algebraic structures. The
product of topological spaces is another instance.
=== The Direct Product for Groups ===
In
group theory one defines the direct product of two
groups (''G'', *) and (''H'', o) , denoted by ''G''×''H'', as follows:
* as
set of the elements of the new group, take the ''
cartesian product'' of the sets of elements of ''G'' and ''H'', that is {(''g'', ''h''): ''g'' in ''G'', ''h'' in ''H''};
* on these elements put an operation, defined elementwise:
(''g'', ''h'') × (''g' '', ''h' '') = (''g'' * ''g' '', ''h'' o ''h' '')
(Note the operation * may be the same as o.)
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'' = ''G'' x ''H''. A relaxation of these conditions 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
:<math>\pi_1 : G \times H \to G\quad \mathrm{by} \quad \pi_1(g, h) = g</math>,
:<math>\pi_2 : G \times H \to H\quad \mathrm{by} \quad \pi_2(g, h) = h</math>
called the '''coordinate functions'''.
Also, every homomorphism ''f'' on the direct product is totally determined by its component functions
<math>f_i = \pi_i \circ f</math>.
=== Direct Product for Vector Spaces ===
The product for
vector spaces 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 (an easy generalization of how it is defined for '''R'''
''n'').
Note that a direct product for a finite index <math>\prod_{i=1}^n X_i </math> is identical to the
direct sum <math>\oplus_{i=1}^n X_i </math>. The direct sum and direct product only differs for infinte indicies, where the elements of a direct sum is zero for all but for a finite number of entries.
=== Direct Product for Topological Spaces ===
The direct product for a collection of
topological spaces ''X
i'' for ''i'' in ''I'', some index set, once again makes use of the cartesian product
:<math>\prod_{i \in I} X_i </math>
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:
:<math>\mathcal B = \{ U_1 \times \cdots \times U_n\ |\ U_i\ \mathrm{open\ in}\ X_i \}</math>
This topology is called the '''product topology'''. For example, directly defining the product topology on '''R'''
2 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 factors are the entire space:
:<math>\mathcal B = \left\{ \prod_{i \in I} U_i\ |\ (\exists j_1,\ldots,j_n)(U_{j_i}\ \mathrm{open\ in}\ X_{j_i})\ \mathrm{and}\ (\forall i \neq j_1,\ldots,j_n)(U_i = X_i) \right\}</math>
(Not a very pretty sight!). 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.
=== Partially Ordered Sets ===
''Still working on this... add stuff here if you like... thought it was going to be fast but have to make sure I understand the definitions just right...''
=== Categorical Product ===
''Main article:
Product (category theory)''
The direct product can be abstracted to an arbitrary
category. In a general category, given a collection of objects ''A
i'' ''and'' a collection of
morphisms ''p
i'' from ''A'' to ''A
i'' 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 ''f
i'' from ''B'' to ''A
i'', there exists a unique morphism ''f'' from ''B'' to ''A'' such that ''f
i = p
i f'' and this object ''A'' is unique. This not only works for two factors, but arbitrarily (possibly infinitely) many.
For groups we similarly define the direct product of a more general, arbitrary collection of groups ''G
i'' 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 ''p
i'' in the definition above are the projection maps
:<math>\pi_i : G \to G_i\quad \mathrm{by} \quad \pi_i(g) = g_i</math>,
the functions that take ''g'' to its ''i''th component (''g
i'').
=== Related topics ===
*
Direct sum
*
Cartesian product
*
Coproduct
*
Free product
=== References ===
*Lang, S. ''Algebra''. New York: Springer-Verlag, 2002.
*''Add some more...''
fr:Produit direct
