Direct product (original) (raw)

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:

(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

,

called the coordinate functions.

Also, every homomorphism f on the direct product is totally determined by its component functions .

Categorical Product

The preceding two observations of the direct product makes the direct product the product in the category of groups. 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 (possibly 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 g to its _i_th component (gi).

More abstract formulations and generalizations of the categorical product (as if this weren't abstract or general enough!) can be found in the separate entry on categorical products.

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 Rn).

Direct Product for Topological Spaces

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 toplogy).

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:

(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 Munkres for an example). 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...

Products vs. Coproducts

In general category theory, as described above in the groups section, a related and "dual" concept to the product is what is called the categorical sum or coproduct, which basically reverses all the arrows in the diagrams. In full detail, given a collection of objects Aj and a collection of morphisms ij from Aj to A, with j ranging in some index set J, an object A is said to be a coproduct in the category if, for any object B and any collection of morphisms gj from Aj to B, there exists a unique morphism g from A to B such that gj = g ij and this object A is unique.

Despite this innocuous-looking change in the name and notation, coproducts can be dramatically different from products. The coproduct in the category of sets is simply the disjoint union, the maps ij being the inclusions. Unlike direct products, coproducts in other categories are not all obviously based on the notion for sets, because unions don't behave well with respect to preserving operations (e.g. the union of two groups need not be a group), and so coproducts in different categories can be dramatically different from each other. For example, the coproduct in the category of groups, called the free product is quite complicated. On the other hand, in the category of abelian groups (including vector spaces), the coproduct, called the direct sum, consists of the elements of the direct product which have only finitely many nonzero terms (this completely coincides with the direct product in the case of finitely many factors--so much for "dramatically different"--as a consequence, since most introductory linear algebra courses deal with only finite-dimensional vector spaces, nobody really hears much about direct sums until later on, perhaps for the first time on an algebra qualifying exam!).

But despite all this dissmilarity, there is still, at the heart of the whole thing, a disjoint union: the direct sum of abelian groups is the group generated by the "almost" disjoint union (disjoint union of all nonzero elements, together with a common zero), similarly for vector spaces: the space spanned by the "almost" disjoint union; the free product for groups is generated by the set of all letters from a similar "almost disjoint" union where no two elements from different sets are allowed to commute.

See also: Free product, Cartesian product, Group, Direct sum.

References

Lang, S. Algebra. New York: Springer-Verlag, 2002.Add some more...