Direct product

Direct product

In mathematics, one can often define a direct product of objects already known, giving a new one. Examples are the product of sets (see Cartesian product), groups (described below), the product of rings and of other algebraic structures. The product of topological spaces is another instance.

Examples

  • If we think of mathbb{R} as the set of real numbers, then the direct product mathbb{R}times mathbb{R} is precisely just the cartesian product, { (x,y) | x,y in mathbb{R} }.
  • If we think of mathbb{R} as the group of real numbers under addition, then the direct product mathbb{R}times mathbb{R} still consists of { (x,y) | x,y in mathbb{R} }. The difference between this and the preceding example is that mathbb{R}times mathbb{R} is now a group. We have to also say how to add their elements. This is done by letting (a,b) + (c,d) = (a+c, b+d).
  • If we think of mathbb{R} as the ring of real numbers, then the direct product mathbb{R}times mathbb{R} again consists of { (x,y) | x,y in mathbb{R} }. To make this a ring, we say how their elements are added, (a,b) + (c,d) = (a+c, b+d), and how they are multiplied (a,b) (c,d) = (ac, bd).
  • However, if we think of mathbb{R} as the field of real numbers, then the direct product mathbb{R}times mathbb{R} does not exist! Naively defining { (x,y) | x,y in mathbb{R} } in a similar manner to the above examples would not result in a field since the element (1,0) does not have a multiplicative inverse.

In a similar manner, we can talk about the product of more than two objects, e.g. mathbb{R} times mathbb{R} times mathbb{R} times mathbb{R}. We can even talk about product of infinitely many objects, e.g. mathbb{R} times mathbb{R} times mathbb{R} times dotsb.

Group direct product

In group theory one can define the direct product of two groups (G, *) and (H, o), denoted by G × H. For abelian groups which are written additively, it may also be called the direct sum of two groups, denoted by G oplus H.

It is defined as follows:

  • the set of the elements of the new group is 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, C2: say {1, a} and {1, b}. Then C2×C2 = {(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,b2) = (1,1).

With a direct product, we get some natural group homomorphisms for free: the projection maps

pi_1 colon G times H to Gquad mathrm{by} quad pi_1(g, h) = g,
pi_2 colon G times H to Hquad mathrm{by} quad pi_2(g, h) = h
called the coordinate functions.

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

For any group (G, *), and any integer n ≥ 0, multiple application of the direct product gives the group of all n-tuples Gn (for n=0 the trivial group). Examples:

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 Rn, the prototypical example of a real n-dimensional vector space. The direct product of Rm and Rn is Rm + n.

Note that a direct product for a finite index prod_{i=1}^n X_i is identical to the direct sum bigoplus_{i=1}^n X_i . 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: the direct sum is the coproduct, while the direct product is the product.

For example, consider X=prod_{i=1}^infty mathcal{R} and Y=bigoplus_{i=1}^infty mathcal{R}, 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, YX).

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

prod_{i in I} X_i

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:

mathcal B = { U_1 times cdots times U_n | U_i mathrm{open in} X_i }

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:

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}

(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.

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, relation T has the same property. Combining properties it follows that this also applies for being a preorder and being an equivalence relation. However, if R and S are total relations, T is in general not.

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 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

pi_i colon G to G_iquad mathrm{by} quad pi_i(g) = g_i,

the functions that take (g_j)_{j in I} to its ith component gi.

Metric and norm

A metric on a Cartesian product of metric spaces, and a norm on a direct product of normed vector spaces, can be defined in various ways, see for example .

See also

Notes

References

  • Lang, S. Algebra. New York: Springer-Verlag, 2002.

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