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 (see Cartesian product), groups (described below), the product of rings and of other algebraic structures. The product of topological spaces is another instance.

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

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

Main article: Direct product of groups

In group theory one can define the direct product of two groups (G, ∘) and (H, ∙), 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 ∈ G, h ∈ H};
on these elements put an operation, defined element-wise:
(g, h) × (g', h' ) = (g ∘ g', h ∙ h')

(Note (G, ∘) may be the same as (H, ∙).)

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 is isomorphic to G × H. A relaxation of these conditions, requiring only one subgroup to be normal, 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

{\begin{aligned}\pi _{1}:G\times H\to G\quad {\text{by}}\quad \pi _{1}(g,h)&=g\\\pi _{2}:G\times H\to H\quad {\text{by}}\quad \pi _{2}(g,h)&=h\end{aligned}}

called the coordinate functions.

Also, every homomorphism f to 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:

Zⁿ
Rⁿ (with additional vector space structure this is called Euclidean space, see below)

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 R^m and Rⁿ is R^{m + 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 in the sense of category theory: the direct sum is the coproduct, while the direct product is the product.

For example, consider $X=\prod _{i=1}^{\infty }\mathbb {R}$ and $Y=\bigoplus _{i=1}^{\infty }\mathbb {R}$ , 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, Y ⊂ X).

Topological space direct product

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

\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}\ {\Big |}\ (\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\}.

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.^[1] 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.

Direct product in universal algebra

If Σ is a fixed signature, I is an arbitrary (possibly infinite) index set, and (A_i)_i∈I is an indexed family of Σ algebras, the direct product A = ∏_i∈I A_i is a Σ algebra defined as follows:

The universe set A of A is the Cartesian product of the universe sets A_i of A_i, formally: A = ∏_i∈I A_i;
For each n and each n-ary operation symbol f ∈ Σ, its interpretation f^A in A is defined componentwise, formally: for all a₁, …, a_n ∈ A and each i ∈ I, the ith component of f^A(a₁, …, a_n) is defined as f^A_i(a₁(i), …, a_n(i)).

For each i ∈ I, the ith projection π_i : A → A_i is defined by π_i(a) = a(i). It is a surjective homomorphism between the Σ algebras A and A_i.^[2]

As a special case, if the index set I = { 1, 2 }, the direct product of two Σ algebras A₁ and A₂ is obtained, written as A = A₁ × A₂. If Σ just contains one binary operation f, the above definition of the direct product of groups is obtained, using the notation A₁ = G, A₂ = H, f^A₁ = ∘, f^A₂ = ∙, and f^A = ×. Similarly, the definition of the direct product of modules is subsumed here.

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 (even 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

\pi _{i}\colon G\to G_{i}\quad \mathrm {by} \quad \pi _{i}(g)=g_{i}

the functions that take $(g_{j})_{j\in I}$ to its ith component g_i.

Internal and external direct product

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 p-norm.

Notes

↑ Equivalence and Order
↑ 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)

References

Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556

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