Direct product

In mathematics, one can often define a direct product of objects already known, giving a new one. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Examples are the product of sets (see Cartesian product), groups (described below), the product of rings and of other algebraic structures. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. In Mathematics, it is possible to combine several rings into one large product ring. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules The product of topological spaces is another instance. In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural

1 Examples
2 Group direct product
3 Direct product of modules
4 Topological space direct product
5 Direct product of binary relations
6 Categorical product
7 Metric and norm
8 See also
9 References

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, $\{ (x,y) | x,y \in \mathbb{R} \}$ . Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory.

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 $\{ (x,y) | x,y \in \mathbb{R} \}$ . In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element 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 $\{ (x,y) | x,y \in \mathbb{R} \}$ . In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real 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 $(a, b)(c, d) = (a c, b d)$ .

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 Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division

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. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element For abelian groups which are written additively, it may also be called the direct sum of two groups, denoted by $G \oplus H$ . An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction

It is defined 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. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. )

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)). In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

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. In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can

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

$\pi_1 \colon G \times H \to G\quad \mathrm{by} \quad \pi_1(g, h) = g$ ,

$\pi_2 \colon G \times H \to H\quad \mathrm{by} \quad \pi_2(g, h) = h$

called the coordinate functions. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function

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). In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple 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. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. 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 symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction 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. In Category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological

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, 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. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of 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. In Mathematics, a base (or basis) B for a Topological space X with topology T is a collection of Open sets 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). In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined

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:

$\mathcal B = \left\{ \prod_{i \in I} U_i\ |\ (\exists j_1,\ldots,j_n)(U_{j_i}\ \mathrm{open\ in}\ X_{j_i})\ \mathrm{and}\ (\forall i \neq j_1,\ldots,j_n)(U_i = X_i) \right\}$

(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). In Topology, the Cartesian product of Topological spaces can be given several different topologies 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. In Mathematics, Tychonoff's theorem states that the product of any collection of compact Topological spaces is compact In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory.

For more properties and equivalent formulations, see the separate entry product topology. In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural

Direct product of binary relations

On the Cartesian product of two sets with binary relations R and S, the product order is defined as (a, b) T (c, d) as a R c and b S d. In Mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of In Mathematics, given two Ordered sets A and B, one can induce an ordering on the Cartesian product A × B. If R and S are both reflexive, irreflexive, transitive, symmetric, or antisymmetric, relation T has the same property. In Set theory, a Binary relation can have among other properties reflexivity or irreflexivity. In Set theory, a Binary relation can have among other properties reflexivity or irreflexivity. In Mathematics, a Binary relation R over a set X is transitive if whenever an element a is related to an element b In Mathematics, a Binary relation R over a set X is symmetric if it holds for all a and b in X that In Mathematics, a Binary relation R on a set X is antisymmetric if for all a and b in X, if ^[1] Combining properties it follows that this also applies for being a preorder and being an equivalence relation. In Mathematics, especially in Order theory, preorders are Binary relations that satisfy certain conditions In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" However, if R and S are total relations, T is in general not. In Mathematics, a Binary relation R over a set X is total if it holds for all a and b in X that

Categorical product

Main article: Product (category theory)

The direct product can be abstracted to an arbitrary category. In Mathematics, given two Ordered sets A and B, one can induce an ordering on the Cartesian product A × B. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation In Category theory, the product of two (or more objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets 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. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and 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.

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. In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length

References

^ Equivalence and Order

Lang, S. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. In Category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological In Abstract algebra, the free product of groups constructs a group from two or more given ones In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can In Mathematics, especially Group theory, the Zappa-Szep product (also known as the knit product) describes a way in which a group can be constructed In Graph theory, the tensor product G × H of graphs G and H is a graph such that the vertex set of G × Algebra. New York: Springer-Verlag, 2002.

Dictionary

-noun

(set theory) The direct product of an indexed family of sets is the set of functions from the indexing set to the union of the family, whose values at any given index lie in the set indexed thereby.
(mathematics) The generalization of direct product (1) to arbitrary categories.

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