Direct product
Encyclopedia
In mathematics
, one can often define a direct product of objects
already known, giving a new one. This is generally 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.
In a similar manner, we can talk about the product of more than two objects, e.g.
. We can even talk about product of infinitely many objects, e.g. 
.
one can define the direct product of two
groups (G, *) and (H, ●), denoted by G × H. For abelian group
s which are written additively, it may also be called the direct sum of two groups, denoted by
.
It is defined as follows:
(Note the operation * may be the same as ●.)
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 x 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 homomorphism
s for free: the projection maps
,
called the coordinate functions.
Also, every homomorphism f on the direct product is totally determined by its component functions
.
For any group (G, *), and any integer n ≥ 0, multiple application of the direct product gives the group of all n-tuple
s Gn (for n=0 the trivial group). Examples:
(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
is identical to the direct sum
. 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
and 
, 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).
s 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
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:
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
.
s 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 relation
s, T is in general not.
. In a general category, given a collection of objects Ai and a collection of morphism
s 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
,
the functions that take
to its ith component gi.
and 
, then we say that X is an internal direct product (of A and B); if A and B are not subobjects, then we say that this is an external direct product.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, one can often define a direct product of objects
already known, giving a new one. This is generally the Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of the underlying sets, together with a suitably defined structure on the product set.
More abstractly, one talks about the product in category theory
Product (category theory)
In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...
, which formalizes these notions.
Examples are the product of sets (see Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
), groups (described below), the product of rings
Product of rings
In mathematics, it is possible to combine several rings into one large product ring. This is done as follows: if I is some index set and Ri is a ring for every i in I, then the cartesian product Πi in I Ri can be turned into a ring by defining the operations coordinatewise, i.e...
and of other algebraic structures
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
. The product of topological spaces
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 topology called the product topology...
is another instance.
There is also the direct sum
Direct sum
In mathematics, one can often define a direct sum of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets , together with a suitably defined structure. More abstractly, the direct sum is often, but not always, the coproduct in the category in question...
– in some areas this is used interchangeably, in others it is a different concept.
Examples
- If we think of
as the set of real numbers, then the direct product 
is precisely just the cartesian productCartesian productIn mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
,
.
- If we think of
as the groupGroup (mathematics)In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
of real numbers under addition, then the direct product
still consists of 
. The difference between this and the preceding example is that 
is now a group. We have to also say how to add their elements. This is done by letting 
.
- If we think of
as the ringRing (mathematics)In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
of real numbers, then the direct product
again consists of 
. To make this a ring, we say how their elements are added, 
, and how they are multiplied 
.
- However, if we think of
as the fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
of real numbers, then the direct product
does not exist - naively defining 
in a similar manner to the above examples would not result in a field since the element 
does not have a multiplicative inverse.
In a similar manner, we can talk about the product of more than two objects, e.g.
. We can even talk about product of infinitely many objects, e.g. .Group direct product
In group theoryGroup (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
one can define the direct product of two
groups (G, *) and (H, ●), denoted by G × H. For abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
s which are written additively, it may also be called the direct sum of two groups, denoted by
.It is defined as follows:
- the set of the elements of the new group is the cartesian productCartesian productIn mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
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 ● h' )
(Note the operation * may be the same as ●.)
This construction gives a new group. It has a normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
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 x H. A relaxation of these conditions, requiring only one subgroup to be normal, gives the semidirect product
Semidirect product
In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct 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 homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
s for free: the projection maps
,called the coordinate functions.
Also, every homomorphism f on the direct product is totally determined by its component functions
.For any group (G, *), and any integer n ≥ 0, multiple application of the direct product gives the group of all n-tuple
Tuple
In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...
s Gn (for n=0 the trivial group). Examples:
- Zn
- Rn (with additional vector spaceVector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
structure this is called Euclidean spaceEuclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, see below)
Direct product of modules
The direct product for modulesModule (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
(not to be confused with the tensor product
Tensor product of modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps to be carried out in terms of linear maps...
) is very similar to the one defined for groups above, using the cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
with the operation of addition being componentwise, and the scalar multiplication just distributing over all the components. Starting from R we get Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
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
is identical to the direct sumDirect sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...
. 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 TheoryCategory theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
: the direct sum is the coproduct
Coproduct
In category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...
, while the direct product is the product.
For example, consider
and , 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 spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s Xi for i in I, some index set, once again makes use of the cartesian product
Defining the topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
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
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 topology 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
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
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:
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
Box topology
In topology, the cartesian product of topological spaces can be given several different topologies. The canonical one is the product topology, because it fits rather nicely with the categorical notion of a product. Another possibility is the box topology. The box topology has a somewhat more...
. 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
Tychonoff's theorem
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact. The theorem is named after Andrey Nikolayevich Tychonoff, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the...
, is yet another equivalence to the axiom of choice.
For more properties and equivalent formulations, see the separate entry product topology
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 topology called the product topology...
.
Direct product of binary relations
On the Cartesian product of two sets with binary relationBinary relation
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
s R and S, define (a, b) T (c, d) as a R c and b S d. If R and S are both reflexive
Reflexive relation
In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S. For example, ~ could be "is equal to".-Related terms:...
, irreflexive, transitive
Transitive relation
In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
, symmetric
Symmetric relation
In mathematics, a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a.In mathematical notation, this is:...
, or antisymmetric
Antisymmetric relation
In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in Xor, equivalently,In mathematical notation, this is:\forall a, b \in X,\ R \and R \; \Rightarrow \; a = bor, equivalently,...
, relation T has the same property. Combining properties it follows that this also applies for being a preorder
Preorder
In mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...
and being an equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
. However, if R and S are total relation
Total relation
In mathematics, a binary relation R over a set X is total if for all a and b in X, a is related to b or b is related to a .In mathematical notation, this is\forall a, b \in X,\ a R b \or b R a....
s, T is in general not.
Categorical product
The direct product can be abstracted to an arbitrary categoryCategory theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
. In a general category, given a collection of objects Ai and a collection of morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s 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
,the functions that take
to its ith component gi.Internal and External direct product
Some authors draw a distinction between an internal direct product and an external direct product. If and , then we say that X is an internal direct product (of A and B); if A and B are not subobjects, then we say that this is an 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.See also
- Direct sumDirect sumIn mathematics, one can often define a direct sum of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets , together with a suitably defined structure. More abstractly, the direct sum is often, but not always, the coproduct in the category in question...
- Cartesian productCartesian productIn mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
- CoproductCoproductIn category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...
- Free productFree productIn mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties...
- Semidirect productSemidirect productIn mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
- Zappa-Szep productZappa-Szep productIn mathematics, especially group theory, the Zappa–Szép product describes a way in which a group can be constructed from two subgroups. It is a generalization of the direct and semidirect products...
- Tensor product of graphsTensor product of graphsIn graph theory, the tensor product G × H of graphs G and H is a graph such that* the vertex set of G × H is the Cartesian product V × V; and...
- Orders on the Cartesian product of totally ordered sets