In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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,
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
(described below),
rings, and other
algebraic structures. The
product of
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
s is another instance.
There is also the
direct sum – in some areas this is used interchangeably, while in others it is a different concept.
Examples
* If we think of
as the set of real numbers, then the direct product
is just the Cartesian product
* If we think of
as the
group of real numbers under addition, then the direct product
still has
as its underlying set. The difference between this and the preceding example is that
is now a group, and so we have to also say how to add their elements. This is done by defining
* If we think of
as the
ring of real numbers, then the direct product
again has
as its underlying set. The ring structure consists of addition defined by
and multiplication defined by
* Although the ring
is a
field,
is not one, because the element
does not have a
multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
.
In a similar manner, we can talk about the direct product of finitely many algebraic structures, for example,
This relies on the fact that the direct product is
associative up to isomorphism. That is,
for any algebraic structures
and
of the same kind. The direct product is also
commutative up to isomorphism, that is,
for any algebraic structures
and
of the same kind. We can even talk about the direct product of infinitely many algebraic structures; for example we can take the direct product of
countably many copies of
which we write as
Group direct product
In
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
one can define the direct product of two groups
and
denoted by
For
abelian group
In mathematics, 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 the order in which they are written. That is, the group operation is comm ...
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
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of the elements of the new group is the ''Cartesian product'' of the sets of elements of
that is
* on these elements put an operation, defined element-wise:
Note that
may be the same as
This construction gives a new group. It has a
normal subgroup isomorphic to
(given by the elements of the form
), and one isomorphic to
(comprising the elements
).
The reverse also holds. There is the following recognition theorem: If a group
contains two normal subgroups
such that
and the intersection of
contains only the identity, then
is isomorphic to
A relaxation of these conditions, requiring only one subgroup to be normal, gives the
semidirect product.
As an example, take as
two copies of the unique (up to isomorphisms) group of order 2,
say
Then
with the operation element by element. For instance,
and
With a direct product, we get some natural
group homomorphisms for free: the projection maps defined by
are called the coordinate functions.
Also, every homomorphism
to the direct product is totally determined by its component functions
For any group
and any integer
repeated application of the direct product gives the group of all
-
tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
s
(for
this is the
trivial group), for example
and
Direct product of modules
The direct product for
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
(not to be confused with the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
) 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
we get
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
the prototypical example of a real
-dimensional vector space. The direct product of
and
is
Note that a direct product for a finite index
is canonically isomorphic to the
direct sum The direct sum and direct product are not isomorphic 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
For example,
is in
but
is not. Both of these sequences are in the direct product
in fact,
is a proper subset of
(that is,
).
Topological space direct product
The direct product for a collection of
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
s
for
in
some index set, once again makes use of the Cartesian product
Defining the
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
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
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. This topology differs from another, perhaps more natural-seem ...
. For example, directly defining the product topology on
by the open sets of
(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 (that is, 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 In topology, the cartesian product of topological spaces can be given several different topologies. One of the more obvious choices is the box topology, where a base is given by the Cartesian products of open sets in the component spaces. Another p ...
. 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
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
.
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 topology called the product topology. This topology differs from another, perhaps more natural-seem ...
.
Direct product of binary relations
On the Cartesian product of two sets with
binary relations
define
as
If
are both
reflexive,
irreflexive
In mathematics, a binary relation ''R'' on a set ''X'' is reflexive if it relates every element of ''X'' to itself.
An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal ...
,
transitive,
symmetric, or
antisymmetric, then
will be also. Similarly,
totality of
is inherited from
Combining properties it follows that this also applies for being a
preorder and being an
equivalence relation. However if
are
connected relation
In mathematics, a relation on a set is called connected or total if it relates (or "compares") all pairs of elements of the set in one direction or the other while it is called strongly connected if it relates pairs of elements.
As described i ...
s,
need not be connected; for example, the direct product of
on
with itself does not relate
Direct product in universal algebra
If
is a fixed
signature
A signature (; from la, signare, "to sign") is a Handwriting, handwritten (and often Stylization, stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and ...
,
is an arbitrary (possibly infinite) index set, and
is an
indexed family of
algebras, the direct product
is a
algebra defined as follows:
* The universe set
of
is the Cartesian product of the universe sets
of
formally:
* For each
and each
-ary operation symbol
its interpretation
in
is defined componentwise, formally: for all
and each
the
th component of
is defined as
For each
the
th projection
is defined by
It is a
surjective homomorphism between the
algebras
[Stanley N. Burris and H.P. Sankappanavar, 1981. ]
A Course in Universal Algebra.
' Springer-Verlag. . Here: Def.7.8, p.53 (=p. 67 in pdf file)
As a special case, if the index set
the direct product of two
algebras
is obtained, written as
If
just contains one binary operation
the
above definition of the direct product of groups is obtained, using the notation
Similarly, the definition of the direct product of modules is subsumed here.
Categorical product
The direct product can be abstracted to an arbitrary
category. In a category, given a collection of objects
indexed by a set
, a product of these objects is an object
together with
morphisms
for all
, such that if
is any other object with morphisms
for all
, there exists a unique morphism
whose composition with
equals
for every
.
Such
and
do not always exist. If they do exist, then
is unique up to isomorphism, and
is denoted
.
In the special case of the category of groups, a product always exists: the underlying set of
is the Cartesian product of the underlying sets of the
, the group operation is componentwise multiplication, and the (homo)morphism
is the projection sending each tuple to its
th coordinate.
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
is an ''internal'' direct product of
while if
are not subobjects then we say that this is an ''external'' direct product.
See also
*
*
*
*
*
*
*
*
Notes
References
*
{{DEFAULTSORT:Direct Product
Abstract algebra
ru:Прямое произведение#Прямое произведение групп