In

_{V} and b_{W} denote the bases of ''V'' and ''W'', and ''v_{i}'' denotes the _{V}^{''i''}, and _{ij}'', is ''f^{j}_{i}'', and ''G_{ij}=g^{j}_{i}''.
The composition of more than two linear mappings can be similarly represented by a chain of matrix multiplication.

^{*}'' and ''W^{*}'' denote the

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...

, a product is the result of multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary Operation (mathematics), mathematical operations of arithmetic, with the ...

, or an expression that identifies factor
FACTOR (the Foundation to Assist Canadian Talent on Records) is a private non-profit organization "dedicated to providing assistance toward the growth and development of the Music of Canada, Canadian music industry".
FACTOR was founded in 1982 by r ...

s to be multiplied. For example, 30 is the product of 6 and 5 (the result of multiplication), and $x\backslash cdot\; (2+x)$ is the product of $x$ and $(2+x)$ (indicating that the two factors should be multiplied together).
The order in which real
Real may refer to:
* Reality
Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...

or complex
The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public university, Public rese ...

numbers are multiplied has no bearing on the product; this is known as the commutative law
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

of multiplication. When matrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...

or members of various other associative algebra
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s are multiplied, the product usually depends on the order of the factors. Matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the nu ...

, for example, is non-commutative, and so is multiplication in other algebras in general as well.
There are many different kinds of products in mathematics: besides being able to multiply just numbers, polynomials or matrices, one can also define products on many different algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s.
Product of two numbers

Product of two natural numbers

Placing several stones into a rectangular pattern with $r$ rows and $s$ columns gives :$r\; \backslash cdot\; s\; =\; \backslash sum\_^s\; r\; =\; \backslash underbrace\_=\; \backslash sum\_^r\; s\; =\; \backslash underbrace\_$ stones.Product of two integers

Integers allow positive and negative numbers. Their product is determined by the product of their positive amounts, combined with the sign derived from the following rule: :$\backslash begin\; \backslash hline\; \backslash cdot\; \&\; -\; \&\; +\; \backslash \backslash \; \backslash hline\; -\; \&\; +\; \&\; -\; \backslash \backslash \; +\; \&\; -\; \&\; +\; \backslash \backslash \; \backslash hline\; \backslash end$ (This rule is a necessary consequence of demandingdistributivity
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of multiplication over addition, and is not an ''additional rule''.)
In words, we have:
* Minus times Minus gives Plus
* Minus times Plus gives Minus
* Plus times Minus gives Minus
* Plus times Plus gives Plus
Product of two fractions

Two fractions can be multiplied by multiplying their numerators and denominators: :$\backslash frac\; \backslash cdot\; \backslash frac\; =\; \backslash frac$Product of two real numbers

The rigorous definition of the product of two real numbers is a byproduct of theConstruction of the real numbers
In mathematics, there are several ways of defining the real number system as an ordered field. The ''synthetic'' approach gives a list of axioms for the real numbers as a ''complete ordered field (mathematics), field''. Under the usual axioms of axi ...

. This construction implies that, for every real number there is a set of rational number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

such that is the least upper bound
are equal.
Image:Supremum illustration.svg, 250px, A set ''A'' of real numbers (blue circles), a set of upper bounds of ''A'' (red diamond and circles), and the smallest such upper bound, that is, the supremum of ''A'' (red diamond).
In mathematic ...

of the elements of :
:$a=\backslash sup\_\; x.$
If is another real number that is the least upper bound of , the product $a\backslash cdot\; b$
is defined as
:$a\backslash cdot\; b=\backslash sup\_x\backslash cdot\; y.$
This definition does not depend of a particular choice of and . That is, if they are changed without changing their least upper bound, then the least upper bound defining $a\backslash cdot\; b$ is not changed.
Product of two complex numbers

Two complex numbers can be multiplied by the distributive law and the fact that $i^2=-1$, as follows: :$\backslash begin\; (a\; +\; b\backslash ,\; i)\; \backslash cdot\; (c\; +\; d\backslash ,\; i)\; \&=\; a\; \backslash cdot\; c\; +\; a\; \backslash cdot\; d\backslash ,\; i\; +\; b\; \backslash ,\; i\; \backslash cdot\; c\; +\; b\; \backslash cdot\; d\; \backslash cdot\; i^2\backslash \backslash \; \&=\; (a\; \backslash cdot\; c\; -\; b\; \backslash cdot\; d)\; +\; (a\; \backslash cdot\; d\; +\; b\; \backslash cdot\; c)\; \backslash ,\; i\; \backslash end$Geometric meaning of complex multiplication

Complex numbers can be written inpolar coordinates
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

:
:$a\; +\; b\backslash ,\; i\; =\; r\; \backslash cdot\; (\; \backslash cos(\backslash varphi)\; +\; i\; \backslash sin(\backslash varphi)\; )\; =\; r\; \backslash cdot\; e\; ^$
Furthermore,
:$c\; +\; d\backslash ,\; i\; =\; s\; \backslash cdot\; (\; \backslash cos(\backslash psi)\; +\; i\backslash sin(\backslash psi)\; )\; =\; s\; \backslash cdot\; e^,$
from which one obtains
:$(a\; \backslash cdot\; c\; -\; b\; \backslash cdot\; d)\; +\; (a\; \backslash cdot\; d\; +\; b\; \backslash cdot\; c)\; i\; =\; r\; \backslash cdot\; s\; \backslash cdot\; e^.$
The geometric meaning is that the magnitudes are multiplied and the arguments are added.
Product of two quaternions

The product of twoquaternion
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s can be found in the article on quaternions
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion a ...

. Note, in this case, that $a\; \backslash cdot\; b$ and $b\; \backslash cdot\; a$ are in general different.
Product of a sequence

The product operator for theproduct of a sequence
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on Computer, computers, by an asterisk ) is one of the four Elementary arithmetic, eleme ...

is denoted by the capital Greek letter pi Π (in analogy to the use of the capital Sigma Σ as summation
In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: function (mathematics), fun ...

symbol). For example, the expression $\backslash textstyle\; \backslash prod\_^i^2$is another way of writing $1\; \backslash cdot\; 4\; \backslash cdot\; 9\; \backslash cdot\; 16\; \backslash cdot\; 25\; \backslash cdot\; 36$.
The product of a sequence consisting of only one number is just that number itself; the product of no factors at all is known as the empty product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, and is equal to 1.
Commutative rings

Commutative ring
In ring theory
In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies ...

s have a product operation.
Residue classes of integers

Residue classes in the rings $\backslash Z/N\backslash Z$ can be added: :$(a\; +\; N\backslash Z)\; +\; (b\; +\; N\backslash Z)\; =\; a\; +\; b\; +\; N\backslash Z$ and multiplied: :$(a\; +\; N\backslash Z)\; \backslash cdot\; (b\; +\; N\backslash Z)\; =\; a\; \backslash cdot\; b\; +\; N\backslash Z$Convolution

Two functions from the reals to itself can be multiplied in another way, called theconvolution
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

.
If
:$\backslash int\backslash limits\_^\backslash infty\; ,\; f(t),\; \backslash ,\backslash mathrmt\; <\; \backslash infty\backslash qquad\backslash mbox\backslash qquad\; \backslash int\backslash limits\_^\backslash infty\; ,\; g(t),\; \backslash ,\backslash mathrmt\; <\; \backslash infty,$
then the integral
:$(f*g)\; (t)\; \backslash ;:=\; \backslash int\backslash limits\_^\backslash infty\; f(\backslash tau)\backslash cdot\; g(t\; -\; \backslash tau)\backslash ,\backslash mathrm\backslash tau$
is well defined and is called the convolution.
Under the Fourier transform#REDIRECT Fourier transform
In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...

, convolution becomes point-wise function multiplication.
Polynomial rings

The product of two polynomials is given by the following: :$\backslash left(\backslash sum\_^n\; a\_i\; X^i\backslash right)\; \backslash cdot\; \backslash left(\backslash sum\_^m\; b\_j\; X^j\backslash right)\; =\; \backslash sum\_^\; c\_k\; X^k$ with :$c\_k\; =\; \backslash sum\_\; a\_i\; \backslash cdot\; b\_j$Products in linear algebra

There are many different kinds of products in linear algebra. Some of these have confusingly similar names (outer product
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and t ...

, exterior product
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...

) with very different meanings, while others have very different names (outer product, tensor product, Kronecker product) and yet convey essentially the same idea. A brief overview of these is given in the following sections.
Scalar multiplication

By the very definition of a vector space, one can form the product of any scalar with any vector, giving a map $\backslash R\; \backslash times\; V\; \backslash rightarrow\; V$.Scalar product

Ascalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

is a bi-linear map:
:$\backslash cdot\; :\; V\; \backslash times\; V\; \backslash rightarrow\; \backslash R$
with the following conditions, that $v\; \backslash cdot\; v\; >\; 0$ for all $0\; \backslash not=\; v\; \backslash in\; V$.
From the scalar product, one can define a norm
Norm, the Norm or NORM may refer to:
In academic disciplines
* Norm (geology), an estimate of the idealised mineral content of a rock
* Norm (philosophy)
Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...

by letting $\backslash ,\; v\backslash ,\; :=\; \backslash sqrt$.
The scalar product also allows one to define an angle between two vectors:
:$\backslash cos\backslash angle(v,\; w)\; =\; \backslash frac$
In $n$-dimensional Euclidean space, the standard scalar product (called the dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

) is given by:
:$\backslash left(\backslash sum\_^n\; \backslash alpha\_i\; e\_i\backslash right)\; \backslash cdot\; \backslash left(\backslash sum\_^n\; \backslash beta\_i\; e\_i\backslash right)\; =\; \backslash sum\_^n\; \backslash alpha\_i\backslash ,\backslash beta\_i$
Cross product in 3-dimensional space

Thecross product
In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

of two vectors in 3-dimensions is a vector perpendicular to the two factors, with length equal to the area of the parallelogram spanned by the two factors.
The cross product can also be expressed as the formal
Formal, formality, informal or informality imply the complying with, or not complying with, some set theory, set of requirements (substantial form, forms, in Ancient Greek). They may refer to:
Dress code and events
* Formal wear, attire for forma ...

determinant
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

:
:$\backslash mathbf\; =\; \backslash begin\; \backslash mathbf\; \&\; \backslash mathbf\; \&\; \backslash mathbf\; \backslash \backslash \; u\_1\; \&\; u\_2\; \&\; u\_3\; \backslash \backslash \; v\_1\; \&\; v\_2\; \&\; v\_3\; \backslash \backslash \; \backslash end$
Composition of linear mappings

A linear mapping can be defined as a function ''f'' between two vector spaces ''V'' and ''W'' with underlying field F, satisfying :$f(t\_1\; x\_1\; +\; t\_2\; x\_2)\; =\; t\_1\; f(x\_1)\; +\; t\_2\; f(x\_2),\; \backslash forall\; x\_1,\; x\_2\; \backslash in\; V,\; \backslash forall\; t\_1,\; t\_2\; \backslash in\; \backslash mathbb.$ If one only considers finite dimensional vector spaces, then :$f(\backslash mathbf)\; =\; f\backslash left(v\_i\; \backslash mathbf^i\backslash right)\; =\; v\_i\; f\backslash left(\backslash mathbf^i\backslash right)\; =\; \_j\; v\_i\; \backslash mathbf^j,$ in which bcomponent
Component may refer to:
In engineering, science, and technology Generic systems
*System components, an entity with discrete structure, such as an assembly or software module, within a system considered at a particular level of analysis
*Lumped ele ...

of v on bEinstein summation convention
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is applied.
Now we consider the composition of two linear mappings between finite dimensional vector spaces. Let the linear mapping ''f'' map ''V'' to ''W'', and let the linear mapping ''g'' map ''W'' to ''U''. Then one can get
:$g\; \backslash circ\; f(\backslash mathbf)\; =\; g\backslash left(\_j\; v\_i\; \backslash mathbf^j\backslash right)\; =\; \_k\; \_j\; v\_i\; \backslash mathbf^k.$
Or in matrix form:
:$g\; \backslash circ\; f(\backslash mathbf)\; =\; \backslash mathbf\; \backslash mathbf\; \backslash mathbf,$
in which the ''i''-row, ''j''-column element of F, denoted by ''FProduct of two matrices

Given two matrices :$A\; =\; (a\_)\_\; \backslash in\; \backslash R^$ and $B\; =\; (b\_)\_\backslash in\; \backslash R^$ their product is given by :$B\; \backslash cdot\; A\; =\; \backslash left(\; \backslash sum\_^r\; a\_\; \backslash cdot\; b\_\; \backslash right)\_\; \backslash ;\backslash in\backslash R^$Composition of linear functions as matrix product

There is a relationship between the composition of linear functions and the product of two matrices. To see this, let r = dim(U), s = dim(V) and t = dim(W) be the (finite)dimensions
thumb
, 236px
, The first four spatial dimensions, represented in a two-dimensional picture.
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature ...

of vector spaces U, V and W. Let
$\backslash mathcal\; U\; =\; \backslash $ be a basis
Basis may refer to:
Finance and accounting
*Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items.
Adjusted Basis or Adjusted Tax Basis refers to the original cost or other ba ...

of U,
$\backslash mathcal\; V\; =\; \backslash $ be a basis of V and
$\backslash mathcal\; W\; =\; \backslash $ be a basis of W. In terms of this basis, let
$A\; =\; M^\_(f)\; \backslash in\; \backslash R^$
be the matrix representing f : U → V and
$B\; =\; M^\_(g)\; \backslash in\; \backslash R^$
be the matrix representing g : V → W. Then
:$B\backslash cdot\; A\; =\; M^\_\; (g\; \backslash circ\; f)\; \backslash in\; \backslash R^$
is the matrix representing $g\; \backslash circ\; f\; :\; U\; \backslash rightarrow\; W$.
In other words: the matrix product is the description in coordinates of the composition of linear functions.
Tensor product of vector spaces

Given two finite dimensional vector spaces ''V'' and ''W'', the tensor product of them can be defined as a (2,0)-tensor satisfying: :$V\; \backslash otimes\; W(v,\; m)\; =\; V(v)\; W(w),\; \backslash forall\; v\; \backslash in\; V^*,\; \backslash forall\; w\; \backslash in\; W^*,$ where ''Vdual space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s of ''V'' and ''W''.
For infinite-dimensional vector spaces, one also has the:
* Tensor product of Hilbert spacesIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

* Topological tensor productIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

.
The tensor product, outer product
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and t ...

and Kronecker product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

all convey the same general idea. The differences between these are that the Kronecker product is just a tensor product of matrices, with respect to a previously-fixed basis, whereas the tensor product is usually given in its intrinsic definition. The outer product is simply the Kronecker product, limited to vectors (instead of matrices).
The class of all objects with a tensor product

In general, whenever one has two mathematical objects that can be combined in a way that behaves like a linear algebra tensor product, then this can be most generally understood as the internal product of amonoidal category
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

. That is, the monoidal category captures precisely the meaning of a tensor product; it captures exactly the notion of why it is that tensor products behave the way they do. More precisely, a monoidal category is the class
Class or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used differently f ...

of all things (of a given type
Type may refer to:
Science and technology Computing
* Typing, producing text via a keyboard, typewriter, etc.
* Data type, collection of values used for computations.
* File type
* TYPE (DOS command), a command to display contents of a file.
* Type ...

) that have a tensor product.
Other products in linear algebra

Other kinds of products in linear algebra include: * Hadamard product *Kronecker product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

* The product of tensor
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s:
** Wedge product or exterior product
** Interior product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

** Outer product
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and t ...

** Tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space that can be thought of as the ''space of all tensors'' that can be built from vectors from its constituent spac ...

Cartesian product

Inset theory
Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that are relevant to ...

, a Cartesian product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

is a mathematical operation
In mathematics, an operation is a Function (mathematics), function which takes zero or more input values (called ''operands'') to a well-defined output value. The number of operands is the arity of the operation.
The most commonly studied operati ...

which returns a set (or product set) from multiple sets. That is, for sets ''A'' and ''B'', the Cartesian product is the set of all ordered pair
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s —where and .
The class of all things (of a given type
Type may refer to:
Science and technology Computing
* Typing, producing text via a keyboard, typewriter, etc.
* Data type, collection of values used for computations.
* File type
* TYPE (DOS command), a command to display contents of a file.
* Type ...

) that have Cartesian products is called a Cartesian category. Many of these are Cartesian closed categories. Sets are an example of such objects.
Empty product

Theempty product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

on numbers and most algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s has the value of 1 (the identity element of multiplication), just like the empty sum
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

has the value of 0 (the identity element of addition). However, the concept of the empty product is more general, and requires special treatment in logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit ...

, set theory
Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that are relevant to ...

, computer programming
Computer programming is the process of designing and building an executable
In computing, executable code, an executable file, or an executable program, sometimes simply referred to as an executable or binary, causes a computer "to perform in ...

and category theory
Category theory formalizes and its concepts in terms of a called a ', whose nodes are called ''objects'', and whose labelled directed edges are called ''arrows'' (or s). A has two basic properties: the ability to the arrows , and the exi ...

.
Products over other algebraic structures

Products over other kinds ofalgebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s include:
* the Cartesian product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

of sets
* the direct product of groups
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, and also the semidirect product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, knit product and wreath product
* the free product of groups
* the product of rings
* the product of ideals
* the product topology, product of topological spaces
* the Wick product of random variables
* the cap product, cap, cup product, cup, Massey product, Massey and slant product in algebraic topology
* the smash product and wedge sum (sometimes called the wedge product) in homotopy
A few of the above products are examples of the general notion of an internal product in a monoidal category
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

; the rest are describable by the general notion of a product (category theory), product in category theory.
Products in category theory

All of the previous examples are special cases or examples of the general notion of a product. For the general treatment of the concept of a product, see product (category theory), which describes how to combine two objects of some kind to create an object, possibly of a different kind. But also, in category theory, one has: * the fiber product or pullback, * the product category, a category that is the product of categories. * the ultraproduct, in model theory. * the internal product of amonoidal category
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

, which captures the essence of a tensor product.
Other products

* A function's product integral (as a continuous equivalent to the product of a sequence or as the multiplicative version of the normal/standard/additive integral. The product integral is also known as "continuous product" or "multiplical". * Complex multiplication, a theory of elliptic curves.See also

* * Indefinite product * Infinite product * *Notes

References

Bibliography

* {{DEFAULTSORT:Product (Mathematics) Multiplication