TheInfoList

OR:

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 In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\tim ...
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 (described below),
rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
, 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 point ...
s is another instance. There is also the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...
– in some areas this is used interchangeably, while in others it is a different concept.

# Examples

* If we think of $\R$ as the set of real numbers, then the direct product $\R \times \R$ is just the Cartesian product $\.$ * If we think of $\R$ as the group of real numbers under addition, then the direct product $\R\times \R$ still has $\$ as its underlying set. The difference between this and the preceding example is that $\R \times \R$ is now a group, and so we have to also say how to add their elements. This is done by defining $\left(a,b\right) + \left(c,d\right) = \left(a+c, b+d\right).$ * If we think of $\R$ as the ring of real numbers, then the direct product $\R\times \R$ again has $\$ as its underlying set. The ring structure consists of addition defined by $\left(a,b\right) + \left(c,d\right) = \left(a+c, b+d\right)$ and multiplication defined by $\left(a,b\right) \left(c,d\right) = \left(ac, bd\right).$ * Although the ring $\R$ is a field, $\R \times \R$ is not one, because the element $\left(1,0\right)$ 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''/''b ...
. In a similar manner, we can talk about the direct product of finitely many algebraic structures, for example, $\R \times \R \times \R \times \R.$ This relies on the fact that the direct product is
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. That is, $\left(A \times B\right) \times C \cong A \times \left(B \times C\right)$ for any algebraic structures $A,$ $B,$ and $C$ of the same kind. The direct product is also
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
up to isomorphism, that is, $A \times B \cong B \times A$ for any algebraic structures $A$ and $B$ 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 $\mathbb R,$ which we write as $\R \times \R \times \R \times \dotsb.$

# 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 as ...
one can define the direct product of two groups $\left(G, \circ\right)$ and $\left(H, \cdot\right),$ denoted by $G \times H.$ 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 commu ...
s 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 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 $G \text H,$ that is $\;$ * on these elements put an operation, defined element-wise: $(g, h) \times \left(g', h'\right) = \left(g \circ g', h \cdot h'\right)$ Note that $\left(G, \circ\right)$ may be the same as $\left(H, \cdot\right).$ This construction gives a new group. It has a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
isomorphic to $G$ (given by the elements of the form $\left(g, 1\right)$), and one isomorphic to $H$ (comprising the elements $\left(1, h\right)$). The reverse also holds. There is the following recognition theorem: If a group $K$ contains two normal subgroups $G \text H,$ such that $K = GH$ and the intersection of $G \text H$ contains only the identity, then $K$ is isomorphic to $G \times H.$ A relaxation of these conditions, requiring only one subgroup to be normal, gives the
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in wh ...
. As an example, take as $G \text H$ two copies of the unique (up to isomorphisms) group of order 2, $C^2:$ say $\ \text \.$ Then $C_2 \times C_2 = \,$ with the operation element by element. For instance, $\left(1,b\right)^* \left(a,1\right) = \left\left(1^* a, b^* 1\right\right) = \left(a, b\right),$ and$\left(1,b\right)^* \left(1, b\right) = \left\left(1, b^2\right\right) = \left(1, 1\right).$ With a direct product, we get some natural
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...
s for free: the projection maps defined by $\begin \pi_1: G \times H \to G, \ \ \pi_1(g, h) &= g \\ \pi_2: G \times H \to H, \ \ \pi_2(g, h) &= h \end$ are 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 $\left(G, \circ\right)$ and any integer $n \geq 0,$ repeated application of the direct product gives the group of all $n$-
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 de ...
s $G^n$ (for $n = 0,$ this is the
trivial group In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually ...
), for example $\Z^n$ and $\R^n.$

# Direct product of modules

The direct product for modules (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 \otimes ...
) 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 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 Euclidean ...
$\R^n,$ the prototypical example of a real $n$-dimensional vector space. The direct product of $\R^m$ and $\R^n$ is $\R^.$ Note that a direct product for a finite index $\prod_^n X_i$ is canonically isomorphic to the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...
$\bigoplus_^n X_i.$ 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 Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
: the direct sum is the
coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The copro ...
, while the direct product is the product. For example, consider $X = \prod_^\infty \R$ and $Y = \bigoplus_^\infty \R,$ 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, $\left(1, 0, 0, 0, \ldots\right)$ is in $Y$ but $\left(1, 1, 1, 1, \ldots\right)$ is not. Both of these sequences are in the direct product $X;$ in fact, $Y$ is a proper subset of $X$ (that is, $Y \subset X$).

# 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 point ...
s $X_i$ for $i$ in $I,$ some index set, once again makes use of the Cartesian product $\prod_ X_i.$ 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 Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates *Basis trading, a trading strategy consisting of ...
of open sets to be the collection of all Cartesian products of open subsets from each factor: $\mathcal B = \left\.$ 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-seemin ...
. For example, directly defining the product topology on $\R^2$ 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 (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: $\mathcal B = \left\.$ 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 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 o ...
. 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-seemin ...
.

# Direct product of binary relations

On the Cartesian product of two sets with
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...
s $R \text S,$ define $\left(a, b\right) T \left(c, d\right)$ as $a R c \text b S d.$ If $R \text S$ are both reflexive, irreflexive, transitive,
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
, or antisymmetric, then $T$ will be also. Similarly, totality of $T$ is inherited from $R \text S.$ Combining properties it follows that this also applies for being a
preorder In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cas ...
and being an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
. However if $R \text S$ are connected relations, $T$ need not be connected; for example, the direct product of $\,\leq\,$ on $\N$ with itself does not relate $\left(1, 2\right) \text \left(2, 1\right).$

# Direct product in universal algebra

If $\Sigma$ is a fixed
signature A signature (; from la, signare, "to sign") is a handwritten (and often 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 intent. The writer of a ...
, $I$ is an arbitrary (possibly infinite) index set, and $\left\left(\mathbf_i\right\right)_$ is an
indexed family In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a ''family of real numbers, indexed by the set of integers'' is a collection of real numbers, whe ...
of $\Sigma$ algebras, the direct product $\mathbf = \prod_ \mathbf_i$ is a $\Sigma$ algebra defined as follows: * The universe set $A$ of $\mathbf$ is the Cartesian product of the universe sets $A_i$ of $\mathbf_i,$ formally: $A = \prod_ A_i.$ * For each $n$ and each $n$-ary operation symbol $f \in \Sigma,$ its interpretation $f^$ in $\mathbf$ is defined componentwise, formally: for all $a_1, \dotsc, a_n \in A$ and each $i \in I,$ the $i$th component of $f^\!\left\left(a_1, \dotsc, a_n\right\right)$ is defined as $f^\!\left\left(a_1\left(i\right), \dotsc, a_n\left(i\right)\right\right).$ For each $i \in I,$ the $i$th projection $\pi_i : A \to A_i$ is defined by $\pi_i\left(a\right) = a\left(i\right).$ It is a surjective homomorphism between the $\Sigma$ algebras $\mathbf \text \mathbf_i.$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 $I = \,$ the direct product of two $\Sigma$ algebras $\mathbf_1 \text \mathbf_2$ is obtained, written as $\mathbf = \mathbf_1 \times \mathbf_2.$ If $\Sigma$ just contains one binary operation $f,$ the above definition of the direct product of groups is obtained, using the notation $A_1 = G, A_2 = H,$ $f^ = \circ, \ f^ = \cdot, \ \text f^A = \times.$ Similarly, the definition of the direct product of modules is subsumed here.

# Categorical product

The direct product can be abstracted to an arbitrary
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) *Categories (Peirce) ...
. In a category, given a collection of objects $\left(A_i\right)_$ indexed by a set $I$, a product of these objects is an object $A$ together with
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s $p_i \colon A \to A_i$ for all $i \in I$, such that if $B$ is any other object with morphisms $f_i \colon B \to A_i$ for all $i \in I$, there exists a unique morphism $B \to A$ whose composition with $p_i$ equals $f_i$ for every $i$. Such $A$ and $\left(p_i\right)_$ do not always exist. If they do exist, then $\left(A,\left(p_i\right)_\right)$ is unique up to isomorphism, and $A$ is denoted $\prod_ A_i$. In the special case of the category of groups, a product always exists: the underlying set of $\prod_ A_i$ is the Cartesian product of the underlying sets of the $A_i$, the group operation is componentwise multiplication, and the (homo)morphism $p_i \colon A \to A_i$ is the projection sending each tuple to its $i$th coordinate.

# Internal and external direct product

Some authors draw a distinction between an internal direct product and an external direct product. If $A, B \subseteq X$ and $A \times B \cong X,$ then we say that $X$ is an ''internal'' direct product of $A \text B,$ while if $A \text B$ are not subobjects then we say that this is an ''external'' direct product.