Inmathematics, adirect product of objects already known can often be defined by giving a new one. That induces a structure on theCartesian product of the underlyingsets from that of the contributing objects. More abstractly, theproduct in category theory is mentioned, which formalizes those notions.

Examples are the product of sets,groups (described below),rings, and otheralgebraic structures. Theproduct oftopological spaces is another instance.

There is also thedirect sum, which in some areas used interchangeably but in others is a different concept.

• If${\displaystyle \mathbb{R}}$ is thought of as the set ofreal numbers without further structure, the direct product${\displaystyle \mathbb{R}\times \mathbb{R}}$ is just the Cartesian product${\displaystyle \{(x,y):x,y\in \mathbb{R}\}.}$
• If${\displaystyle \mathbb{R}}$ is thought of as thegroup of real numbers under addition, the direct product${\displaystyle \mathbb{R}\times \mathbb{R}}$ still has${\displaystyle \{(x,y):x,y\in \mathbb{R}\}}$ as its underlying set. The difference between this and the preceding examples is that${\displaystyle \mathbb{R}\times \mathbb{R}}$ is now a group and so how to add their elements must also be stated. That is done by defining${\displaystyle (a,b)+(c,d)=(a+c,b+d).}$
• If${\displaystyle \mathbb{R}}$ is thought of as thering of real numbers, the direct product${\displaystyle \mathbb{R}\times \mathbb{R}}$ again has${\displaystyle \{(x,y):x,y\in \mathbb{R}\}}$ as its underlying set. The ring structure consists of addition defined by${\displaystyle (a,b)+(c,d)=(a+c,b+d)}$ and multiplication defined by${\displaystyle (a,b)(c,d)=(ac,bd).}$
• Although the ring${\displaystyle \mathbb{R}}$ is afield,${\displaystyle \mathbb{R}\times \mathbb{R}}$ is not because the nonzero element${\displaystyle (1,0)}$ does not have amultiplicative inverse.

In a similar manner, the direct product of finitely many algebraic structures can be talked about; for example,${\displaystyle \mathbb{R}\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}.}$ That relies on the direct product beingassociativeup toisomorphism. That is,${\displaystyle (A\times B)\times C\cong A\times (B\times C)}$ for any algebraic structures${\displaystyle A,}$${\displaystyle B,}$ and${\displaystyle C}$ of the same kind. The direct product is alsocommutative up to isomorphism; that is,${\displaystyle A\times B\cong B\times A}$ for any algebraic structures${\displaystyle A}$ and${\displaystyle B}$ of the same kind. Even the direct product of infinitely many algebraic structures can be talked about; for example, the direct product ofcountably many copies of${\displaystyle \mathbb{R},}$ is written as${\displaystyle \mathbb{R}\times \mathbb{R}\times \mathbb{R}\times \cdots .}$

Ingroup theory, define the direct product of two groups${\displaystyle (G,\circ )}$ and${\displaystyle (H,\cdot ),}$ can be denoted by${\displaystyle G\times H.}$ Forabelian groups that are written additively, it may also be called thedirect sum of two groups, denoted by${\displaystyle G\oplus H.}$

It is defined as follows:

• theset of the elements of the new group is theCartesian product of the sets of elements of${\displaystyle G\text{ and }H,}$ that is${\displaystyle \{(g,h):g\in G,h\in H\};}$
• on tse elements put an operation, defined element-wise:$${\displaystyle (g,h)\times \left(g^{\prime },h^{\prime }\right)=\left(g\circ g^{\prime },h\cdot h^{\prime }\right)}$$

Note that${\displaystyle (G,\circ )}$ may be the same as${\displaystyle (H,\cdot ).}$

The construction gives a new group, which has anormal subgroup that is isomorphic to${\displaystyle G}$ (given by the elements of the form${\displaystyle (g,1)}$) and one that is isomorphic to${\displaystyle H}$ (comprising the elements${\displaystyle (1,h)}$).

The reverse also holds in the recognition theorem. If a group${\displaystyle K}$ contains two normal subgroups${\displaystyle G\text{ and }H,}$ such that${\displaystyle K=GH}$ and the intersection of${\displaystyle G\text{ and }H}$ contains only the identity,${\displaystyle K}$ is isomorphic to${\displaystyle G\times H.}$ A relaxation of those conditions by requiring only one subgroup to be normal gives thesemidirect product.

For example,${\displaystyle G\text{ and }H}$ are taken as two copies of the unique (up to isomorphisms) group of order 2,${\displaystyle C^2:}$ say${\displaystyle \{1,a\}\text{ and }\{1,b\}.}$ Then,${\displaystyle C_2\times C_2=\{(1,1),(1,b),(a,1),(a,b)\},}$ with the operation element by element. For instance,${\displaystyle (1,b{)}^{\ast }(a,1)=\left(1^{\ast }a,b^{\ast }1\right)=(a,b),}$ and${\displaystyle (1,b{)}^{\ast }(1,b)=\left(1,b^2\right)=(1,1).}$

With a direct product, some naturalgroup homomorphisms are obtained for free: the projection maps defined byare called thecoordinate functions.

Also, every homomorphism${\displaystyle f}$ to the direct product is totally determined by its component functions${\displaystyle f_i={\pi }_i\circ f.}$

For any group${\displaystyle (G,\circ )}$ and any integer${\displaystyle n\ge 0,}$ repeated application of the direct product gives the group of all${\displaystyle n}$-tuples${\displaystyle G^n}$ (for${\displaystyle n=0,}$ that is thetrivial group); for example,${\displaystyle {\mathbb{Z}}^n}$ and${\displaystyle {\mathbb{R}}^n.}$

The direct product formodules (not to be confused with thetensor product) is very similar to the one that is defined for groups above by using the Cartesian product with the operation of addition being componentwise, and the scalar multiplication just distributing over all the components. Starting from${\displaystyle \mathbb{R}}$,Euclidean space${\displaystyle {\mathbb{R}}^n}$ is gotten, the prototypical example of a real${\displaystyle n}$-dimensional vector space. The direct product of${\displaystyle {\mathbb{R}}^m}$ and${\displaystyle {\mathbb{R}}^n}$ is${\displaystyle {\mathbb{R}}^{m+n}.}$

A direct product for a finite index$\prod _{i=1}^n{X}_i$ is canonically isomorphic to thedirect sum$\underset{i=1}{\overset{n}{⨁}}{X}_i.$ The direct sum and the direct product are not isomorphic for infinite indices for which the elements of a direct sum are zero for all but for a finite number of entries. They are dual in the sense ofcategory theory: the direct sum is thecoproduct, and the direct product is the product.

For example, for$X=\prod _{i=1}^{\mathrm{\infty }}\mathbb{R}$ and$Y=\underset{i=1}{\overset{\mathrm{\infty }}{⨁}}\mathbb{R},$ the infinite direct product and direct sum of the real numbers. Only sequences with a finite number of non-zero elements are in${\displaystyle Y.}$ For example,${\displaystyle (1,0,0,0,\dots )}$ is in${\displaystyle Y}$ but${\displaystyle (1,1,1,1,\dots )}$ is not. Both sequences are in the direct product${\displaystyle X;}$ in fact,${\displaystyle Y}$ is a proper subset of${\displaystyle X}$ (that is,${\displaystyle Y\subset X}$).^[1][2]

The direct product for a collection oftopological spaces${\displaystyle X_i}$ for${\displaystyle i}$ in${\displaystyle I,}$ some index set, once again makes use of the Cartesian product$${\displaystyle \prod _{i\in I}{X}_i.}$$

Defining thetopology is a little tricky. For finitely many factors, it is the obvious and natural thing to do: simply take as abasis of open sets to be the collection of all Cartesian products of open subsets from each factor:$${\displaystyle \mathcal{B}=\left\{U_1\times \cdots \times U_n:U_i\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{i}\mathrm{n}{X}_i\right\}.}$$

That topology is called theproduct topology. For example, by directly defining the product topology on${\displaystyle {\mathbb{R}}^2}$ by the open sets of${\displaystyle \mathbb{R}}$ (disjoint unions of open intervals), the basis for that topology would consist of all disjoint unions of open rectangles in the plane (as it turns out, it coincides with the usualmetric topology).

The product topology for infinite products has a twist, which 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). The basis of open sets is taken 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:$${\displaystyle \mathcal{B}=\left\{\prod _{i\in I}{U}_i:(\mathrm{\exists }{j}_1,\dots ,j_n)(U_{j_i}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{i}\mathrm{n}{X}_{j_i})\mathrm{a}\mathrm{n}\mathrm{d}(\mathrm{\forall }i\ne j_1,\dots ,j_n)(U_i=X_i)\right\}.}$$

The more natural-sounding topology would be, in this case, to take products of infinitely many open subsets as before, which yields a somewhat interesting topology, thebox 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 that makes the twist necessary is ultimately rooted in the fact that the intersection of open sets is guaranteed to be open only 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, calledTychonoff's theorem, is yet another equivalence to theaxiom of choice.

For more properties and equivalent formulations, seeproduct topology.

On the Cartesian product of two sets withbinary relations${\displaystyle R\text{ and }S,}$ define${\displaystyle (a,b)T(c,d)}$ as${\displaystyle aRc\text{ and }bSd.}$ If${\displaystyle R\text{ and }S}$ are bothreflexive,irreflexive,transitive,symmetric, orantisymmetric, then${\displaystyle T}$ will be also.^[3] Similarly,totality of${\displaystyle T}$ is inherited from${\displaystyle R\text{ and }S.}$ If the properties are combined, that also applies for being apreorder and being anequivalence relation. However, if${\displaystyle R\text{ and }S}$ areconnected relations,${\displaystyle T}$ need not be connected; for example, the direct product of${\displaystyle \le }$ on${\displaystyle \mathbb{N}}$ with itself does not relate${\displaystyle (1,2)\text{ and }(2,1).}$

If${\displaystyle \mathrm{\Sigma }}$ is a fixedsignature,${\displaystyle I}$ is an arbitrary (possibly infinite) index set, and${\displaystyle {\left({\mathbf{A}}_i\right)}_{i\in I}}$ is anindexed family of${\displaystyle \mathrm{\Sigma }}$ algebras, thedirect product$\mathbf{A}=\prod _{i\in I}{\mathbf{A}}_i$ is a${\displaystyle \mathrm{\Sigma }}$ algebra defined as follows:

• The universe set${\displaystyle A}$ of${\displaystyle \mathbf{A}}$ is the Cartesian product of the universe sets${\displaystyle A_i}$ of${\displaystyle {\mathbf{A}}_i,}$ formally:$A=\prod _{i\in I}{A}_i.$
• For each${\displaystyle n}$ and each${\displaystyle n}$-ary operation symbol${\displaystyle f\in \mathrm{\Sigma },}$ its interpretation${\displaystyle f^{\mathbf{A}}}$ in${\displaystyle \mathbf{A}}$ is defined componentwise, formally. For all${\displaystyle a_1,\dots ,a_n\in A}$ and each${\displaystyle i\in I,}$ the${\displaystyle i}$th component of${\displaystyle f^{\mathbf{A}}\left(a_1,\dots ,a_n\right)}$ is defined as${\displaystyle f^{{\mathbf{A}}_i}\left(a_1(i),\dots ,a_n(i)\right).}$

For each${\displaystyle i\in I,}$ the${\displaystyle i}$th projection${\displaystyle {\pi }_i:A\to A_i}$ is defined by${\displaystyle {\pi }_i(a)=a(i).}$ It is asurjective homomorphism between the${\displaystyle \mathrm{\Sigma }}$ algebras${\displaystyle \mathbf{A}\text{ and }{\mathbf{A}}_i.}$^[4]

As a special case, if the index set${\displaystyle I=\{1,2\},}$ the direct product of two${\displaystyle \mathrm{\Sigma }}$ algebras${\displaystyle {\mathbf{A}}_1\text{ and }{\mathbf{A}}_2}$ is obtained, written as${\displaystyle \mathbf{A}={\mathbf{A}}_1\times {\mathbf{A}}_2.}$ If${\displaystyle \mathrm{\Sigma }}$ contains only one binary operation${\displaystyle f,}$ theabove definition of the direct product of groups is obtained by using the notation${\displaystyle A_1=G,A_2=H,}$${\displaystyle f^{A_1}=\circ ,f^{A_2}=\cdot ,\text{ and }{f}^A=\times .}$ Similarly, the definition of the direct product of modules is subsumed here.

The direct product can be abstracted to an arbitrarycategory. In a category, given a collection of objects${\displaystyle (A_i{)}_{i\in I}}$ indexed by a set${\displaystyle I}$, aproduct of those objects is an object${\displaystyle A}$ together withmorphisms${\displaystyle p_i:A\to A_i}$ for all${\displaystyle i\in I}$, such that if${\displaystyle B}$ is any other object with morphisms${\displaystyle f_i:B\to A_i}$ for all${\displaystyle i\in I}$, there is a unique morphism${\displaystyle B\to A}$ whose composition with${\displaystyle p_i}$ equals${\displaystyle f_i}$ for every${\displaystyle i}$. Such${\displaystyle A}$ and${\displaystyle (p_i{)}_{i\in I}}$ do not always exist. If they exist, then${\displaystyle (A,(p_i{)}_{i\in I})}$ is unique up to isomorphism, and${\displaystyle A}$ is denoted${\displaystyle \prod _{i\in I}{A}_i}$.

In the special case of the category of groups, a product always exists. The underlying set of${\displaystyle \prod _{i\in I}{A}_i}$ is the Cartesian product of the underlying sets of the${\displaystyle A_i}$, the group operation is componentwise multiplication, and the (homo)morphism${\displaystyle p_i:A\to A_i}$ is the projection sending each tuple to its${\displaystyle i}$th coordinate.

Some authors draw a distinction between aninternal direct product and anexternal direct product. For example, if${\displaystyle A}$ and${\displaystyle B}$ are subgroups of an additive abelian group${\displaystyle G}$ such that${\displaystyle A+B=G}$ and${\displaystyle A\cap B=\{0\}}$,${\displaystyle A\times B\cong G,}$ and it is said that${\displaystyle G}$ is theinternal direct product of${\displaystyle A}$ and${\displaystyle B}$. To avoid ambiguity, the set${\displaystyle \{(a,b)\mid a\in A,b\in B\}}$ can be referred to as theexternal direct product of${\displaystyle A}$ and${\displaystyle B}$.

• Direct sum – Operation in abstract algebra composing objects into "more complicated" objects
• Cartesian product – Mathematical set formed from two given sets
• Coproduct – Category-theoretic construction
• Free product – Operation that combines groups
• Semidirect product – Operation in group theory
• Zappa–Szep product – Mathematics conceptPages displaying short descriptions of redirect targets
• Tensor product of graphs – Operation in graph theory
• Orders on the Cartesian product of totally ordered sets – Order whose elements are all comparable

• Lang, Serge (2002),Algebra,Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag,ISBN 978-0-387-95385-4,MR 1878556

• Abstract algebra

• Articles with short description
• Short description matches Wikidata
• Pages displaying short descriptions of redirect targets via Module:Annotated link