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{{MSC|03E}}
 
{{MSC|03E}}
  
A basic general mathematical construction. The idea behind it is due to R. Descartes; therefore the direct product is also called the Cartesian product. The direct product, or simply the product, of two non-empty sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d0327301.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d0327302.png" /> is the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d0327303.png" /> consisting of all [[ordered pair]]s of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d0327304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d0327305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d0327306.png" />:
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A basic general mathematical construction. The idea behind it is due to R. Descartes; therefore the direct product is also called the Cartesian product. The direct product, or simply the product, of two non-empty sets $X$ and $Y$ is the set $X \times Y$ consisting of all [[ordered pair]]s of the form $(x,y)$, $x \in X$, $y \in Y$:
 +
$$
 +
X \times Y = \{ (x,y) : x \in X\,\ y \in Y \} \ .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d0327307.png" /></td> </tr></table>
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If one of the sets $X$ or $Y$ is empty then so is their product. The set $X \times Y$ can be identified with the set of functions defined on the two-element set $\{1,2\}$ and taking the value $1$ to elements of $X$, and the value $2$ to elements of $Y$. This identification leads to a general definition of a direct product of sets. Let $I$ be some index set and suppose that $X_i$ is an arbitrary family of sets, indexed by the elements of $I$. The direct product of the $X_i$, $i \in I$, is the set of functions $f : I \rightarrow X$, where $X = \bigcup_{i\in I} X_i$, such that $f(i) \in X_i$ for every $i \in I$. Usually, the direct product is denoted by $\prod_{i\in I} X_i$, for a finite index set $I = \{1,\ldots,n\}$ one also uses the notations $\prod_{i=1}^n X_i$ and $X_1 \times \cdots \times X_n$. If $I$ consists of the single element $\{1\}$, then $\prod_I X_i = X_1$. Sometimes one defines the direct product of a finite number of factors inductively:
 +
$$
 +
\prod_{i=1} X_i = X_1 \ ;
 +
$$
 +
$$
 +
\prod_{i=1}^{n+1} = \left({ \prod_{i=1}^n X_i }\right) \times X_{n+1} \ .
 +
$$
  
If one of the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d0327308.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d0327309.png" /> is empty then so is their product. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273010.png" /> can be identified with the set of functions defined on the two-element set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273011.png" /> and taking the value 1 for elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273012.png" />, and the value 2 for elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273013.png" />. This identification leads to a general definition of a direct product of sets. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273014.png" /> be some index set and suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273015.png" /> is an arbitrary family of sets, indexed by the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273016.png" />. The direct product of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273018.png" />, is the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273020.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273021.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273022.png" />. Usually, the direct product is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273023.png" />, for a finite index set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273024.png" /> one also uses the notations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273027.png" /> consists of the single element 1, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273028.png" />. Sometimes one defines the direct product of a finite number of factors inductively:
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One merit of the construction of a direct product rests above all in the possibility of naturally introducing supplementary structures in it, if all factors have the same mathematical structure. E.g., if the $X_i$, $i \in I$, are [[algebraic system]]s of the same type, i.e. sets with a common [[signature]] of finitely-placed predicates and operations, then the product $\prod_{i\in I} X_i$ can be made into an algebraic system of the same signature: For functions $f_1,\ldots,f_k : I \rightarrow X$ and a $k$-ary operation $\omega$ the action of the function $f_1,\ldots,f_k\,\omega$ on one element $i\in I$ is defined by
 +
$$
 +
f_1,\ldots,f_k\,\omega (i) = f_1(i),\ldots,f_k(i)\,\omega \ .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273029.png" /></td> </tr></table>
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The value of a predicate $P(f_1,\ldots,f_k)$ is true if for every $i \in I$ the value of $P(f_1(i),\ldots,f_k(i))$ is true. Moreover, if in all $X_i$ an equation is satisfied, then it is also satisfied in their product. Therefore, the product of semi-groups, groups, rings, vector spaces, etc., is again a semi-group, group, ring, vector space, respectively.
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273030.png" /></td> </tr></table>
 
 
 
One merit of the construction of a direct product rests above all in the possibility of naturally introducing supplementary structures in it, if all factors have the same mathematical structure. E.g., if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273032.png" />, are algebraic systems of the same type, i.e. sets with a common signature of finitely-placed predicates and operations, then the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273033.png" /> can be made into an algebraic system of the same signature: For functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273034.png" /> and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273035.png" />-ary operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273036.png" /> the action of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273037.png" /> on one element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273038.png" /> is defined by
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273039.png" /></td> </tr></table>
 
 
 
The value of a predicate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273040.png" /> is true if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273041.png" /> the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273042.png" /> is true. Moreover, if in all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273043.png" /> an equation is satisfied, then it is also satisfied in their product. Therefore, the product of semi-groups, groups, rings, vector spaces, etc., is again a semi-group, group, ring, vector space, respectively.
 
  
 
For an arbitrary factor of a direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273044.png" /> there exists a natural projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273045.png" />, defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273046.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273047.png" /> and the family of projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273049.png" />, have the following universal property: For every family of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273050.png" /> there exists a unique mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273051.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273052.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273053.png" />. This property also holds if all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273054.png" /> are algebraic systems of one type, and makes it possible to define a suitable topology on a direct product of topological spaces. The property formulated is the basis for the definition of the [[product of a family of objects in a category]].
 
For an arbitrary factor of a direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273044.png" /> there exists a natural projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273045.png" />, defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273046.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273047.png" /> and the family of projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273049.png" />, have the following universal property: For every family of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273050.png" /> there exists a unique mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273051.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273052.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273053.png" />. This property also holds if all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032730/d03273054.png" /> are algebraic systems of one type, and makes it possible to define a suitable topology on a direct product of topological spaces. The property formulated is the basis for the definition of the [[product of a family of objects in a category]].
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The direct product is sometimes called the complete direct product, to distinguish it from the [[restricted direct product]], which is defined when there is a supplementary structure in the factors: an important case of this is the discrete direct product (or [[direct sum]]), which is defined when the supplementary structures are a one-element substructures (e.g. base points of [[pointed set]]s and [[pointed space]]s, unit subgroups of groups, zero subspaces, etc.). As a rule, the direct product of a finite number of factors coincides with the discrete product.
 
The direct product is sometimes called the complete direct product, to distinguish it from the [[restricted direct product]], which is defined when there is a supplementary structure in the factors: an important case of this is the discrete direct product (or [[direct sum]]), which is defined when the supplementary structures are a one-element substructures (e.g. base points of [[pointed set]]s and [[pointed space]]s, unit subgroups of groups, zero subspaces, etc.). As a rule, the direct product of a finite number of factors coincides with the discrete product.
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{{TEX|part}}

Revision as of 20:11, 7 February 2017

2020 Mathematics Subject Classification: Primary: 03E [MSN][ZBL]

A basic general mathematical construction. The idea behind it is due to R. Descartes; therefore the direct product is also called the Cartesian product. The direct product, or simply the product, of two non-empty sets $X$ and $Y$ is the set $X \times Y$ consisting of all ordered pairs of the form $(x,y)$, $x \in X$, $y \in Y$: $$ X \times Y = \{ (x,y) : x \in X\,\ y \in Y \} \ . $$

If one of the sets $X$ or $Y$ is empty then so is their product. The set $X \times Y$ can be identified with the set of functions defined on the two-element set $\{1,2\}$ and taking the value $1$ to elements of $X$, and the value $2$ to elements of $Y$. This identification leads to a general definition of a direct product of sets. Let $I$ be some index set and suppose that $X_i$ is an arbitrary family of sets, indexed by the elements of $I$. The direct product of the $X_i$, $i \in I$, is the set of functions $f : I \rightarrow X$, where $X = \bigcup_{i\in I} X_i$, such that $f(i) \in X_i$ for every $i \in I$. Usually, the direct product is denoted by $\prod_{i\in I} X_i$, for a finite index set $I = \{1,\ldots,n\}$ one also uses the notations $\prod_{i=1}^n X_i$ and $X_1 \times \cdots \times X_n$. If $I$ consists of the single element $\{1\}$, then $\prod_I X_i = X_1$. Sometimes one defines the direct product of a finite number of factors inductively: $$ \prod_{i=1} X_i = X_1 \ ; $$ $$ \prod_{i=1}^{n+1} = \left({ \prod_{i=1}^n X_i }\right) \times X_{n+1} \ . $$

One merit of the construction of a direct product rests above all in the possibility of naturally introducing supplementary structures in it, if all factors have the same mathematical structure. E.g., if the $X_i$, $i \in I$, are algebraic systems of the same type, i.e. sets with a common signature of finitely-placed predicates and operations, then the product $\prod_{i\in I} X_i$ can be made into an algebraic system of the same signature: For functions $f_1,\ldots,f_k : I \rightarrow X$ and a $k$-ary operation $\omega$ the action of the function $f_1,\ldots,f_k\,\omega$ on one element $i\in I$ is defined by $$ f_1,\ldots,f_k\,\omega (i) = f_1(i),\ldots,f_k(i)\,\omega \ . $$

The value of a predicate $P(f_1,\ldots,f_k)$ is true if for every $i \in I$ the value of $P(f_1(i),\ldots,f_k(i))$ is true. Moreover, if in all $X_i$ an equation is satisfied, then it is also satisfied in their product. Therefore, the product of semi-groups, groups, rings, vector spaces, etc., is again a semi-group, group, ring, vector space, respectively.

For an arbitrary factor of a direct product there exists a natural projection , defined by . The set and the family of projections , , have the following universal property: For every family of mappings there exists a unique mapping such that for every . This property also holds if all are algebraic systems of one type, and makes it possible to define a suitable topology on a direct product of topological spaces. The property formulated is the basis for the definition of the product of a family of objects in a category.

One often encounters problems of describing mathematical objects that cannot be decomposed into a direct product, and of stating conditions under which the factors of a direct product are uniquely determined up to an isomorphism. Classical results in this respect are the theorem on the structure of finitely-generated modules over principal ideal rings and the Remak–Schmidt theorem on the central isomorphism of direct decompositions of a group with a principal series.

The direct product is sometimes called the complete direct product, to distinguish it from the restricted direct product, which is defined when there is a supplementary structure in the factors: an important case of this is the discrete direct product (or direct sum), which is defined when the supplementary structures are a one-element substructures (e.g. base points of pointed sets and pointed spaces, unit subgroups of groups, zero subspaces, etc.). As a rule, the direct product of a finite number of factors coincides with the discrete product.

How to Cite This Entry:
Direct product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Direct_product&oldid=40201
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article