# direct product

Definition from Wiktionary, the free dictionary

## English[edit]

### Noun[edit]

**direct product** (*plural* **direct products**)

- (set theory) The set of all possible tuples whose elements are elements of given, separately specified, sets.
*If A and B are sets, their***direct product**is the set of ordered pairs (a,b) with a in A and b in B.

- (group theory) Such a set of tuples formed from two or more groups, forming another group whose group operation is the component-wise application of the original group operations and of which the original groups are normal subgroups.
**1976**, Marshall Hall, Jr.,*The Theory of Groups*, 2nd edition, page 40,- Theorem 3.2.3.
*A periodic Abelian group is the***direct product**of its Sylow subgroups, S(p).

- Theorem 3.2.3.

- (ring theory) Such a set of tuples formed from two or more rings, forming another ring whose operations arise from the component-wise application of the corresponding original ring operations.
- (topology) A topological space analogously formed from two or more (up to an infinite number of) topological spaces.
- (mathematics) Any of a number of mathematical objects analogously derived from a given ordered set of objects.
**1978**, K. Itô,*An Introduction to Probability Theory*, page 53,- Let us start with the definition of the
**direct product**of two probability measures. Let and be probability measures on and , respectively, and denote by . A probability measure on with is called the**direct product**of and (written ) if- .

- The probability space is called the
**direct product**of and , written- .

- For example, the Lebesgue measure on [0, 1]
^{2}is the**direct product**of that on [0, 1] and itself.

- Let us start with the definition of the

- (category theory) A high-level generalization of the preceding that applies to objects in an arbitrary category and produces a new object constructable by morphisms from each of the the original objects.

#### Usage notes[edit]

In the cases of abelian groups and of rings, the term * direct product* is synonymous with

*direct sum*.

In the case of topological spaces, in order for the resultant space to be regarded as a **categorical product** (i.e., a **direct product** in the category theory sense), the space should be equipped with the product topology (rather than the box topology, which is more intuitively derived from the topologies of the component spaces).

#### Related terms[edit]

- direct sum (synonymous in specific domains)
- product ring
- product space
- product topology

#### Synonyms[edit]

- (product of sets): Cartesian product
- (product of objects in a category): categorical product, product

#### Translations[edit]

product of sets

product of objects —

*see*categorical product