Direct product

English

The direct product of sets A={x, y, z} and B={1, 2, 3}

Noun

1. Template:Lb The set of all possible tuples whose elements are elements of given, separately specified, sets.

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2. Template:Lb 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.
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3. Template:Lb 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.

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4. Template:Lb A topological space analogously formed from two or more (up to an infinite number of) topological spaces.
5. Template:Lb Any of a number of mathematical objects analogously derived from a given ordered set of objects.
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     $P(B_1\times B_2)=P\left(B_1\right)P\left(B_2\right)$    $B_i\in 𝔇\left(P_i\right);i=1,2$.

     The probability space $(\mathrm{\Omega },P)$ is called the direct product of $({\mathrm{\Omega }}_1,P_1)$ and $({\mathrm{\Omega }}_2,P_2)$, written

     $(\mathrm{\Omega },P)=({\mathrm{\Omega }}_1,P_1)\times ({\mathrm{\Omega }}_2,P_2)$.

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

6. Template:Lb 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 original objects.

Usage notes

In the cases of abelian groups and of rings, the term direct product is synonymous with Template:M.

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 Template:W, which is more intuitively derived from the topologies of the component spaces).

Synonyms

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Related terms

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Translations

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