direct product

Noun

 * 1)  The set of all possible tuples whose elements are elements of given, separately specified, sets.
 * 2)  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.
 * 3)  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.
 * A Boolean ring of order $$2^n$$ (or dimension $$n$$) may be constructed as the direct product of $$n$$ Boolean rings of dimension one.
 * 1)  A topological space analogously formed from two or more (up to an infinite number of) topological spaces.
 * 2)  Any of a number of mathematical objects analogously derived from a given ordered set of objects.
 * $$P \left ( B_1 \times B_2 \right ) = P \left ( B_1 \right ) P \left ( B_2 \right )$$   $$B_i \in \mathfrak{D} \left ( P_i \right ) ;i = 1, 2$$.
 * The probability space $$\left ( \Omega, P \right )$$ is called the direct product of $$\left ( \Omega_1, P_1 \right )$$ and $$\left ( \Omega_2, P_2 \right )$$, written
 * $$\left ( \Omega, P \right ) = \left ( \Omega_1, P_1 \right ) \times \left ( \Omega_2, P_2 \right )$$.
 * For example, the Lebesgue measure on [0, 1]2 is the direct product of that on [0, 1] and itself.
 * 1)  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.
 * $$\left ( \Omega, P \right ) = \left ( \Omega_1, P_1 \right ) \times \left ( \Omega_2, P_2 \right )$$.
 * For example, the Lebesgue measure on [0, 1]2 is the direct product of that on [0, 1] and itself.
 * 1)  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.

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

Translations

 * Italian: prodotto diretto