direct product

direct product (plural direct products)

1. (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.

2. (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).

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.

   A Boolean ring of order ${\displaystyle 2^{n}}$ (or dimension ${\displaystyle n}$) may be constructed as the direct product of ${\displaystyle n}$ Boolean rings of dimension one.

4. (topology) A topological space analogously formed from two or more (up to an infinite number of) topological spaces.
5. (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 ${\displaystyle P_{1}}$ and ${\displaystyle P_{2}}$ be probability measures on ${\displaystyle \Omega _{1}}$ and ${\displaystyle \Omega _{2}}$, respectively, and denote ${\displaystyle \Omega _{1}\times \Omega _{2}}$ by ${\displaystyle \Omega }$. A probability measure ${\displaystyle P}$ on ${\displaystyle \Omega }$ with ${\displaystyle {\mathfrak {D}}\left(P\right)={\mathfrak {D}}\left(P_{1}\right)\times {\mathfrak {D}}\left(P_{2}\right)}$ is called the direct product of ${\displaystyle P_{1}}$ and ${\displaystyle P_{2}}$ (written ${\displaystyle P_{1}\times P_{2}}$) if

     ${\displaystyle P\left(B_{1}\times B_{2}\right)=P\left(B_{1}\right)P\left(B_{2}\right)}$    ${\displaystyle B_{i}\in {\mathfrak {D}}\left(P_{i}\right);i=1,2}$.

     The probability space ${\displaystyle \left(\Omega ,P\right)}$ is called the direct product of ${\displaystyle \left(\Omega _{1},P_{1}\right)}$ and ${\displaystyle \left(\Omega _{2},P_{2}\right)}$, written

     ${\displaystyle \left(\Omega ,P\right)=\left(\Omega _{1},P_{1}\right)\times \left(\Omega _{2},P_{2}\right)}$.

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

6. (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 original objects.

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).

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

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