direct product

plural direct products

1. mathematicsThe 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.

   Synonyms: Cartesian product

2. group theorySuch 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.

   Theorem 3.2.3. A periodic Abelian group is the direct product of its Sylow subgroups, S(p).

   1976, Marshall Hall, Jr., The Theory of Groups, 2nd edition, page 40

3. ring theorySuch 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 2ⁿ (or dimension n) may be constructed as the direct product of n Boolean rings of dimension one.

4. mathematicsA topological space analogously formed from two or more (up to an infinite number of) topological spaces.
5. mathematicsAny of a number of mathematical objects analogously derived from a given ordered set of objects.

   1 more example

   Let us start with the definition of the direct product of two probability measures. Let P#95;1 and P#95;2 be probability measures on #92;Omega#95;1 and #92;Omega#95;2, respectively, and denote #92;Omega#95;1#92;times#92;Omega#95;2 by #92;Omega. A probability measure P on #92;Omega with #92;mathfrak#123;D#125;#92;left(P#92;right)#61;#92;mathfrak#123;D#125;#92;left(P#95;1#92;right)#92;times#92;mathfrak#123;D#125;#92;left(P#95;2#92;right) is called the direct product of P#95;1 and P#95;2 (written P#95;1#92;timesP#95;2) if P(B_1×B_2)=P(B_1)P(B_2) B_i∈𝔇(P_i);i=1,2. The probability space (Ω,P) is called the direct product of (Ω₁,P_1) and (Ω₂,P_2), written (Ω,P)=(Ω₁,P_1)×(Ω₂,P_2). For example, the Lebesgue measure on [0, 1]² is the direct product of that on [0, 1] and itself.

   1978, K. Itô, An Introduction to Probability Theory, page 53

6. category theoryA 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.

   Synonyms: categorical product, product