direct products

Noun

direct product (plural direct products)

1. 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.

   Synonyms:

   Cartesian product

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.
   • 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 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 2ⁿ (or dimension n) may be constructed as the direct product of n Boolean rings of dimension one.
4. A topological space analogously formed from two or more (up to an infinite number of) topological spaces.
5. Any of a number of mathematical objects analogously derived from a given ordered set of objects.
   • 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...1978, K. Itô, An Introduction to Probability Theory, page 53
6. 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.

   Synonyms:

   categorical product, product