direct product

noun6

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.
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
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 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.
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.