direct product
A "direct product" is a way to combine two or more mathematical structures into a new structure, defined by operations performed component-wise.
Definition
C1Group Theory
(technical, academic)A group formed by ordered pairs of elements from two groups, with the group operation applied to each component separately.
Example
- The direct product of the groups Z2 and Z3 results in a group of order 6.
B2Set Theory
(technical, academic)The set of all ordered pairs formed by taking one element from each of two sets.
Example
- The direct product of the sets {1, 2} and {a, b} is {(1, a), (1, b), (2, a), (2, b)}.
C1Ring Theory
(technical, academic)A ring formed by ordered pairs of elements from two rings, with addition and multiplication applied to each component separately.
Example
- In ring theory, the direct product of two rings R and S is denoted as R × S.
C1Linear Algebra
(technical, academic)A vector space consisting of ordered pairs of vectors from two vector spaces, with vector addition and scalar multiplication applied to each component separately.
Example
- The direct product of vector spaces V and W is used to construct new vector spaces.
C2Category Theory
(technical, academic)An object formed from two objects in a category, along with projection morphisms to each of the original objects.
Example
- In category theory, the direct product generalizes the concept of Cartesian products.
Similar
Terms that have similar or relatively close meanings to "direct product":