direct product
Noun
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Noun
direct product (plural direct products)
- (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.
- (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).
- 1976, Marshall Hall, Jr., The Theory of Groups, 2nd edition, page 40 ↗,
- (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 2^n (or dimension n) may be constructed as the direct product of n Boolean rings of dimension one.
- (topology) A topological space analogously formed from two or more (up to an infinite number of) topological spaces.
- (mathematics) Any of a number of mathematical objects analogously derived from a given ordered set of objects.
- P \left ( B_1 \times B_2 \right ) = P \left ( B_1 \right ) P \left ( B_2 \right ) B_i \in \mathfrak{D} \left ( P_i \right ) ;i = 1, 2.
- The probability space \left ( \Omega, P \right ) is called the direct product of \left ( \Omega_1, P_1 \right ) and \left ( \Omega_2, P_2 \right ), written
- \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.
- (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.
- (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
- Italian: prodotto diretto
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