English Dictionary

direct product

名詞

(set theory) The set of all possible tuples whose elements are elements of given, separately specified, sets.

日本語の意味

直積：別々に指定された集合それぞれの要素からなるタプル全体の集合（集合論における用語）

The professor explained how the direct product of the sets consists of all ordered tuples whose components belong to the respective sets.

教授は、集合の直積が、それぞれの集合の要素から成るすべての順序付きタプルで構成されることを説明した。

意味（1）

(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.

意味（2）

(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.

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

意味（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.

意味（4）

(topology) A topological space analogously formed from two or more (up to an infinite number of) topological spaces.

意味（5）

(mathematics) 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_1 and P_2 be probability measures on 𝛺₁ and 𝛺₂, respectively, and denote 𝛺₁⨯𝛺₂ by 𝛺. A probability measure P on 𝛺 with 𝔇(P)=𝔇(P_1)⨯𝔇(P_2) is called the direct product of P_1 and P_2 (written P_1⨯P_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.

意味（6）

(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.

direct products （ plural ）