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In mathematics, given two partially ordered sets A and B, the product order (also called the coordinatewise order or componentwise order) is a partial ordering on the cartesian product A × B. Given two pairs (a1, b1) and (a2, b2) in A × B, one defines (a1, b1) ≤ (a2, b2) if and only if a1 ≤ a2 and b1 ≤ b2. The cartesian product with the product order is the categorical product in the category of partially ordered sets with monotone functions.

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  • Product order
  • 直積順序
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  • 数学において、二つの順序集合 A と B が与えられたとき、そのデカルト積 A × B に対して、一つの半順序を以下のように導入することが出来る。 A × B 内の与えられた二つのペア (a1,b1) および (a2,b2) に対して、a1 ≤ a2 および b1 ≤ b2 が成り立つとき、そしてそのときに限り (a1,b1) ≤ (a2,b2) と定義する。 この順序は直積順序(ちょくせきじゅんじょ、英: product order)と呼ばれる。A × B 上の他の順序として、辞書式順序がある。 直積順序を伴うデカルト積は、単調函数を射とする半順序集合の圏における積である。
  • In mathematics, given two partially ordered sets A and B, the product order (also called the coordinatewise order or componentwise order) is a partial ordering on the cartesian product A × B. Given two pairs (a1, b1) and (a2, b2) in A × B, one defines (a1, b1) ≤ (a2, b2) if and only if a1 ≤ a2 and b1 ≤ b2. The cartesian product with the product order is the categorical product in the category of partially ordered sets with monotone functions.
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  • In mathematics, given two partially ordered sets A and B, the product order (also called the coordinatewise order or componentwise order) is a partial ordering on the cartesian product A × B. Given two pairs (a1, b1) and (a2, b2) in A × B, one defines (a1, b1) ≤ (a2, b2) if and only if a1 ≤ a2 and b1 ≤ b2. Another possible ordering on A × B is the lexicographical order, which is a total ordering. However the product order of two totally ordered sets is not in general total; for example, the pairs (0, 1) and (1, 0) are incomparable in the product order of the ordering 0 < 1 with itself. The lexicographic order of totally ordered sets is a linear extension of their product order, and thus the product order is a subrelation of the lexicographic order. The cartesian product with the product order is the categorical product in the category of partially ordered sets with monotone functions. The product order generalizes to arbitrary (possibly infinitary) cartesian products. Furthermore, given a set A, the product order over the cartesian product can be identified with the inclusion ordering of subsets of A. The notion applies equally well to preorders. The product order is also the categorical product in a number of richer categories, including lattices and Boolean algebras.
  • 数学において、二つの順序集合 A と B が与えられたとき、そのデカルト積 A × B に対して、一つの半順序を以下のように導入することが出来る。 A × B 内の与えられた二つのペア (a1,b1) および (a2,b2) に対して、a1 ≤ a2 および b1 ≤ b2 が成り立つとき、そしてそのときに限り (a1,b1) ≤ (a2,b2) と定義する。 この順序は直積順序(ちょくせきじゅんじょ、英: product order)と呼ばれる。A × B 上の他の順序として、辞書式順序がある。 直積順序を伴うデカルト積は、単調函数を射とする半順序集合の圏における積である。
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