ordered ring

名詞

(algebra, order theory, ring theory) A ring, R, equipped with a partial order, ≤, such that for arbitrary a, b, c ∈ R, if a ≤ b then a + c ≤ b + c, and if, additionally, 0 ≤ c, then both ca ≤ cb and ac ≤ bc.

日本語の意味

順序環：部分順序が定義された環（代数系）であり、任意の a, b, c ∈ R に対して、a ≤ b なら a + c ≤ b + c が成立し、さらに 0 ≤ c のとき c·a ≤ c·b および a·c ≤ b·c も成立する性質を持つ。

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Google翻訳 / DeepL翻訳

コンパクト空間上の連続実数値関数の順序付き環は、正性や順序を保つ演算を比較するための自然な枠組みを提供します。

詳細

意味（1）

If 0, and x is strictly positive if x>0.

The set of all positive elements of an ordered ring A is denoted by A_+, and the set of all strictly positive elements of A is denoted by A^*₊.

If (A,+,·,<) is an ordered ring and if < is a total ordering, we shall, of course, call (A,+,·,<) a totally ordered ring; if (A,+,·) is a field, we shall call (A,+,·,<) an ordered field, and if, moreover, < is a total ordering, we shal call (A,+·,<) a totally ordered field.

Google翻訳 / DeepL翻訳

詳細

意味（2）

(algebra, order theory, ring theory) A ring, R, equipped with a total order, ≤, such that for arbitrary a, b, c ∈ R, if a ≤ b then a + c ≤ b + c, and if, additionally, 0 ≤ c, then both ca ≤ cb and ac ≤ bc.

The positive elements in an ordered ring allow us to compare elements to 0, but we know in the integers that we can compare any two elements to each other. For example, we know that 4>2 because 4-2>0. We can extend this idea to any ordered ring. If R is an ordered ring and a,b∈R, then we know by trichotomy that exactly one of the following must be true: a-b>0, a-b=0, or -(a-b)>0.

Google翻訳 / DeepL翻訳

詳細

ordered rings

（ plural ）

詳細

ordered ring

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