Heyting algebra

英語（米国） (en-US) 英語（イギリス） (en-GB)

名詞

(algebra, order theory) A bounded lattice, L, modified to serve as a model for a logical calculus by being equipped with a binary operation called "implies", denoted → (sometimes ⊃ or ⇒), defined such that (a→b)∧a ≤ b and, moreover, that x = a→b is the greatest element such that x∧a ≤ b (in the sense that if c∧a ≤ b then c ≤ a→b).

日本語の意味

ヒーティング代数：論理計算のモデルとして利用される有界格子であり、二項演算「含意」（→、または⊃や⇒）を備える。具体的には、(a→b)∧a ≤ bを満たし、さらにx = a→bがx∧a ≤ bとなるような最大の元となるように定義される代数構造。

英語の意味

(algebra, order theory) A bounded lattice, L, modified to serve as a model for a logical calculus by being equipped with a binary operation called "implies", denoted → (sometimes ⊃ or ⇒), defined such that (a→b)∧a ≤ b and, moreover, that x = a→b is the greatest element such that x∧a ≤ b (in the sense that if c∧a ≤ b then c ≤ a→b).

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A Heyting algebra provides a semantic model for intuitionistic logic by interpreting implication as the greatest element x such that x ∧ a ≤ b.

含意演算を備えた有界束は，含意を x∧a ≤ b を満たす最大要素 x と解釈することで，直観主義命題論理の意味論的枠組みを提供する。

詳細

意味（1）

(algebra, order theory) A bounded lattice, L, modified to serve as a model for a logical calculus by being equipped with a binary operation called "implies", denoted → (sometimes ⊃ or ⇒), defined such that (a→b)∧a ≤ b and, moreover, that x = a→b is the greatest element such that x∧a ≤ b (in the sense that if c∧a ≤ b then c ≤ a→b).

The laws of Heyting algebra embody a rich and profound mathematical structure that is manifest in a variety of contexts. It arises from the epistemological deliberations of Brouwer, the topologisation (localisation) of set-theoretic notions, and the categorial formulation of set theory, all of which, although interrelated, are independently motivated. The ubiquity lends weight, not to the suggestion that the correct logic is in fact intuitionistic instead of classical, but rather to the recognition that thinking in such terms is simply inappropriate — in the same way that it is inappropriate to speak without qualification about the correct geometry.

詳細

Heyting algebras （ plural ）