英和和英

名詞

Boolean algebra (plural Boolean algebras)

1. (algebra) An algebraic structure ${\displaystyle (\Sigma ,\vee ,\wedge ,\sim ,0,1)}$ where ${\displaystyle \vee }$ and ${\displaystyle \wedge }$ are idempotent binary operators, ${\displaystyle \sim }$ is a unary involutory operator (called "complement"), and 0 and 1 are nullary operators (i.e., constants), such that ${\displaystyle (\Sigma ,\vee ,0)}$ is a commutative monoid, ${\displaystyle (\Sigma ,\wedge ,1)}$ is a commutative monoid, ${\displaystyle \wedge }$ and ${\displaystyle \vee }$ distribute with respect to each other, and such that combining two complementary elements through one binary operator yields the identity of the other binary operator. (See Boolean algebra (structure)#Axiomatics.)

   The set of divisors of 30, with binary operators: g.c.d. and l.c.m., unary operator: division into 30, and identity elements: 1 and 30, forms a Boolean algebra.

   A Boolean algebra is a De Morgan algebra which also satisfies the law of excluded middle and the law of noncontradiction.

2. (algebra, logic, computing) Specifically, an algebra in which all elements can take only one of two values (typically 0 and 1, or "true" and "false") and are subject to operations based on AND, OR and NOT
3. (mathematics) The study of such algebras; Boolean logic, classical logic.

同意語

• (Specifically ...): switching algebra

上位語

• Kleene algebra
  • De Morgan algebra
    • Ockham algebra
      • distributive lattice
• Heyting algebra
  • residuated lattice
• MV-algebra

下位語

• complete Boolean algebra

派生語

• free Boolean algebra

参考

• Boolean lattice
• Boolean ring