Boolean algebra

Etymology
Named after (1815–1864), an English mathematician, educator, philosopher and logician.

Noun

 * 1)  An algebraic structure $$(\Sigma, \vee, \wedge, \sim, 0, 1)$$ where $$\vee$$ and $$\wedge$$ are idempotent binary operators, $$\sim$$ is a unary involutory operator (called "complement"), and 0 and 1 are nullary operators (i.e., constants), such that $$(\Sigma, \vee, 0)$$ is a commutative monoid, $$(\Sigma, \wedge, 1)$$ is a commutative monoid, $$\wedge$$ and $$\vee$$ 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 .)
 * 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.
 * 1)  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
 * 2)  The study of such algebras; Boolean logic, classical logic.

Synonyms

 * switching algebra

Hypernyms

 * De Morgan algebra
 * Ockham algebra
 * distributive lattice
 * residuated lattice
 * MV-algebra
 * residuated lattice
 * MV-algebra

Derived terms

 * free Boolean algebra

Translations

 * Chinese:
 * Mandarin:
 * Czech: Booleova algebra, booleovská algebra
 * French:, algèbre booléenne
 * German: boolesche Algebra
 * Hungarian:
 * Italian: algebra booleana, reticolo booleano,
 * Japanese: ブール代数
 * Korean: ^불 대수, ^부울 대수
 * Macedonian: Булова алге́бра
 * Romanian: algebră booleană
 * Russian: бу́лева а́лгебра
 * Serbo-Croatian: Booleova algebra
 * Swedish: Boolesk algebra