Boolean algebra

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

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.

• (Specifically ...): switching algebra

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

• complete Boolean algebra

• free Boolean algebra

• Boolean lattice
• Boolean ring