Boolean algebras canonically defined

Boolean algebra is a mathematically rich branch of abstract algebra. Stanford Encyclopaedia of Philosophy defines Boolean algebra as 'the algebra of two-valued logic with only sentential connectives, or equivalently of algebras of sets under union and complementation.' Just as group theory deals with groups, and linear algebra with vector spaces, Boolean algebras are models of the equational theory of the two values 0 and 1 (whose interpretation need not be numerical). Common to Boolean algebras, groups, and vector spaces is the notion of an algebraic structure, a set closed under some operations satisfying certain equations.

Boolean algebras are models of the equational theory of two values; this definition is equivalent to the lattice and ring definitions.

Just as there are basic examples of groups, such as the group ${\displaystyle \mathbb {Z} }$ of integers and the symmetric group S_n of permutations of n objects, there are also basic examples of Boolean algebras such as the following.

• The algebra of binary digits or bits 0 and 1 under the logical operations including disjunction, conjunction, and negation. Applications include the propositional calculus and the theory of digital circuits.
• The algebra of sets under the set operations including union, intersection, and complement. Applications are far-reaching because set theory is the standard foundations of mathematics.

Boolean algebra thus permits applying the methods of abstract algebra to mathematical logic and digital logic.

Unlike groups of finite order, which exhibit complexity and diversity and whose first-order theory is decidable only in special cases, all finite Boolean algebras share the same theorems and have a decidable first-order theory. Instead, the intricacies of Boolean algebra are divided between the structure of infinite algebras and the algorithmic complexity of their syntactic structure.