Boolean algebras canonically defined

Boolean algebras have been formally defined variously as a kind of lattice and as a kind of ring. This article presents them, equally formally, as simply the models of the equational theory of two values, and observes the equivalence of both the lattice and ring definitions to this more elementary one.Boolean algebra is a mathematically rich branch of abstract algebra. 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 zero or more operations satisfying certain equations.Just as there are basic examples of groups, such as the group Z of integers and the permutation group Sn of permutations of n objects, there are also basic examples of Boolean algebra 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 include any area of mathematics for which sets form a natural foundation.Boolean algebra thus permits applying the methods of abstract algebra to mathematical logic, digital logic, and the set-theoretic foundations of mathematics.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.