Boolean Algebra (structure) - Definition

Definition

A Boolean algebra is a six-tuple consisting of a set A, equipped with two binary operations ∧ (called "meet" or "and"), ∨ (called "join" or "or"), a unary operation ¬ (called "complement" or "not") and two elements 0 and 1 (sometimes denoted by the symbols ⊥ and ⊤, respectively), such that for all elements a, b and c of A, the following axioms hold:

a ∨ (b ∨ c) = (a ∨ b) ∨ c	a ∧ (b ∧ c) = (a ∧ b) ∧ c	associativity
a ∨ b = b ∨ a	a ∧ b = b ∧ a	commutativity
a ∨ 0 = a	a ∧ 1 = a	identity
a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)	a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)	distributivity
a ∨ ¬a = 1	a ∧ ¬a = 0	complements

A Boolean algebra with only one element is called a trivial Boolean algebra or a degenerate Boolean algebra. (Some authors require 0 and 1 to be distinct elements in order to exclude this case.)

It follows from the last three pairs of axioms above (identity, distributivity and complements) that

a = b ∧ a if and only if a ∨ b = b.

The relation ≤ defined by a ≤ b if and only if the above equivalent conditions hold, is a partial order with least element 0 and greatest element 1. The meet a ∧ b and the join a ∨ b of two elements coincide with their infimum and supremum, respectively, with respect to ≤.

As in every bounded lattice, the relations ∧ and ∨ satisfy the first three pairs of axioms above; the fourth pair is just distributivity. Since the complements in a distributive lattice are unique, to define the involution ¬ it suffices to define ¬a as the complement of a.

The set of axioms is self-dual in the sense that if one exchanges ∨ with ∧ and 0 with 1 in an axiom, the result is again an axiom. Therefore by applying this operation to a Boolean algebra (or Boolean lattice), one obtains another Boolean algebra with the same elements; it is called its dual.