0. Definitions

Definition. In a lattice $L$ with 0 and 1, $y$ is a complement of $x$ if $x \wedge y = 0$, $x \vee y = 1$.

Proposition. In a distributive lattice, complements are unique.

Definition. A Boolean lattice is a distributive lattice with 0 and 1 in which every element has a complement and $0 \neq 1$.

The complement of $x$ is denoted $x'$.

Definition. A Boolean algebra is a Boolean lattice in which complementation is regarded as an operation. It is best to regard 0 and 1 as constant operations. Thus the algebra has the form $\langle B ; \vee, \wedge, 0, 1, ' \rangle$.

Thinking of complementation as an operation makes a difference when subalgebras or homomorphisms are considered. Thus a Boolean subalgebra is a sublattice that is also closed under complementation and contains 0 and 1. (In particular, a Boolean subalgebra cannot be empty, unlike a sublattice.)

$Proposition.$ In a Boolean lattice $B$, the complementation map $x \mapsto x'$ is a ``dual isomorphism'', meaning an isomorphism of $B$ with its dual (i.e., $B$ upside-down). In other words, the complementation map is one-to-one and obeys de Morgan's laws $(x \vee y)' = x' \wedge y'$ and $(x \wedge y)' = x' \vee y'$.