Definition:Boolean Algebra

Order-Theoretic Definition
A Boolean algebra is a poset $\left({S, \preceq}\right)$ with:


 * a binary operation $\vee$, called meet;
 * a binary operation $\wedge$, called join;
 * a unary operation $'$, called complement;
 * two distinguished elements $\bot_S$ and $\top_S$, sometimes called bottom and top;

which are subject to the following axioms:

A Boolean algebra may be denoted as $\left({S, \vee, \wedge}\right)$, or $\left({S, \vee, \wedge, \preceq}\right)$ if the ordering is to be explicated.

The latter notation is reminiscent of that for an ordered structure; this is justified by Boolean Algebra is Ordered Structure.

Algebraic Definition
A Boolean algebra is an algebraic structure $\left({S, \vee, \wedge}\right)$ such that:

The element $a'$ in $(HA \ 4)$ is often called the complement of $a$.