Definition:Boolean Algebra/Definition 3

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A Boolean algebra is an algebraic structure $\struct {S, \vee, \wedge}$ such that:

\((\text {BA} 0)\)   $:$   $S$ is closed under both $\vee$ and $\wedge$             
\((\text {BA} 1)\)   $:$   Both $\vee$ and $\wedge$ are commutative             
\((\text {BA} 2)\)   $:$   Both $\vee$ and $\wedge$ distribute over the other             
\((\text {BA} 3)\)   $:$   Both $\vee$ and $\wedge$ have identities $\bot$ and $\top$ respectively             
\((\text {BA} 4)\)   $:$   $\forall a \in S: \exists \neg a \in S: a \vee \neg a = \top, a \wedge \neg a = \bot$             

The operations $\vee$ and $\wedge$ are called join and meet, respectively.

The identities $\bot$ and $\top$ are called bottom and top, respectively.

Also, $\neg a$ is called the complement of $a$.

Also defined as

Some sources define a Boolean algebra to be what on $\mathsf{Pr} \infty \mathsf{fWiki}$ is called a Boolean lattice.

Some sources define a Boolean algebra to be a set with two elements (typically $\set {0, 1}$ together with the two operations addition $+$ and multiplication $\times$ defined as follows:

$\begin{array}{c|cc} + & 0 & 1 \\ \hline 0 & 0 & 1 \\ 1 & 1 & 0 \\ \end{array} \qquad \begin{array}{c|cc} \times & 0 & 1 \\ \hline 0 & 0 & 0 \\ 1 & 0 & 1 \\ \end{array}$

Also known as

Some sources refer to a Boolean algebra as:

a Boolean ring


a Huntington algebra

both of which terms already have a different definition on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Other common notations for the elements of a Boolean algebra include:

$0$ and $1$ for $\bot$ and $\top$, respectively
$a'$ for $\neg a$.

When this convention is used, $0$ is called zero, and $1$ is called one or unit.

Also see

  • Results about Boolean algebras can be found here.

Source of Name

This entry was named for George Boole.