# Definition:Boolean Lattice

## Definition

### Definition 1

A **Boolean lattice** is a complemented distributive lattice.

### Definition 2

An ordered structure $\left({S, \vee, \wedge, \preceq}\right)$ is a **Boolean lattice** if and only if:

$(1): \quad \left({S, \vee, \wedge}\right)$ is a Boolean algebra

$(2): \quad$ For all $a, b \in S$: $a \wedge b \preceq a \vee b$

### Definition 3

A **Boolean lattice** is a bounded lattice $\left({S, \vee, \wedge, \preceq, \bot, \top}\right)$ together with a unary operation $\neg$ called **complementation**, subject to:

$(1): \quad$ For all $a, b \in S$, $a \preceq \neg b$ if and only if $a \wedge b = \bot$

$(2): \quad$ For all $a \in S$, $\neg \neg a = a$.

## Equivalence of Definitions

That the definitions given above are equivalent is shown on Equivalence of Definitions of Boolean Lattice.

## Also known as

Some sources refer to a Boolean lattice as a **Boolean algebra**.

However, the latter has a different meaning on $\mathsf{Pr} \infty \mathsf{fWiki}$: see Definition:Boolean Algebra.

## Also see

- Definition:Boolean Algebra
- Results about
**Boolean lattices**can be found here.

## Source of Name

This entry was named for George Boole.

## Sources

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- 1997: Rudolf Lidl and Günter Pilz:
*Applied Abstract Algebra*: $\S 2$