# 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

This page may be the result of a refactoring operation.As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.In particular: Decide which of the two defs is given, or if another needs to be addedIf you have access to any of these works, then you are invited to review this list, and make any necessary corrections.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{SourceReview}}` from the code. |

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