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: $\S 2$