Definition:Boolean Lattice/Definition 3

Definition

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 $a \in S$, $\neg \neg a = a$.

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

• Equivalence of Definitions of Boolean Lattice

• Results about Boolean lattices can be found here.

Source of Name

This entry was named for George Boole.

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.2$: Example $2.11$: $4$