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 $a \in S$, $\neg \neg a = a$.
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
- 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: $\S 2$