5. Facts valid in Boolean lattices

A lattice is Boolean if it has a top element 1 and bottom element 0 and every element $x$ has a complement--an element $y$ with $x \wedge y=0$, $x \vee y = 1$.

Theorem. In a Boolean lattice $B$, for an ideal $I$ the following are equivalent.

(1) $I$ is a prime ideal;

(2) $I$ separates elements from their complements; in other words, for each $b \in B$, either $b \in I$ or $b' \in I$ but not both;

(3) $I$ is a maximal ideal.