Ideal frame of a Boolean algebra ($\mathrm{Idl}(B)$)

The lattice ideals of a Boolean algebra form a frame under inclusion. Meets are intersections and joins are ideals generated by unions. A complemented ideal is principal: if $I\vee J=B$ and $I\cap J=\{0\}$, choose $b\in I,c\in J$ with $b\vee c=1$; then $c=\neg b$ and $I=\downarrow b$. This ideal frame defines an extremally disconnected locale exactly when $B$ is complete.