Boolean algebra

A Boolean algebra is a bounded distributive lattice in which every element has a complement. Its meet, join, and complement abstract conjunction, disjunction, and negation.

Every proper lattice ideal of a Boolean algebra extends to a prime lattice ideal. More generally, an ideal disjoint from a lattice filter extends to a prime ideal still disjoint from that filter. This Boolean case of the Stone prime filter theorem is equivalent to the ultrafilter lemma over Zermelo–Fraenkel set theory and is weaker than the axiom of choice.

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.

A Boolean algebra in which every set-indexed family has a join and a meet. Double-negation Boolean algebra is an intuitionistic example, with joins formed by regularizing unions. Completeness makes truth values for quantifiers possible in a Boolean-valued inner universe over intuitionistic set theory.

Inside IZF, truth values are subsets of a singleton. Their double-negation-stable elements form a complete Boolean algebra: arbitrary meets are intersections and arbitrary joins are double-negation closures of unions. Negation gives Boolean complement. The join of $p$ and its complement is top although their ordinary union need not be top. The closure here is taken inside the fixed singleton; it does not assume that the class of all objects double-negatively belonging to an arbitrary set is itself a set.