Complete Boolean algebra

Remarks

Then every subset also has an infimum, $\bigwedge S = \neg \bigvee \set{\neg s : s \in S}$, so $B$ is a complete lattice. The infinite distributive laws $a \wedge \bigvee S = \bigvee \set{a \wedge s : s \in S}$ hold in every complete Boolean algebra.

Example: the power set of a set $X$, ordered by inclusion. The supremum of a family of subsets is its union, which is again a subset of $X$.

Non-example: the algebra $B$ of finite and cofinite subsets of $\N$ is not complete. Let $S = \set{\set{2n} : n \in \N}$. An upper bound of $S$ in $B$ contains all even numbers, so it is infinite and hence cofinite, and it contains infinitely many odd numbers. Removing one odd number gives a smaller upper bound in $B$, so there is no least one.

In Mathlib the notion is CompleteBooleanAlgebra.