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.
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Complete Boolean algebra is an extension of Boolean algebra that includes additional properties to ensure that every subset of its elements has a supremum (least upper bound) and an infimum (greatest lower bound).