Every Boolean algebra $\mathbb{B}$ embeds densely in its completion as a Boolean algebra, which is a complete Boolean algebra (more than just countably complete). The completion $\bar\mathbb{B}$ can be constructed as the regular open algebra, the set of all regular open subsets of $\mathbb{B}-\{0\}$, where the topology is generated by the basic open sets consisting of lower cones. A set is regular open if it is the interior of its closure.

Indeed, the completion operation is extremely general, and is used pervasively in set theory in the context of the forcing technique. Every separative partial order $\mathbb{P}$, and this includes any Boolean algebra (minus $0$), embeds densely in its regular open algebra, and this is always a complete Boolean algebra. This fact is the main connection between the poset-based account of forcing and the Boolean-algebra based account of forcing.

The wikipedia link above lists several universal properties of this completion.

Every Boolean algebra $\mathbb{B}$ embeds densely in its completion as a Boolean algebra, which is a complete Boolean algebra (more than just countably complete). The completion $\bar{\mathbb{B}}$ can be constructed as the regular open algebra, the set of all regular open subsets of $\mathbb{B}-\{0\}$, where the topology is generated by the basic open sets consisting of lower cones. A set is regular open if it is the interior of its closure.

Indeed, the completion operation is extremely general, and is used pervasively in set theory in the context of the forcing technique. Every separative partial order $\mathbb{P}$, and this includes any Boolean algebra (minus $0$), embeds densely in its regular open algebra, and this is always a complete Boolean algebra. This fact is the main connection between the poset-based account of forcing and the Boolean-algebra based account of forcing.

The wikipedia link above lists several universal properties of this completion.

Every Boolean algebra $\mathbb{B}$ embeds densely in its completion as a Boolean algebra, which is a complete Boolean algebra (more than just countably complete). The completion $\bar\mathbb{B}$ can be constructed as the regular open algebra, the set of all regular open subsets of $\mathbb{B}-\{0\}$, where the topology is generated by the basic open sets consisting of lower cones. A set is regular open if it is the interior of its closure.

Indeed, the completion operation is extremely general, and is used pervasively in set theory in the context of the forcing technique. Every separative partial order $\mathbb{P}$, and this includes any Boolean algebra (minus $0$), embeds densely in its regular open algebra, and this is always a complete Boolean algebra. This fact is the main connection between the poset-based account of forcing and the Boolean-algebra based account of forcing.

Every Boolean algebra $\mathbb{B}$ embeds densely in its completion as a Boolean algebra, which is a complete Boolean algebra (more than just countably complete). The completion $\bar\mathbb{B}$ can be constructed as the regular open algebra, the set of all regular open subsets of $\mathbb{B}-\{0\}$, where the topology is generated by the basic open sets consisting of lower cones. A set is regular open if it is the interior of its closure.

Indeed, the completion operation is extremely general, and is used pervasively in set theory in the context of the forcing technique. Every separative partial order $\mathbb{P}$, and this includes any Boolean algebra (minus $0$), embeds densely in its regular open algebra, and this is always a complete Boolean algebra. This fact is the main connection between the poset-based account of forcing and the Boolean-algebra based account of forcing.

The wikipedia link above lists several universal properties of this completion.