Dedekind–MacNeille completion

Remarks

$\operatorname{DM}(P)$ is a complete lattice, whose meets are intersections, and $\iota$ is an order embedding: $x \le y$ if and only if $\iota(x) \subseteq \iota(y)$. The image of $\iota$ is join-dense and meet-dense: each element of $\operatorname{DM}(P)$ is a supremum and an infimum of elements of $\iota(P)$. These properties characterize $\operatorname{DM}(P)$ up to isomorphism, and it embeds in every complete lattice that contains $P$ as an ordered subset. Example: for $P = \Q$, the sets $A = (A^u)^l$ are $\varnothing$, $\Q$, and the sets $\set{q \in \Q : q \le r}$ for $r \in \R$. So $\operatorname{DM}(\Q)$ is the extended real line, and its proper nonempty elements are closely related to Dedekind cuts. Non-example: in $\Q$, the set $A = \set{q \in \Q : q < 0}$ is not in $\operatorname{DM}(\Q)$, because $A^u$ is the set of nonnegative rationals and $(A^u)^l = \set{q : q \le 0} \ne A$.