Definition:Dedekind-MacNeille Completion

Definition
Let $\left({S, \preceq}\right)$ be an ordered set.

For a subset $A \subseteq S$, let $A_+$ and $A_-$ be the sets of all upper and lower bounds for $A$ in $S$, respectively.

The Dedekind–MacNeille completion of $\left({S, \preceq}\right)$ is defined as the set:
 * $\widehat S := \left\{{A \subseteq S: A = \left({A_+}\right)_-}\right\}$

ordered by inclusion ($\subseteq$).

Also see
This is not to be confused with the Dedekind completion.