Definition:Dedekind-MacNeille Completion

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

For a subset $A \subseteq S$, let $\mathop{\bar \uparrow} \left({A}\right)$ and $\mathop{\bar \downarrow} \left({A}\right)$ be the sets of all upper and lower bounds of $A$ in $S$, respectively.

The Dedekind–MacNeille completion of $\left({S, \preceq}\right)$ is defined as the set:
 * $\widehat{S} = \left\{{A \subseteq S: A = \mathop{\bar \downarrow} \left({\mathop{\bar \uparrow} \left({A}\right)}\right)}\right\}$

ordered by inclusion ($\subseteq$).