Definition:Dedekind-MacNeille Completion

Definition
Let $\struct {S, \preceq}$ 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 $\struct {S, \preceq}$ is defined as the set:
 * $\widehat S := \set {A \subseteq S: A = \paren {A_+}_-}$

ordered by inclusion ($\subseteq$).

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