\begin{definition}[Definition:Dedekind-MacNeille Completion]
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.
{{explain|Review notation for upper and lower bounds}}
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 ($\subseteq$).
{{explain|ordered by inclusion in this context - exactly how does this work?}}
\end{definition}