Definition:Dedekind-MacNeille Completion

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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.


Source of Name

This entry was named for Julius Wilhelm Richard Dedekind and Holbrook Mann MacNeille.