Dedekind cut
An alternative definition of the Dedekind–MacNeille completion that more closely resembles the definition of a Dedekind cut is sometimes used. In a partially ordered set , define a "cut" to be a pair of sets for which and . If is a cut then "A" satisfies the equation , and conversely if then is a cut. Therefore, the set of cuts, partially ordered by inclusion on the lower set of the cut (or the reverse of the inclusion relation on the upper set), gives an equivalent definition of the Dedekind–MacNeille completion.