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Dedekind-MacNeille completion
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William Elliot
Oct 6, 2008, 7:21:05 AM10/6/08
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Let S be an (partially) ordered set. For A subset S, let
above A = set of upper bounds of A.
below A = set of lower bounds of A.
above A = set of upper bounds of A.
below A = set of lower bounds of A.
The Dedekind-MacNeille completion of S is
M = { below above A | A subset S }
If S is a Boolean algebra, then M is a Boolean algebra.
In particular, when A subset S, what set B is there for which
below above A /\ below above B = {0}
sup{ below above A, below above B } = S.
Alas it is not S\A.
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