MacNeille completion (original) (raw)

In a first course on real analysis, one is generally introduced to the concept of a Dedekind cut. It is a way of constructing the set of real numbers from the rationals. This is a process commonly known as the completion of the rationals. Three key features of this completion are:

• •
  the rationals can be embedded in its completion (the reals)
• •
  every subset with an upper bound has a least upper bound
• •

If we extend the reals by adjoining +∞ and -∞ and define the appropriate ordering relations on this new extended set (the extended real numbers), then it is a set where every subset has a least upper bound and a greatest lower bound.

When we deal with the rationals and the reals (and extended reals), we are working with linearly ordered sets. So the next question is: can the procedure of a completion be generalized to an arbitrary poset? In other words, if P is a poset ordered by ≤, does there exist another poset Q ordered by ≤Q such that

1. 1. P can be embedded in Q as a poset (so that ≤ is compatible with ≤Q), and
2. 1. every subset of Q has both a least upper bound and a greatest lower bound

In 1937, MacNeille answered this question in the affirmative by the following construction:

Given a poset P with order ≤, define for every subset A of P, two subsets of P as follows:

Au={p∈P∣a≤p⁢ for all ⁢a∈A}⁢ and ⁢Aℓ={q∈P∣q≤a⁢ for all ⁢a∈A}.

Then M⁢(P):={A∈2P∣(Au)ℓ=A} ordered by the usual set inclusion is a poset satisfying conditions (1) and (2) above.

This is known as the MacNeille completion M⁢(P) of a poset P. In M⁢(P), since lub and glb exist for any subset, M⁢(P) is a complete lattice. So this process can be readily applied to any lattice, if we define a completion of a lattice to follow the two conditions above.

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