MacNeille completion

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

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. 2.

    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={pPap for all aA} and A={qPqa for all aA}.

Then M(P):={A2P(Au)=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 latticeMathworldPlanetmath. So this process can be readily applied to any latticeMathworldPlanetmath, if we define a completion of a lattice to follow the two conditions above.


  • 1 H. M. MacNeille, Partially Ordered SetsMathworldPlanetmath. Trans. Amer. Math. Soc. 42 (1937), pp 416-460
  • 2 B. A. Davey, H. A. Priestley, Introduction to Lattices and Order, 2nd edition, Cambridge (2003)
Title MacNeille completion
Canonical name MacNeilleCompletion
Date of creation 2013-03-22 16:05:27
Last modified on 2013-03-22 16:05:27
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 06B23
Synonym Dedekind-MacNeille completion
Synonym normal completion
Related topic DedekindCuts