MacNeille completion
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

•
every subset with a lower bound has a greatest lower bound^{}
If we extend the reals by adjoining $+\mathrm{\infty}$ and $\mathrm{\infty}$ 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 $\le $, does there exist another poset $Q$ ordered by ${\le}_{Q}$ such that

1.
$P$ can be embedded in $Q$ as a poset (so that $\le $ is compatible with ${\le}_{Q}$), and

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 $\le $, define for every subset $A$ of $P$, two subsets of $P$ as follows:
$${A}^{u}=\{p\in P\mid a\le p\text{for all}a\in A\}\text{and}{A}^{\mathrm{\ell}}=\{q\in P\mid q\le a\text{for all}a\in A\}.$$ Then $M(P):=\{A\in {2}^{P}\mid {({A}^{u})}^{\mathrm{\ell}}=A\}$ 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.
References
 1 H. M. MacNeille, Partially Ordered Sets^{}. Trans. Amer. Math. Soc. 42 (1937), pp 416460
 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  20130322 16:05:27 
Last modified on  20130322 16:05:27 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  8 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06B23 
Synonym  DedekindMacNeille completion 
Synonym  normal completion 
Related topic  DedekindCuts 