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 $+\infty$ and $-\infty$ 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: $$A^u=\lbrace p\in P\mid a\le p\mbox{ for all }a\in A\rbrace\mbox{ and }A^{\ell}=\lbrace q\in P\mid q\le a\mbox{ for all }a\in A\rbrace.$$ Then $M(P):=\lbrace A\in 2^P \mid (A^u)^{\ell}=A\rbrace$ 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.

Bibliography

1

H. M. MacNeille, Partially Ordered Sets. 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)