MacNeille completionMacNeilleCompletion2006-07-19 19:13:502006-10-21 12:20:29Definition\usepackage{amssymb,amscd}
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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:
\begin{itemize}
\item the rationals can be embedded in its completion (the reals)
\item every subset with an upper bound has a least upper bound
\item every subset with a lower bound has a greatest lower bound
\end{itemize}
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
\begin{enumerate}
\item $P$ can be embedded in $Q$ as a poset (so that $\le$ is compatible with $\le_Q$), and
\item every subset of $Q$ has both a least upper bound and a greatest lower bound
\end{enumerate}
In 1937, MacNeille answered this question in the affirmative by the following construction:
\begin{quote}
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.
\end{quote}
This is known as the \emph{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.
\begin{thebibliography}{8}
\bibitem{hmm} H. M. MacNeille, {\it Partially Ordered Sets}. Trans. Amer. Math. Soc. 42 (1937), pp 416-460
\bibitem{dp} B. A. Davey, H. A. Priestley, {\it Introduction to Lattices and Order}, 2nd edition, Cambridge (2003)
\end{thebibliography}