You are here
HomeMacNeille completion
Primary tabs
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 $+\infty$ 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 $\leq$, does there exist another poset $Q$ ordered by $\leq_{Q}$ such that
1. $P$ can be embedded in $Q$ as a poset (so that $\leq$ is compatible with $\leq_{Q}$), 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 $\leq$, define for every subset $A$ of $P$, two subsets of $P$ as follows:
$A^{u}=\{p\in P\mid a\leq p\mbox{ for all }a\in A\}\mbox{ and }A^{{\ell}}=\{q% \in P\mid q\leq a\mbox{ for all }a\in A\}.$ Then $M(P):=\{A\in 2^{P}\mid(A^{u})^{{\ell}}=A\}$ 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)
Mathematics Subject Classification
06B23 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
Recent Activity
new correction: Error in proof of Proposition 2 by alex2907
Jun 24
new question: A good question by Ron Castillo
Jun 23
new question: A trascendental number. by Ron Castillo
Jun 19
new question: Banach lattice valued Bochner integrals by math ias
Jun 13
new question: young tableau and young projectors by zmth
Jun 11
new question: binomial coefficients: is this a known relation? by pfb
Jun 6
new question: difference of a function and a finite sum by pfb