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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)
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