How is a MacNeille completion "universal" like a beta-compactification is "universal"? The beta-compactification of a topological space is characterized as the largest space such that every mapping from the original space to another (range) space can be extended through to a mapping from its beta-Cpt to the range space (Engelking, Outline, Chapter 5.3). For these reasons, it has performed well as the "universal" space in many applications, notably in Ellis' theory of topological dynamics. What "like" theories are known for the MacNeille completion of a partially ordered set? For example, is the MacNeille completion the "biggest" completion in the same spirit as the Stone-Cech compactification? Can every monotone map from the original POSET to another POSET be extended up through the MacNeille completion? Please accept my apologies for not knowing the partial order theory as well as I know the topological theory. Thank you.
 A: There are some differences on the categorical level.
The compact Hausdorff spaces are a reflective subcategory of topological spaces, and the Stone-Cech compactification is left adjoint to the inclusion of compact Hausdorff spaces into topological spaces. This basically encodes the universality and is enough reasons for ubiquitous appearance.
For the Dedekind-MacNeille completion, things are a bit different. As a first aside, the Dedekind-MacNeille completion is not the "biggest" something, it is the smallest complete lattice that contains the given partially ordered set. The category of complete lattices is not reflective inside the category of partially ordered sets with monotone maps. What is true is that the complete lattices are injective objects for order-embeddings, and the Dedekind-MacNeille completion is the injective hull of a poset, see the Wikipedia article.
If you want the Dedekind-MacNeille completion as a reflector resp. adjoint functor, you have to consider so-called cut-stable maps, see

*

*M. Erné. The Dedekind-MacNeille completion as a reflector. Order 8 (1991), 159-173.

A: A subset $A$ of a complete lattice $L$ is said to be join-dense if $L=\{\bigvee^{L} R\mid R\subseteq A\}$ and $A$ is said to be meet-dense in $L$ if $L=\{\bigwedge^{L}R\mid R\subseteq A\}$. It turns out that the Dedekind–MacNeille completion of a poset $P$ is up to an isomorphism preserving $P$ the only complete lattice $L$ with $P\subseteq L$ and where $P$ is both join-dense and meet-dense in $L$.
As was mentioned by Matthias Wendt, the Dedekind–MacNeille completion is the smallest completion of $P$. Let me formalize what I mean by smallest.
Proposition Suppose that $P$ is a poset and $L$ is a complete lattice such that $P\subseteq L$ and $P$ is join-dense in $L$. Then $L$ is up to an isomorphism preserving $P$ the Dedekind–MacNeille completion of $P$ if and only if whenever $M$ is a complete lattice where $P\subseteq M$ and $P$ is join-dense in $M$, then there is a $j\colon M\rightarrow L$ such that $j(p)=p$ and $j(\bigvee^{M}R)=\bigvee^{L}j[R]$ whenever $R\subseteq M$ (from these properties, one can immediately deduce that the mapping $j$ is always surjective).
In other words, the Dedekind–MacNeille completion of a poset $P$ is the smallest complete lattice in the lattice of completions $L$ of $P$ such that $P$ is join-dense in $L$. And yes, the completions $L$ of $P$ such that $P$ is join-dense in $L$ do form a complete lattice.
There is also another sense in which the Dedekind–MacNeille completion of a poset is the smallest completion.
We say that a poset $(Y,\leq')$ is a minimal completion of a poset $(X,\leq)$ if

*

*$X\subseteq Y$,


*${\leq}=X^{2}\cap{\leq'}$, and


*whenever $X\subseteq Z\subseteq Y$ and $(Z,{\leq'}\cap Z^{2})$ is a complete lattice, then $X=Z$.
I showed in this answer that the Dedekind–MacNeille completion of a poset is the unique minimal completion of a poset.
A: For a more recent perspective on the universal property of the Dedekind–MacNeille (extended from posets to categories), see the paper Tight limits and completions from Dedekind-MacNeille to Lambek-Isbell by Pavlovic and Hughes that goes into some detail about this, by introducing the concept of "tight limits".
