The betacompactification 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 betaCpt 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 StoneCech 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.

$\begingroup$ See mathoverflow.net/questions/59291/completionofacategory. $\endgroup$– Qiaochu YuanFeb 17, 2015 at 20:09

$\begingroup$ I also found answers to my questions in the work of Gehrke & colleagues. Gehrke & Priestley published a paper titled "Canonical extensions and completions of posets and lattices" which answered parts of my questions, in particular, the one I so poorly described as the betacompactificationlike question. Proposition 2.1 and their discussion presents compact completions with the universal mapping property, and they show where the MacNeille completion fits in. The G&P paper is strongly related to a 2nd paper by Gehrke, Jansana, and Palmigiano in which these ideas are also illustrated in detail. $\endgroup$– James BrewerMar 4, 2015 at 21:35
3 Answers
A subset $A$ of a complete lattice $L$ is said to be joindense if $L=\{\bigvee^{L} R\mid R\subseteq A\}$ and $A$ is said to be meetdense 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 joindense and meetdense 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 joindense 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 joindense 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 joindense in $L$. And yes, the completions $L$ of $P$ such that $P$ is joindense 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.

$\begingroup$ What? I can think of at least two other reasonable completions, namely the free cocompletion $P^{op} \Rightarrow 2$, and the free completion $(P \Rightarrow 2)^{op}$. These correspond to looking at downwardclosed and upwardclosed sets respectively. $\endgroup$ Feb 18, 2015 at 18:02

$\begingroup$ Qiaochu Yuan. I was not being clear about what I meant saying that the DedekindMacNeille completion is the only reasonable completion of a poset. I apologize for that. I only meant to say that the DedekindMacNeille completion is only completion in which the original poset is both meet dense and join dense and if one wants the original poset to be meetdense and joindense in the complete lattice, then the completion is the DedekindMacNielle completion. This is the sense in which the DedekindMacNielle completion is the only completion of a poset. $\endgroup$ Feb 18, 2015 at 23:19

$\begingroup$ Qiaochu Yuan. I agree that for different purposes there are other reasonable notions of a completion of a poset. For instance, anyone interested in forcing would be interested in the Boolean completion of a partially ordered set where the Boolean completion of a separative poset is the unique complete Boolean algebra that contains the poset as a joindense subset. $\endgroup$ Feb 19, 2015 at 0:39
There are some differences on the categorical level.
The compact Hausdorff spaces are a reflective subcategory of topological spaces, and the StoneCech 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 DedekindMacNeille completion, things are a bit different. As a first aside, the DedekindMacNeille 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 orderembeddings, and the DedekindMacNeille completion is the injective hull of a poset, see the Wikipedia article. If you want the DedekindMacNeille completion as a reflector resp. adjoint functor, you have to consider socalled cutstable maps, see
 M. Erné. The DedekindMacNeille completion as a reflector. Order 8 (1991), 159173.

$\begingroup$ Matthias, I was reading research by Twuenissen and Venema (staff.science.uva.nl/~yde/papers/mcnle.pdf). I had not found research describing "sizes" of completions,so I look forward to reading Erné and understanding what you mean by invectives objects and that the DM is an injective hull of a poset. TY. $\endgroup$ Feb 17, 2015 at 20:50

$\begingroup$ Just posting this comment to make the URL in the comment above appear as a clickable link: staff.fnwi.uva.nl/y.venema/papers/mcnle.pdf. Full citation: Theunissen, Mark; Venema, Yde, MacNeille completions of lattice expansions, Algebra Univers. 57, No. 2, 143193 (2007). Zbl 1133.06005. $\endgroup$ Feb 5 at 18:03
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 DedekindMacNeille to LambekIsbell by Pavlovic and Hughes that goes into some detail about this, by introducing the concept of "tight limits".