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.



How is a MacNeille completion "universal" like a beta-compactification is "universal"?
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


1. $X\subseteq Y$,
2. ${\leq}=X^{2}\cap{\leq'}$, and
3. whenever $X\subseteq Z\subseteq Y$ and $(Z,{\leq'}\cap Z^{2})$ is a complete lattice, then $X=Z$.


I showed in [this answer](https://mathoverflow.net/a/153007/22277) that the Dedekind–MacNeille completion of a poset is the unique minimal completion of a poset.


