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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.

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  • $\begingroup$ See mathoverflow.net/questions/59291/completion-of-a-category. $\endgroup$ – Qiaochu Yuan Feb 17 '15 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 beta-compactification-like 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 Brewer Mar 4 '15 at 21:35
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A subset $A$ of a complete lattice $L$ is said to be join-dense if $L=\{\bigvee^{L} R|R\subseteq A\}$ and $A$ is said to be meet dense in $L$ if $L=\{\bigwedge^{L}R|R\subseteq A\}$. 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 it was mentioned by Matthias Wendt, the Dedekind-MacNeille completion is the smallest completion of $P$. Let me formalize what I mean by smallest.

$\mathbf{Proposition}$ 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 $j:M\rightarrow L$ 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 $P$ so that $P$ is join-dense in $L$. And yes, the completions $L$ of $P$ so 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 that the Dedekind-MacNeille completion of a poset is the unique minimal completion of a poset.

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  • $\begingroup$ 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 downward-closed and upward-closed sets respectively. $\endgroup$ – Qiaochu Yuan Feb 18 '15 at 18:02
  • $\begingroup$ Qiaochu Yuan. I was not being clear about what I meant saying that the Dedekind-MacNeille completion is the only reasonable completion of a poset. I apologize for that. I only meant to say that the Dedekind-MacNeille 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 meet-dense and join-dense in the complete lattice, then the completion is the Dedekind-MacNielle completion. This is the sense in which the Dedekind-MacNielle completion is the only completion of a poset. $\endgroup$ – Joseph Van Name Feb 18 '15 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 join-dense subset. $\endgroup$ – Joseph Van Name Feb 19 '15 at 0:39
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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.
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  • $\begingroup$ 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 D-M is an injective hull of a poset. TY. $\endgroup$ – James Brewer Feb 17 '15 at 20:50

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