Dedekind-MacNeille Completion

Question

Let $\mathbb{N}$ be the set of natural numbers.

a) Describe the Dedekind-MacNeille completion of the poset of finite subsets of $\mathbb{N}$.

b) Describe the Dedekind-MacNeille completion of the poset of cofinite subsets of $\mathbb{N}$.

c) Describe the Dedekind-MacNeille completion of the poset of finite and cofinite subsets of $\mathbb{N}$.

Here is my efforts to do with part a. Call $P$ is the poset of finite subsets of $\mathbb{N}$. Suppose $A$ is an element in the power set of $P$. Then $A^{u}$, which is the set of all upper bound of $A$. Then we have $A^u$ is the set of elements of the form $A \cup B$ where $B$ is an element in the power set of $P$. And then take $\left(A^u\right)^l = A^{ul}$, which is the set of all lower bounds of $A^u$, we have $A^{ul}$ contains the $\emptyset$ and all subsets of $A$. Then by the definition of Dedekind-MacNeille, I conclude $A = \left\{\emptyset\right\}$ and $A = \left\{\emptyset, \left\{n\right\}\right\}$, where $n$ is a natural number.

Do I messed up something? And I still stuck with b and c. So any hint is appreciated. Thanks.

Answer 1

I think the best way to tackle this problem is to use the following result, which you can find, for example, in Davey and Priestley, Introduction to Lattices and Order.

7.41 Theorem. Let $P$ be a poset and let $\varphi:P\to\mathbf{DM}(P)$ be the order-embedding of $P$ into its Dedekind—MacNeille completion given by $\varphi(x) = \,\downarrow\!x$.
(i) $\varphi(P)$ is both join-dense and meet-dense in $\mathbf{DM}(P)$.
(ii) Let $\mathbf L$ be a complete lattice and assume that $P$ is a subset of $L$ which is both join-dense and meet-dense in $L$. Then $\mathbf L \cong \mathbf{DM}(P)$ via an order-isomorphism which agrees with $\varphi$ on $P$.

Now, let us see how we could use this result to answer (a) and (c). (Notice that (b) is similar to (a), so I'll leave it to you.)

If $P$ is the poset of finite subsets of $\mathbb N$, then I claim that $P\cup\{\mathbb N\}$ is its Dedekind—MacNeille completion. Let us use (ii) from the Theorem to show it.
To start with, you can easily show that $P\cup\{\mathbb N\}$, with the $\subseteq$ order relation, is a complete lattice.
Now, given an element of $P\cup\{\mathbb N\}$, we show that it is the join and the meet of some subset of $P$.
For the element $\mathbb N$, we have that $\mathbb N = \bigwedge \varnothing = \bigvee \{\{n\}:n \in \mathbb N\}$, so we're done; any other element $A$ belongs to $P$, so we just say $A = \bigwedge\{A\}=\bigvee\{A\}$.

In the case of (c), that is, $P$ is the poset of finite and co-finite subsets of $\mathbb N$, I claim that the Dedekind—MacNeille completion of the finite and co-finite subsets of $\mathbb N$ is the powerset of $\mathbb N$.
Indeed, any full powerset lattice is a complete lattice.
Now take $A \subseteq \mathbb N$. Then $$A = \bigcup \{ \{a\} : a \in A \} = \bigcap \{ \mathbb N \setminus \{b\} : b \notin A \}.$$ Since $\{a\}$ is a finite subset of $\mathbb N$, for each $a \in A$ and $\mathbb N \setminus \{b\}$ is a co-finite one, for each $b \notin A$, we get that $P$ is both join-dense and meet-dense in the powerset of $\mathbb N$.