I'll start by writing the answers to my questions. The answers to **MD 2** and **MD 3** are given distinctly by:

$$\vee \dashv \Delta \dashv \wedge$$

In **MD 4** I ask about the special case of \$$\mathbb{N}\$$ ordered by the *evenly divides* relation \$$\cdot\ |\ \cdot\$$, and the answer is

$$lcm \dashv \Delta \dashv gcd$$

As I was saying, the poset \$$(\mathbb{N}, \cdot\ |\ \cdot)\$$ does not have infima or suprema, so you can't use them directly to figure all of this out.

It's often nice to operate as if we have a infima and suprema for a preorder \$$(P,\leq_P)\$$ even if it doesn't have them. Also, it would be nice if it was a poset!

We can have all of this by constructing the smallest poset that has them and embeds \$$P\$$. It is called the [*Dedekind–MacNeille completion*](https://en.wikipedia.org/wiki/Dedekind%E2%80%93MacNeille_completion) of \$$P\$$. It is related to the Dedekind cut construction of the real numbers.

Dedekind–MacNeille gives rise to a *monad* \$$\mathbf{DM}\$$ on the category of preorders with monotone maps as morphisms.

> **Definition.** For a given preorder \$$(P,\leq_P)\$$, let
>
> - \$$A^u := \\{p \in P\ :\ \forall a \in A. a \leq p\\}\$$ and
> - \$$A^d := \\{p \in P\ :\ \forall a \in A. p \leq a\\}\$$
>
> Define \$$\mathbf{DM}(P) := \\{A \subseteq P\ :\ A = (A^u)^d\\}\$$.
>
> The structure \$$(\mathbf{DM}(P), \subseteq, \bigcup, \bigcap)\$$ is the Dedekind–MacNeille completion of \$$P\$$.
>
> The *principle ideal* function \$$(\cdot \downarrow) : P \to \mathbf{DM}(P)\$$ takes every element to its completion \$$x \downarrow\;:= \\{x\\}^d\$$.
>
> Finally, we can lift every function \$$f: A \to B\$$ between two posets \$$A\$$ and \$$B\$$ into a function between their completions \$$f^{\mathbf{DM}} : \mathbf{DM}(A) \to \mathbf{DM}(B)\$$ using:
> $$f^{\mathbf{DM}}(X) := ((f_!(X))^u)^d$$

By convention its nice to distinguish objects in the completed structures with the Fracktur font \$$\mathfrak{a}, \mathfrak{b}, \ldots\$$

> **Lemma.** Let \$$f: A \to B\$$ and \$$g: B \to A\$$ be maps on the preorders \$$A\$$ and \$$B\$$. Then:
>
> $$\begin{eqnarray} f^{\mathbf{DM}} \dashv g^{\mathbf{DM}} & \Longleftrightarrow & \forall \mathfrak{b}. g^{\mathbf{DM}}(\mathfrak{b}) = \bigcup\{ \mathfrak{a} \in \mathbf{DM}(A)\ :\ f^{\mathbf{DM}}(\mathfrak{a}) \subseteq \mathfrak{b} \} \\ & \Longleftrightarrow & \forall \mathfrak{a}. f^{\mathbf{DM}}(\mathfrak{a}) = \bigcap\{ \mathfrak{b} \in \mathbf{DM}(B)\ :\ \mathfrak{b} \subseteq g^{\mathbf{DM}}(\mathfrak{a}) \} \end{eqnarray}$$
> and
> $$f^{\mathbf{DM}} \dashv g^{\mathbf{DM}} \Longrightarrow f \dashv g$$
**Proof.** \$$f^{\mathbf{DM}} \dashv g^{\mathbf{DM}} \Longrightarrow f \dashv g\$$ follows by naturality of the principle ideal operation \$$(\cdot\downarrow)\$$. See [Davey and Priestley (2002), §7.38 The Dedekind–MacNeille completion](https://books.google.com/books?id=vVVTxeuiyvQC&pg=PA166#v=onepage&q&f=false). \$$\Box\$$

So certainly proving a Galois adjunction in the Dedekind–MacNeille completion *suffices* to show a Galois connection.

I think the converse of this Lemma is true too but I can't find a reference:

> **Conjecture**. \$$f \dashv g \Longrightarrow f^{\mathbf{DM}} \dashv g^{\mathbf{DM}} \$$

This would give a full on *transfer theorem*.

If I find the time I will tackle this, but I also wanted to do some Haskell in another thread today, so I might not get around to it until the weekend.