Matthew: I was being pretty vague when I wrote

> Everything Matthew and Owen just said is true for posets, but not for preorders.

I didn't mean _nothing_ you said was true for preorders. For example, I think the alternative characterization of Galois connections works fine for preorders. Looking over what you said, this is the only thing that I'm sure is false for preorders:

> Moreover, if a monotone function has a left (or right) Galois adjoint it is unique.

I tried to hint at the reason why:

> The left or right adjoint of a monotone function between posets is unique if it exists. This need not be true for preorders.

> The issue can be seen clearly in the phrases "the smallest \\(b\\) with \\(a \le g(b)\\)". In a poset, such an \\(a\\) is unique if it exists. In a preorder, that's not true, since we could have \\(a \le a'\\) and \\(a' \le a\\) yet still \\(a \ne a'\\).

Do you see how to cook up a monotone function between preorders that has more than one left adjoint?

> Everything Matthew and Owen just said is true for posets, but not for preorders.

I didn't mean _nothing_ you said was true for preorders. For example, I think the alternative characterization of Galois connections works fine for preorders. Looking over what you said, this is the only thing that I'm sure is false for preorders:

> Moreover, if a monotone function has a left (or right) Galois adjoint it is unique.

I tried to hint at the reason why:

> The left or right adjoint of a monotone function between posets is unique if it exists. This need not be true for preorders.

> The issue can be seen clearly in the phrases "the smallest \\(b\\) with \\(a \le g(b)\\)". In a poset, such an \\(a\\) is unique if it exists. In a preorder, that's not true, since we could have \\(a \le a'\\) and \\(a' \le a\\) yet still \\(a \ne a'\\).

Do you see how to cook up a monotone function between preorders that has more than one left adjoint?