John, continuing the discussion from Lecture 21, I became interested in the following, maybe you'll be able to help me here. Let's say we have a monotone function \\(f: X \times X \to X\\) and the condition \\(C^\times: x \le x' \textrm{ and } y \le y' \Rightarrow f(x, y) \le f(x', y') .\\)

As we learned, the condition implies that the function is monotone: \\(C^\times \Rightarrow f \in M^\times\\), where \\(M^\times\\) is the set of monotone functions \\(X \times X \to X\\). The question is whether converse \\(f \in M^\times \Rightarrow C^\times\\) is also true, that is the function \\(f\\) being monotonic implies that it supports the condition \\(C^\times\\)?

The easiest way to show that converse is not true, is to find a monotone function \\(g \in M^\times\\) which doesn't support \\(C^\times\\). The lexicographic order function looks like a good candidate. If it is monotonic (it looks like so, but I'm not sure: \\(l(x, y) = x\\)) and doesn't obey \\(C^\times\\), then the conclusion is that the condition \\(C^\times\\) is supported only by a subset \\(F \subset M^\times\\).

So the question is whether any monotone will do to automatically transform \\((X \times X)\\) into preorder, or the condition brings something new to the table? Still figuring it out.