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.