And if my last argument in 38 is somewhat valid, then
>What's another way to state this last condition using ideas we've already discussed?

is somewhat redundant, since all \$$\rho: X \times X \to X\$$ are automatically monotone, but only some of them additionally support product order condition.

**Edit**: it seems that my travel in the dark continues; in the set of all possible preorders, defined on the set \$$X \times X\$$, there is the subset of preorders with "natural order" induced by \$$\rho \$$, and the subset of preorders possesing \$$C^\times\$$ property. And these 2 subsets actually may be in any relation with each other - containment, intersection, or even disjoint. So John's argument is valid, don't know why I decided that preorders with \$$C^\times\$$ should be part of "naturally ordered" preorders, probably it was morning.