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.

>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.