Igor, I think conventionally, the "default" order on \$$X \times X\$$ is the product (component-wise) order. When we write a monotone function \$$f : X \times X \to X\$$, we're leaving the orders implicit for both \$$X \times X\$$ and \$$X\$$, but there's still a _fixed_ order for both, which all functions in \$$M^\times\$$ must respect. In that sense, \$$C^\times\$$ is indeed the defining characteristic of \$$M^\times\$$.

Put another way, a function \$$f : X \times X \to X\$$ may be monotone with respect to the product order, but not with respect to the lexicographical order -- or vice versa. Thus, \$$M^\times\$$ _can't_ be defined independent of any particular order.