[Frederick Eisele, #37](https://forum.azimuthproject.org/discussion/comment/18375/#Comment_18375):

> Where did the definition of the relation \$$\le X \times X\$$ in the product preorder come from?
>
> Is it just an alternate formulation of the "pixie dust"?

The product preorder is a way of _producing_ a new preorder given two available preorders. Example 1.42 in the text-book says:

> Given preorders \$$(P, \le_P)\$$ and \$$(Q, \le_Q)\$$, we may define a preorder structure on the product set \$$P \times Q\$$ by setting \$$(p, q) \le_{P \times Q} (p', q')\$$ if and only if \$$p \le_P p'\$$ and \$$q \le_Q q'\$$.

In our case, since \$$X\$$ is a preorder, so is \$$X \times X\$$; given this follows logically, I wouldn't dub it "pixie dust".
If anything, the special ingredient in the definition of monoidal preorders is "monotonicity".

But John has already elaborated on these ideas in the second half of [comment #32](https://forum.azimuthproject.org/discussion/comment/18302/#Comment_18302) – check it out! :-)