[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! :-)

> 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! :-)