Dan wrote:

> Here is my attempt at **Puzzle 66**.

> The condition of the monoidal preorder arises if \\(\otimes\\) is a _monotone_ map between the product preorder \\(X \times X\\) and the original preorder \\(X\\).

Yes, that's my preferred answer to this puzzle! Saying that \\(\otimes : X \times X \to X\\) is monotone is equivalent to the condition

\[ x \le x' \textrm{ and } y \le y' \textrm{ imply } x \otimes y \le x' \otimes y' .\]

for all \\(x,x',y,y' \in X\\). It's nice to compress condition into a simple phrase. So, we can give this more conceptual definition, equivalent to the one given before:

**Definition.** A **monoidal preorder** is a preorder that's also a monoid, for which the multiplication is a monotone function.

Terse and elegant! If we knew more category theory we could do even better:

**Definition.** A **monoidal preorder** is a monoid in the category of preorders.

This is the truly slick way to blend the definitions of "monoid" and "preorder". But we're not at this level of sophistication yet, in this course!

> Here is my attempt at **Puzzle 66**.

> The condition of the monoidal preorder arises if \\(\otimes\\) is a _monotone_ map between the product preorder \\(X \times X\\) and the original preorder \\(X\\).

Yes, that's my preferred answer to this puzzle! Saying that \\(\otimes : X \times X \to X\\) is monotone is equivalent to the condition

\[ x \le x' \textrm{ and } y \le y' \textrm{ imply } x \otimes y \le x' \otimes y' .\]

for all \\(x,x',y,y' \in X\\). It's nice to compress condition into a simple phrase. So, we can give this more conceptual definition, equivalent to the one given before:

**Definition.** A **monoidal preorder** is a preorder that's also a monoid, for which the multiplication is a monotone function.

Terse and elegant! If we knew more category theory we could do even better:

**Definition.** A **monoidal preorder** is a monoid in the category of preorders.

This is the truly slick way to blend the definitions of "monoid" and "preorder". But we're not at this level of sophistication yet, in this course!