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\$$.

If \$$\otimes\$$ is a monotone map then, for all pairs \$$(x, y), (x' , y') \in X \times X \$$, we have:

- \$$(x, y) \le_{X \times X} (x', y')\$$ implies \$$x \otimes y \le_{X} x' \otimes y'\$$.

Using the definition of the relation \$$\le_{X \times X}\$$ in the product preorder, we arrive at the condition of the monoidal preorder:

- \$$x \le_X x'\$$ and \$$y \le_X y'\$$ imply \$$x \otimes y \le_{X} x' \otimes y'\$$.