**Puzzle 80** Let \$$X\$$ be the monoidal preorder described by Anindya in Comment 9 of Lecture 22 . The elements of \$$X\$$ are finite length words, the relation \$$\leq_X\$$ is defined by word length, and \$$\otimes_X\$$ is word concatenation. The map that takes words to their length (e.g. CAT\$$\mapsto 3\$$) is a homomorphism between \$$X\$$ and \$$\mathbb N\$$. It is a homomorphism because the lengths of two words added together is the same as concatenating the words and then taking the length of the result!