**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!