re **Puzzle 67** – how about we take the set of finite "words" on an "alphabet" (including the empty word)

• this is a monoid by string concatenation, eg CAT \\(\otimes\\) FISH \\(=\\) CATFISH

• it is a preorder by word length, eg CAT \\(\leq\\) DOG, DOG \\(\leq\\) CAT, CAT \\(\leq\\) FISH, FISH \\(\nleq\\) DOG

• the monoid structure preserves \\(\leq\\), so it is a monoidal preorder

• it is symmetric, eg CATFISH \\(\leq\\) FISHCAT and FISHCAT \\(\le\\) CATFISH

• but it is not commutative, since CATFISH \\(\neq\\) FISHCAT