Beautiful proofs! I'm glad my hunch was on the right track :D

I don't think **Puzzle TF1** has a positive answer in general. It works out if \$$\otimes\$$ is commutative, in which case it progresses almost exactly as my answer to Puzzle 60. We could also make it work if we get both left-multiplication and right-multiplication, since we'd just pick the side to multiply on, exactly as in my answer to Puzzle 60. But not every monoid has to be commutative, so it seems unlikely that this would be a sufficient condition in general.