Frederick - that's a nice solution to Puzzle 62. Note also that \\(\mathbb{R}^+\\) with \\(\le\\) as partial order, \\(\cdot\\) as monoidal operation and \\(1\\) as unit is _isomorphic_ as a monoidal poset to \\(\mathbb{R}\\) with \\(\le\\) as partial order, \\(+\\) as monoidal operation and \\(0\\) as unit. An isomorphism is

\[ \ln : \mathbb{R}^+ \to \mathbb{R} \]

with inverse

\[ \exp: \mathbb{R} \to \mathbb{R}^+ . \]

Another cute thing: you didn't need to throw out zero! There's another nice monoidal poset: \\( [0,\infty) \\) with \\(\le\\) as partial order, \\(\cdot\\) as monoidal operation and \\(1\\) as unit. This is isomorphic as a monoidal poset to \\( [-\infty, 0) \\) with \\(\le\\) as partial order, \\(+\\) as monoidal operation and \\(0\\) as unit.