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.