**Puzzle 60.** Yes. Suppose we have \$$x \le y\$$ and \$$x' \le y'\$$. We know all of the variables are nonnegative, so we can freely multiply our variables into these inequalities to obtain \$$x \cdot x' \le y \cdot x'\$$ and \$$y \cdot x' \le y \cdot y'\$$; by transitivity, we have \$$x \cdot x' \le y \cdot y'\$$.

**Puzzle 61.** No, because Puzzle 62 says one of Puzzle 60 and Puzzle 61 must have "no" as its answer. ;) Without appealing to outside information, we can merely demonstrate that \$$-1 \le 1\$$ and \$$-2 \le 1\$$ but \$$-1 \cdot -2 = 2 \not\le 1 = 1 \cdot 1\$$.

**Puzzle 62.** Define \$$x \preceq y\$$ by \$$\lvert x \rvert \le \lvert y \rvert\$$. Then if \$$x \preceq y\$$ and \$$x' \preceq y'\$$, we know \$$\lvert x \rvert \le \lvert y \rvert\$$ and \$$\lvert x' \rvert \le \lvert y' \rvert\$$; these are all positive quantities as before, so by the same process we find \$$\lvert x \cdot x' \rvert = \lvert x \rvert \cdot \lvert x' \rvert \le \lvert y \rvert \cdot \lvert y' \rvert = \lvert y \cdot y' \rvert\$$. Therefore, \$$x \cdot x' \preceq y \cdot y'\$$, as desired.

We do get two units out of this (\$$-1\$$ and \$$1\$$), but I think that's okay as long as we pick one and stick to it. They're equivalent in the sense that \$$-1 \preceq 1\$$ and \$$1 \preceq -1\$$.

**Puzzle 63.** The structures \$$\langle \mathcal{P}(X), \subseteq, \cup, \emptyset \rangle\$$ and \$$\langle \mathcal{P}(X), \subseteq, \cap, X \rangle\$$ are monoidal preorders. More generally, any lattice possesses two (dual) monoidal structures.

**Puzzle 64.** We can always apply either the discrete or codiscrete preorders to make a monoid into a monoidal preorder.

**Puzzle 65.** We can always apply the discrete preorder to make a monoid into a monoidal poset.

**Puzzle 66.** This is surprisingly hard to think through. I've got a weak hunch that the poset below cannot be made into a monoidal poset, but I'm still working through it. The poset can't be a lattice, per Puzzle 63, and this is the smallest non-semilattice non-trivial-order at my disposal.

![](https://i.imgur.com/Km2PosD.png)