> **Puzzle 64**. Are there monoids that cannot be given a relation ≤ making them into monoidal preorders?

I don't think so. Following the definition:

> A **monoidal preorder** is a set \$$X\$$ with a relation \$$\le\$$ making it into a preorder, an operation \$$\otimes : X \times X \to X\$$ and element \$$I \in X\$$ making it into a monoid, and obeying:
> $$x \le x' \textrm{ and } y \le y' \textrm{ imply } x \otimes y \le x' \otimes y' .$$

That law is always obeyed for the trivial preorder where \$$\forall xy. x \leq y\$$, right?

> **Puzzle 65**. Are there monoids that cannot be given any relation ≤ making them into monoidal posets?

Every monoid has a partial order that makes it a monoidal poset. It's the dual of the trivial preorder - the discrete partial order.

Assume \$$x \le x'\$$ and \$$y \le y'\$$. If \$$\le\$$ is the discrete partial order, then \$$x = x'\$$ and \$$y = y'\$$. Hence \$$x \otimes y = x' \otimes y'\$$ by substitution, and thus \$$x \otimes y \leq x' \otimes y'\$$ by reflexivity and substitution.