> **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.

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.