Hah, I'm only fast because I was so convinced myself that the 3-element monoid sufficed that I put together a diagram and everything, only to convince myself at the last second that I was wrong. I only had to copy back the stuff I had already written. ;)

I just looked at your own counterexample, and it's quite nice! It's interesting to consider that the four-element monoid we've been alluding to is just a particular instance of your function-based construction, since we can model \\(\\{A, B, C, \varepsilon\\}\\) as a _[left action](https://en.wikipedia.org/wiki/Semigroup_action)_ on itself. We get that \\(A\\), \\(B\\) and \\(C\\) are just constant functions; their corresponding right actions are just the identity function, which trivially preserves the order.

I just looked at your own counterexample, and it's quite nice! It's interesting to consider that the four-element monoid we've been alluding to is just a particular instance of your function-based construction, since we can model \\(\\{A, B, C, \varepsilon\\}\\) as a _[left action](https://en.wikipedia.org/wiki/Semigroup_action)_ on itself. We get that \\(A\\), \\(B\\) and \\(C\\) are just constant functions; their corresponding right actions are just the identity function, which trivially preserves the order.