[@Anindya](https://forum.azimuthproject.org/profile/1950/Anindya%20Bhattacharyya) wrote:
> As for some counterexamples...

Here's another one! Let \$$X\$$ be any set with at least two elements, and endow it with the _codiscrete_ order where \$$x \le y\$$ for all \$$x, y\$$. Then every monoid on \$$X\$$ gives a monoidal preorder. But certainly, _none_ of these monoids give a (unique) join or meet. (I'm solidly in the camp that thinks that if there's no unique [least upper / greatest lower] bound, then the join/meet operation shouldn't be considered defined at all.)

If we do this same trick with the discrete order, we get your first counterexample, but from a slightly different starting point.

> How "lattice-like" can ⊗ get before it is forced to be a join or meet wrt the underlying preorder?

One thing we require of lattices that we don't of arbitrary monoidal preorders is that \$$x \le x \vee y\$$ and \$$y \le x \vee y\$$. Using a monoid where the product is incomparable with at least one of the operands should be sufficient to dodge being a meet or join. (Or heck, try something weird where \$$x \le x \otimes y \le y\$$.)