> What about the poset is making it so that it can't have a monodical product?

I am not sure. If I really knew the theory well, I would have solved Tobias' puzzle by now.

> Can we add or remove elements or arrows to make it possible?

In this particular case, if we added an element \$$\top\$$ above \$$a\$$ and \$$b\$$, we would have a [join-semi-lattice](https://en.wikipedia.org/wiki/Semilattice) and we could use \$$\langle \top, \vee \rangle\$$ for the monoid structure. In category theory, we call \$$\top\$$ an *terminal object*. In lattice theory it's called the *top* or *maximum*.

We could also add a bottom element \$$\bot\$$, and get a *meet-semi-lattice* for Jonathon's bowtie. Bottom elements are also called minimal elements. Category theory calls bottom elements *initial objects*.

Furthermore, if we make \$$b\leq a\$$, then \$$a\$$ would be an end and the structure would be a join-semi-lattice. We could then use \$$\langle a, \vee\rangle\$$ as the monoid.