> 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.