One interesting thing about the bowtie is that it seems to have a lot of internal symmetry. We can swap \\(a\\) and \\(b\\), swap \\(c\\) and \\(d\\), or even take the opposite of the relation while simultaneously swapping \\(a\\) with \\(c\\) and \\(b\\) with \\(d\\). I wonder if there's a connection between the number/kind of automoprphisms and isomorphisms a poset has, and whether or not it can support a monoidal structure. I suspect there might be, if only because each isomorphism should restrict any monoidal structure. (I wish I knew some Galois theory!)