@Michael – re "what about the poset is making it so that it can't have a monodical product?"...

• if a finite non-empty poset is *discrete*, we can make it monoidal by slapping any old monoid structure on it (eg turn it into a cyclic group) – since \\(x\leq y \iff x = y\\) this monoid automatically respects the partial order

• if a poset has *all finite joins*, we can make it monoidal by choosing the bottom element (ie the join of nothing!) to be the unit and the binary join \\(x\vee y\\) to be the monoidal product

• if a poset has *all finite meets*, we can make it monoidal by choosing the top element (ie the meet of nothing!) to be the unit and the binary meet \\(x\wedge y\\) to be the monoidal product

So if a finite non-empty poset *can't* be monoidal, then we know it can't be discrete, or have all finite joins, or have all finite meets.

Jonathan's "bow tie" poset is just the smallest poset abiding by all three of these conditions.

• if a finite non-empty poset is *discrete*, we can make it monoidal by slapping any old monoid structure on it (eg turn it into a cyclic group) – since \\(x\leq y \iff x = y\\) this monoid automatically respects the partial order

• if a poset has *all finite joins*, we can make it monoidal by choosing the bottom element (ie the join of nothing!) to be the unit and the binary join \\(x\vee y\\) to be the monoidal product

• if a poset has *all finite meets*, we can make it monoidal by choosing the top element (ie the meet of nothing!) to be the unit and the binary meet \\(x\wedge y\\) to be the monoidal product

So if a finite non-empty poset *can't* be monoidal, then we know it can't be discrete, or have all finite joins, or have all finite meets.

Jonathan's "bow tie" poset is just the smallest poset abiding by all three of these conditions.