The subject of polynomial monads is well trodden. We know that every monad is generated by an adjunction. What are the special properties of any adjunction that generates a polynomial monad?
Take a domain, $D$, like the interval domain, and look at the set of maps $F = [D,D]$, where $f \in F, f: D \rightarrow D$. Let $F_a \subset F$ be the subset of all $f$ that preserve both meet and join, so they form adjunctions when $D$ is considered a category. Is there a subset $P \subset F_a$ which generate polynomial monads? What are the special properties of such $f' \in P$
