 Under AFA, things are different.   Here is the general picture: An operator on sets F is  monotone if whenever a ⊆ b, then also Fa ⊆ Fb.     This is a very common feature for operators on sets.   The polynomial operators on sets are the smallest collection containing the constant operators,  and closed under cartesian product, disjoint union, and functions from a fixed set.    For example, if A and B are fixed sets, then  Fs  = (A × s)  +  B(s + A) is a polynomial operator on sets. If one also allows the power set operator to occur, then we get the power polynomial operators.   Every power polynomial operator is monotone.    And now, we have the following results due to  Aczel: 