Let $P$ be a finite polynomial functor on $\mathbf{Set}$, i.e. $$ P(X) = A_0 + A_1 \times X + \cdots + A_n \times X^n $$
It is well known that such functors preserve $\omega$-colimits and so their initial algebras can be constructed by iterating $P$ up to $\omega$ starting with the empty set.
Take a concrete example of a functor defining (finite) lists of elements of type $A$ $$L(X) = 1 + A \times X.$$
Let $[A]$ be the set of finite lists with elements drawn from $A$. The set $[A]$ is a carrier of the initial algebra for $L$. Define the rank of a list $xs \in [A]$ to be the smallest $n\in\mathbb{N}$ such that $xs \in L^n(\emptyset)$. For lists this is just the length of the list + 1.
First question
Is there another established name for this "rank" for arbitrary (at least finite) polynomial functors?
Second question
Is there a name for functions that preserve rank? Or rather, don't increase it. So on lists these would be functions $f$ such that length of $f(xs)$ is not greater than the length of $xs$.