Hello, everyone. I want to know whether there are nontrivial results of the following type:\
Given an operation $F$, then for any set $A \neq \emptyset$ that satisfies the property: $\exists K(A)>0$,s.t.,$A\supseteq F(A)\supseteq F\circ F(A)\supseteq \cdots \supseteq \underbrace{F\circ \cdots \circ F}_{K(A)}(A)=\emptyset$,then we can find a precise upper bound of $K(A)$ by using the information of $F,A$.
For example,consider A is the variety corresponding to a finite set of differential polynomials$f_{1},\cdots,f_{n}$ such that the differential ideal it generated is the unit ideal,(here A is obtained by regarding the differential polynomials as ordinary ones in polynomial ring) now take $F$ as the operation that putting $\delta f_{1},\cdots,\delta f_{n}$ to the set $f_{1},\cdots,f_{n}$ (here $\delta$ denotes the derivation operation),we obtain a set $A_{1}$ and then taking its variety,we obtain $F(A)$,continue,putting ${\delta}^{2} f_{1},\cdots,{\delta}^{2} f_{n}$ to set $A_{1}$,we obtain $A_{2}$,then $F\circ F(A)$,$\cdots,$. we know that the set sequence is nonincreasing and will become $\emptyset$ in some step.
Since I want to know how to attack this type of problem, any known results linked to this type may be illuminating.
