Iterative function on set

Question

Is there a name for the procedure/function $ S = f(A) $ such that:

$$ S = \{ x : x \in A \} \cup \{ f(x) : x \in S \} $$

Basically take a set, apply a function to all it's elements, if you get any new elements, apply the function to those elements again etc.

Answer 1

This is an example of a closure operation. Coincidentally, there was a question asked earlier today about such things.

Given a set $S$ and a function $f : S \to S$, the closure of a subset $A \subseteq X$ is a subset $\mathrm{cl}_f(A) \subseteq S$ such that the following three conditions hold:

• $A \subseteq \mathrm{cl}_f(A)$;
• For all $x \in S$, if $x \in \mathrm{cl}_f(A)$, then $f(x) \in \mathrm{cl}_f(A)$; and
• If $C \subseteq S$ is any subset such that $A \subseteq C$ and $f(x) \in C$ for all $x \in C$, then $\mathrm{cl}_f(A) \subseteq C$.

The first two conditions say that $\mathrm{cl}_f(A)$ contains $A$ and contains $f$ of all of its elements; the third condition says that $\mathrm{cl}_f(A)$ is the smallest such subset.