Suppose we have a set $X$ and a collection $\mathcal{F}$ of partial functions from $X$ to $X$. Say that $A\subseteq X$ is $\mathcal{F}$-stable if for every $f\in\mathcal{F}$ with $dom(f)\supseteq A$ we have:
there is some $g\in\mathcal{F}$ with $dom(g)=f[A]$, and
there is some $h\in\mathcal{F}$ with $dom(h)=X\setminus f[A]$.
(Here "$j[C]$" is the set $\{j(c): c\in C\}$.)
My question is:
Does this notion appear anywhere?
... Outside recursion theory, that is.
This notion is an abstraction of the intuition behind "generalized finiteness" in higher recursion theory, where we often take the stance that the most important feature of finite sets in classical recursion theory is that a set is finite iff its image under a recursive function is always recursive. This heuristic, for example, led to the key insight that "hyperarithmetic" is the analogue of finite, not recursive (which led to metarecursion theory). In some contexts it's a bit more complicated than that (e.g. in $\beta$-recursion theory), but the basic idea is definitely an important one.
So the starting point is the situation when $\mathcal{F}$ is the set of partial recursive functions $X\rightarrow X$ in some generalized recursion theory; "domain of a partial computable function" is then the standard analogue of "recursively enumerable," and the recursive sets in this theory are those which are both r.e. and co-r.e. Dropping all assumptions on what a "generalized recursion theory" should consist of, we get the notion above, which is applicable to any family of partial functions; I'm curious whether it shows up anywhere. (Obviously what I'd love is for it to crop up somewhere far from recursion theory where we can then subsequently interpret the situation as a kind of generalized recursion theory.)
At a guess, the two most plausible candidates to me are category theory and universal algebra (hence those tags). But I'm happy for examples coming from anywhere. Indeed, the further away from logic, the better.