Suppose A is a set and S is a collection of subsets closed under arbitrary unions and intersections. Can we find a collection F of functions from A to itself such that a subset B of A is in S if and only if $f(B) \subseteq B$ for all $f \in F$ (in other words, is S precisely the collection of invariant subsets under a collection of functions)?

P.S.: I don't really know what subject tag to give this, so I'm giving it "combinatorics", which seems the closest, though it is more like a question from lattice theory.