If I want to enumerate all independent sets of an independence system, is there a nicer way to do this than to just iterate through all $2^n$ subsets? In particular, if I've found that something is not an independent set, then I obviously don't have to check any supersets of that set. But I don't see how to do that without a lot of bookkeeping.
So, is there a nice/clever way to iterate through the subsets of some set, skipping over supersets of sets I determine to be dependent (and possibly also skipping over subsets of sets I determine to be independent)?