@Zev: yeah, I was once a pure mathematician, but that was 20 years ago :-( I started out thinking about negative rectangles and what that means for intersection, but I would in fact need to make 'as most sense as possible' of a number of operations. If I could convince myself that operationA makes sense because it now builds a Monoid or whatever instead of just a 'dumb set', I'd be happy. Maybe I should just consider each operation, and see whether they can be sensibly made to be associative or reflexive, etc. And see how it goes from there.

I'll update the quesiton with a few more thoughts...