We give a quadratic O(|X|^2) space representation based on a canonical tree for any subset family F\subseteq 2^X closed under the union and the difference of its overlapping members. The cardinality of F is potentially in O(2^|^X|), and the total cardinality of its members even higher. As far as we know this is the first representation result for such families. As an application of this framework we obtain a unique digraph decomposition that not only captures, but also is strictly more powerful than the well-studied modular decomposition. A polynomial time decomposition algorithm for this case is described.