Abstract
A subset family $\cf\subseteq2^X$ is partitive crossing if it is close under the union, the intersection, and the difference of its crossing members; it is a union-difference family if closed under the union and the difference of its overlapping members. In both cases, the cardinality of $\cf$ is potentially in $O(2^{|X|})$, and the total cardinality of its members even higher. We give a linear $O(|X|)$ and a quadratic $O(|X|^2)$ space representation based on a canonical tree for any partitive crossing family and union-difference family, respectively. As an application of this framework we obtain a unique digraph decomposition and a unique decomposition of $2-$structure. Both of them not only captures, but also is strictly more powerful than the well-studied modular decomposition and clan decomposition. Polynomial time decomposition algorithms for both cases are described.