Nested set collection

A nested set collection or nested set family is a collection of sets that consists of chains of subsets forming a hierarchical structure, like Russian dolls. It is used as reference concept in scientific hierarchy definitions, and many technical approaches, like the tree in computational data structures or nested set model of relational databases. Sometimes the concept is confused with a collection of sets with a hereditary property (like finiteness in a hereditarily finite set). Some authors regard a nested set collection as a family of sets. Others prefer to classify it relation as an inclusion order. Let B be a non-empty set and C a collection of subsets of B. Then C is a nested set collection if: (and, for some authors, ) The first condition states that the whole set B, which contains all the elements of every subset, must belong to the nested set collection. Some authors do not assume that B is nonempty, or assume that the empty set is not a member of C. The second condition states that the intersection of every couple of sets in the nested set collection is not the empty set only if one set is a subset of the other. In particular, when scanning all pairs of subsets at the second condition, it is true for any combination with B. Using a set of atomic elements, as the set of the playing card suits: B = {♠, ♥, ♦, ♣}; B1 = {♠, ♥}; B2 = {♦, ♣}; B3 = {♣}; C = {B, B1, B2, B3}. The second condition of the formal definition can be checked by combining all pairs: B1 ∩ B2 = ∅; B1 ∩ B3 = ∅; B3 ⊂ B2. There is a hierarchy that can be expressed by two branches and its nested order: B3 ⊂ B2 ⊂ B; B1 ⊂ B. As sets, that are general abstraction and foundations for many concepts, the nested set is the foundation for "nested hierarchy", "containment hierarchy" and others. A nested hierarchy or inclusion hierarchy is a hierarchical ordering of nested sets. The concept of nesting is exemplified in Russian matryoshka dolls. Each doll is encompassed by another doll, all the way to the outer doll.