As sets, that are general abstraction and foundations for many concepts, the
nested set is the foundation for "nested hierarchy", "containment hierarchy" and others.
Nested hierarchy 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. The outer doll holds all of the inner dolls, the next outer doll holds all the remaining inner dolls, and so on. Matryoshkas represent a nested hierarchy where each level contains only one object, i.e., there is only one of each size of doll; a generalized nested hierarchy allows for multiple objects within levels but with each object having only one parent at each level. Illustrating the general concept: : \text{square} \subset \text{quadrilateral} \subset \text{polygon} \subset \text{shape} \, A square can always also be referred to as a quadrilateral, polygon or shape. In this way, it is a hierarchy. However, consider the set of polygons using this classification. A square can
only be a quadrilateral; it can never be a
triangle,
hexagon, etc. Nested hierarchies are the organizational schemes behind
taxonomies and systematic classifications. For example, using the original
Linnaean taxonomy (the version he laid out in the 10th edition of
Systema Naturae), a human can be formulated as: : \text{H. sapiens} \subset \text{Homo} \subset \text{Primates} \subset \text{Mammalia} \subset \text{Animalia} Taxonomies may change frequently (as seen in
biological taxonomy), but the underlying concept of nested hierarchies is always the same.
Containment hierarchy A containment hierarchy is a direct extrapolation of the
nested hierarchy concept. All of the ordered sets are still nested, but every set must be "
strict" — no two sets can be identical. The shapes example above can be modified to demonstrate this: : \text{square} \subsetneq \text{quadrilateral} \subsetneq \text{polygon} \subsetneq \text{shape} \, The notation x \subsetneq y \, means
x is a subset of
y but is not equal to
y. Containment hierarchy is used in
class inheritance of
object-oriented programming. ==See also==