Nested hierarchy s, also known as
nesting dolls or
Russian dolls. Each doll is encompassed inside another until the smallest one is reached. This is the concept of
nesting. When the concept is applied to
sets, the resulting ordering is a
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. The general concept is both demonstrated and mathematically formulated in the following example: : \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. In many programming taxonomies and syntax models (as well as fractals in mathematics), nested hierarchies, including Russian dolls, are also used to illustrate the properties of
self-similarity and
recursion. Recursion itself is included as a subset of hierarchical programming, and recursive thinking can be synonymous with a form of hierarchical thinking and logic.
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. A general example of a containment hierarchy is demonstrated in
class inheritance in
object-oriented programming. Two types of containment hierarchies are the
subsumptive containment hierarchy and the
compositional containment hierarchy. A subsumptive hierarchy "
subsumes" its children, and a compositional hierarchy is "
composed" of its children. A hierarchy can also be both subsumptive
and compositional.
Subsumptive containment hierarchy A
subsumptive containment hierarchy is a classification of object classes from the general to the specific. Other names for this type of hierarchy are "taxonomic hierarchy" and "
IS-A hierarchy". The last term describes the relationship between each level—a lower-level object "is a" member of the higher class. The taxonomical structure outlined above is a subsumptive containment hierarchy. Using again the example of Linnaean taxonomy, it can be seen that an object that is a member of the level
Mammalia "is a" member of the level
Animalia; more specifically, a human "is a" primate, a primate "is a" mammal, and so on. A subsumptive hierarchy can also be defined abstractly as a hierarchy of "
concepts". Most engineered structures, whether natural or artificial, can be broken down in this manner. The compositional hierarchy that every person encounters at every moment is the
hierarchy of life. Every person can be reduced to
organ systems, which are composed of
organs, which are composed of
tissues, which are composed of
cells, which are composed of
molecules, which are composed of
atoms. In fact, the last two levels apply to all
matter, at least at the
macroscopic scale. Moreover, each of these levels inherit all the properties of their
children. In this particular example, there are also
emergent properties—functions that are not seen at the lower level (e.g.,
cognition is not a property of
neurons but is of the
brain)—and a scalar quality (molecules are bigger than atoms, cells are bigger than molecules, etc.). Both of these concepts commonly exist in compositional hierarchies, but they are not a required general property. These
level hierarchies are characterized by bi-directional
causation.
Upward causation involves lower-level entities causing some property of a higher level entity; children entities may interact to yield parent entities, and parents are composed at least partly by their children.
Downward causation refers to the effect that the incorporation of entity
x into a higher-level entity can have on
x's properties and interactions. Furthermore, the entities found at each level are
autonomous. ==Contexts and applications==