Editing Hierarchy (section)

===Nested hierarchy===
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[[File:Russian-Matroshka no bg.jpg|200px|right|thumb|[[Matryoshka doll]]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 [[set (mathematics)|sets]], the resulting ordering is a ''nested hierarchy''.]]
A nested hierarchy or ''inclusion hierarchy'' is a hierarchical ordering of [[Nested set collection|nested set]]s.<ref name="natsocsci-ch4">{{cite encyclopedia|title=Hierarchy, Complexity, Society|last=Lane|first=David|pages=81–120|encyclopedia=Hierarchy in Natural and Social Sciences|editor=Pumain, Denise|publisher=[[Springer-Verlag]]|location=New York, New York|year=2006|isbn=978-1-4020-4126-6}}</ref> The concept of nesting is exemplified in Russian [[matryoshka doll]]s. 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:

: <math> \text{square} \subset \text{quadrilateral} \subset \text{polygon} \subset \text{shape} \, </math>

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 [[Taxonomy (general)|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:<ref>{{cite book|title=Systema naturae per regna tria naturae :secundum classes, ordines, genera, species, cum characteribus, differentiis, synonymis, locis|last=Linnaei|first=Carl von|author-link=Carl Linnaeus|year=1959|edition=10th|language=la|url=https://www.biodiversitylibrary.org/bibliography/542|location=[[Stockholm]]|publisher=Impensis Direct|isbn=0-665-53008-0|access-date=2011-09-24|archive-date=2008-10-10|archive-url=https://web.archive.org/web/20081010032456/http://www.biodiversitylibrary.org/bibliography/542|url-status=live}}</ref>

: <math>\text{H. sapiens} \subset \text{Homo} \subset \text{Primates} \subset \text{Mammalia} \subset \text{Animalia}</math>

Taxonomies may change frequently (as seen in [[biological classification|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.<ref name=Corballis>{{cite book |first=Michael |last=Corballis |title=The Recursive Mind  |publisher=Princeton University Press |year=2011 |isbn=978-0691145471}}</ref>