Editing Hierarchy (section)

==Subtypes==
===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>

===Containment hierarchy===
[[File:Containment Hierarchy example.png|300px|thumb|right|A diagram illustrating a containment hierarchy. The set of all squares is completely contained in the larger set of quadrilaterals, and so on.]]
A containment hierarchy is a direct extrapolation of the [[#Nested hierarchy|nested hierarchy]] concept. All of the ordered sets are still nested, but every set must be "[[strict subset|strict]]"—no two sets can be identical. The shapes example above can be modified to demonstrate this:

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

The notation <math> x \subsetneq y \, </math> means ''x'' is a subset of ''y'' but is not equal to&nbsp;''y''.

A general example of a containment hierarchy is demonstrated in [[inheritance (object-oriented programming)|class inheritance]] in [[object-oriented programming]].

Two types of containment hierarchies are the ''subsumptive'' containment hierarchy and the ''compositional'' containment hierarchy. A subsumptive hierarchy "[[wikt:subsume|subsumes]]" its children, and a compositional hierarchy is "[[wikt:composed|composed]]" of its children. A hierarchy can also be both subsumptive ''and'' compositional{{example needed|date=August 2018}}.<ref name="AI industrial">{{cite encyclopedia|title=Industrial and Engineering Applications of Artificial Intelligence and Expert Systems|last1=Kopisch|first1=Manfred|last2=Günther|first2=Andreas|doi=10.1007/BFb0024994|editor-last=Belli|editor-first=Fevzi|encyclopedia=Industrial and engineering applications of artificial intelligence and expert systems: 5th international conference, IEA/AIE-92, Paderborn, Germany, June 9–12, 1992 : proceedings|year=1992|publisher=[[Springer Science+Business Media|Springer]]|pages=424–427|isbn=3-540-55601-X|series=Lecture Notes in Computer Science Series|volume=602|issn=0302-9743|editor6-last=Radermacher|editor8-first=Franz-Josef|chapter=Configuration of a passenger aircraft cabin based on conceptual hierarchy, constraints and flexible control}}</ref>

===Subsumptive containment hierarchy===
A ''[[Category theory|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|IS-A]] hierarchy".<ref name="Lehmann"/><ref name="ibm">{{cite web|url=http://publib.boulder.ibm.com/infocenter/wtxdoc/v8r2m0/index.jsp?topic=/com.ibm.websphere.dtx.md.doc/concepts/c_map_design_Compositional_Hierarchy.htm|archive-url=https://archive.today/20130103052727/http://publib.boulder.ibm.com/infocenter/wtxdoc/v8r2m0/index.jsp?topic=/com.ibm.websphere.dtx.md.doc/concepts/c_map_design_Compositional_Hierarchy.htm|url-status=dead|archive-date=3 January 2013|title=Compositional hierarchy|work=WebSphere Transformation Extender Design Studio|access-date=9 October 2009}}</ref><ref name="sys model">{{cite book|chapter=An advanced modeling environment based on a hybrid AI-OR approach|chapter-url=https://books.google.com/books?id=ds2eIQ6XZy0C&pg=PA366|pages=366–75|last1=Funke|first1=Birger|last2=Sebastian|first2=Hans-Jürgen|title=Systems modelling and optimization: proceedings of the 18th IFIP TC7 conference|volume=396|series=Research notes in mathematics series|editor1-last=Polis|editor1-first=Michael P.|editor2-last=Dontchev|editor2-first=Asen L.|editor3-last=Kall|editor3-first=Peter|editor4-last=Lascieka|editor4-first=Irena|editor5-last=Olbrot|editor5-first=Andrzej W.|publisher=[[CRC Press]]|year=1999|isbn=978-0-8493-0607-5}}</ref> 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 "[[concept]]s".<ref name="sys model"/> For example, with the Linnaean hierarchy outlined above, an entity name like ''Animalia'' is a way to group all the species that fit the [[wikt:conceptualization|conceptualization]] of an animal.

===Compositional containment hierarchy===
A ''compositional'' containment hierarchy is an ordering of the parts that make up a system—the system is "composed" of these parts.<ref name="Parsons">{{cite book|last=Parsons|first=David|title=Object Oriented Programming in C++|publisher=Cengage Learning|year=2002|pages=110–185|isbn=0-8264-5428-3}}</ref> 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 system]]s, which are composed of [[organ (anatomy)|organs]], which are composed of [[tissue (biology)|tissues]], which are composed of [[cells (biology)|cells]], which are composed of [[molecule]]s, which are composed of [[atom]]s. 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 [[#Terminology|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 [[neuron]]s but is of the [[Human brain|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 [[Causality|causation]].<ref name="natsocsci-ch4"/> ''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]]''.