TheInfoList

A hierarchy (from
Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...
: , from , 'president of sacred rites') is an arrangement of items (objects, names, values, categories, etc.) that are represented as being "above", "below", or "at the same level as" one another. Hierarchy is an important concept in a wide variety of fields, such as
philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of language, language. Such questio ...

,
architecture upright=1.45, alt=Plan d'exécution du second étage de l'hôtel de Brionne (dessin) De Cotte 2503c – Gallica 2011 (adjusted), Plan of the second floor (attic storey) of the Hôtel de Brionne in Paris – 1734. Architecture (Latin ''archi ...

,
design A design is a plan or specification for the construction of an object or system or for the implementation of an activity or process, or the result of that plan or specification in the form of a prototype A prototype is an early sample, mode ...

,
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
,
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of computation, automation, a ...
,
organizational theory A theory involves concepts or construct Construct, Constructs or constructs may refer to: * Construct (information technology), a collection of logic components forming an interactive agent or environment ** Language construct * Construct (albu ...
,
systems theory Systems theory is the interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic disciplines into one activity (e.g., a research project). It draws knowledge from several other fields ...
,
systematic biology Biology, Biological systematics is the study of the diversification of living forms, both past and present, and the Correlation and dependence, relationships among living things through time. Relationships are visualized as evolutionary trees (sy ...
, and the
social sciences Social science is the branch The branches and leaves of a tree. A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant life and a branch of biol ...

(especially
political philosophy Political philosophy or political theory is the philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical or menta ...

). A hierarchy can link entities either directly or indirectly, and either vertically or diagonally. The only direct links in a hierarchy, insofar as they are hierarchical, are to one's immediate superior or to one of one's subordinates, although a system that is largely hierarchical can also incorporate alternative hierarchies. Hierarchical links can extend "vertically" upwards or downwards via multiple links in the same direction, following a path. All parts of the hierarchy that are not linked vertically to one another nevertheless can be "horizontally" linked through a path by traveling up the hierarchy to find a common direct or indirect superior, and then down again. This is akin to two
co-worker Employment is the relationship between two parties Image:'Hip, Hip, Hurrah! Artist Festival at Skagen', by Peder Severin Krøyer (1888) Demisted with DXO PhotoLab Clearview; cropped away black border edge.jpg, 300px, ''Hip, Hip, Hurrah!'' ( ...
s or
colleague Collegiality is the relationship between ''colleagues''. Colleague is taken to mean a fellow member of the same profession, a group of colleagues united in a common purpose, and used in proper names, such as Electoral college, Electoral College, Co ...
s; each reports to a common superior, but they have the same relative amount of authority. Organizational forms exist that are both alternative and complementary to hierarchy.
Heterarchy A heterarchy is a system of organization where the elements of the organization are unranked (non-hierarchical A hierarchy (from the Greek: , from , 'president of sacred rites') is an arrangement of items (objects, names, values, categories, etc. ...
is one such form.

# Nomenclature

Hierarchies have their own special vocabulary. These terms are easiest to understand when a hierarchy is diagrammed (see
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
). In an organizational context, the following terms are often used related to hierarchies: *
Object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the grasp of the senses ** Object (abstract), an object which does not exist at any particular time or pl ...
: one entity (e.g., a person, department or
concept Concepts are defined as abstract ideas A mental representation (or cognitive representation), in philosophy of mind Philosophy of mind is a branch of philosophy that studies the ontology and nature of the mind and its relationship with the bo ...

or element of arrangement or member of a set) *
System A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purp ...

: the entire set of objects that are being arranged hierarchically (e.g., an administration) *
Dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

: another word for "system" from on-line analytical processing (e.g. cubes) *
Member Member may refer to: * Military juryA United States military "jury" (or "Members", in military parlance) serves a function similar to an American civilian jury, but with several notable differences. Only a Courts-martial in the United States, Gene ...
: an (element or object) at any (level or rank) in a (class-system, taxonomy or dimension) *Terms about Positioning **
Rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking A ranking is a relationship between a set of items such that, for any two items, the first is either "rank ...

: the relative
value Value or values may refer to: * Value (ethics) In ethics Ethics or moral philosophy is a branch of philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, E ...
, worth,
complexity Complexity characterises the behaviour of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environme ...
,
power Power typically refers to: * Power (physics) In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. In older works, p ...
, importance,
authority In the fields of sociology Sociology is the study of society, human social behaviour, patterns of social relationships, social interaction, and culture that surrounds everyday life. It is a social science that uses various methods of Empiric ...

, level etc. of an object **
Level Level or levels may refer to: Engineering * Level (instrument), a device used to measure true horizontal or relative heights *Canal pound or level *Regrading or levelling, the process of raising and/or lowering the levels of land *Storey or level ...

or Tier: a set of objects with the same rank OR importance **
Ordering Order or ORDER or Orders may refer to: * Orderliness, a desire for organization * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements hav ...
: the arrangement of the (ranks or levels) **Hierarchy: the arrangement of a particular set of members into (ranks or levels). Multiple hierarchies are possible per (dimension taxonomy or Classification-system), in which selected levels of the dimension are omitted to flatten the structure *Terms about Placement **
Hierarch An ordinary (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Rom ...
, the apex of the hierarchy, consisting of one single orphan (object or member) in the top level of a dimension. The root of an **
Member Member may refer to: * Military juryA United States military "jury" (or "Members", in military parlance) serves a function similar to an American civilian jury, but with several notable differences. Only a Courts-martial in the United States, Gene ...
, a (member or node) in any level of a hierarchy in a dimension to which (superior and subordinate) members are attached **
Orphan An orphan (from the el, ορφανός, orphanós) is a child whose parents have died, are unknown, or have permanently abandoned them. In common usage, only a child who has lost both parents due to death is called an orphan. When referring ...
, a member in any level of a dimension without a parent member. Often the apex of a disconnected branch. Orphans can be grafted back into the hierarchy by creating a relationship (interaction) with a parent in the immediately superior level **
Leaf A leaf (plural leaves) is the principal lateral appendage of the vascular plant plant stem, stem, usually borne above ground and specialized for photosynthesis. The leaves, stem, flower and fruit together form the shoot system. Leaves are ...
, a member in any level of a dimension without subordinates in the hierarchy **
Neighbour Neighbour, Neighbor, Neighbours or Neighbors may refer to: * Person who lives in one's neighbourhood A neighbourhood (British English British English (BrE) is the standard dialect of the English language English is a West Ge ...
: a member adjacent to another member in the same (level or rank). Always a peer. **
Superior Superior may refer to: *Superior (hierarchy) In a hierarchy A hierarchy (from the Greek: , from , 'president of sacred rites') is an arrangement of items (objects, names, values, categories, etc.) in which the items are represented as being "ab ...
: a higher level or an object ranked at a higher level (A parent or an ancestor) **
Subordinate A hierarchy (from the Greek: , from , 'president of sacred rites') is an arrangement of items (objects, names, values, categories, etc.) in which the items are represented as being "above", "below", or "at the same level as" one another. Hierarch ...
: a lower level or an object ranked at a lower level (A child or a descendant) **
Collection Collection or Collections may refer to: * Cash collection, the function of an accounts receivable department * Collection agency, agency to collect cash * Collections management (museum) ** Collection (artwork), objects in a particular field fo ...
: all of the objects at one level (i.e. Peers) **
Peer Peer may refer to: Sociology * Peer, an equal in age, education or social class; see Peer group * Peer, a member of the peerage Computing * Peer, one of several functional units in the same layer of a network; See Peer group (computer networking) ...
: an object with the same rank (and therefore at the same level) **
Interaction Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way causal effect. Closely related terms are interac ...
: the relationship between an object and its direct superior or subordinate (i.e. a superior/inferior pair) *** a direct interaction occurs when one object is on a level exactly one higher or one lower than the other (i.e., on a
tree In botany, a tree is a perennial plant with an elongated Plant stem, stem, or trunk (botany), trunk, supporting branches and leaves in most species. In some usages, the definition of a tree may be narrower, including only wood plants with se ...
, the two objects have a line between them) **
Distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...
: the minimum number of connections between two objects, i.e., one less than the number of objects that need to be "crossed" to trace a path from one object to another ** Span: a qualitative description of the width of a level when diagrammed, i.e., the number of subordinates an object has *Terms about Nature ** Attribute: a heritable characteristic of (members and their subordinates) in a level (e.g. ''hair-colour'') ** Attribute-value: the specific value of a heritable characteristic (e.g. ''Auburn'') In a mathematical context (in
graph theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
), the general terminology used is different. Most hierarchies use a more specific vocabulary pertaining to their subject, but the idea behind them is the same. For example, with
data structure In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of ...

s, objects are known as
nodes In general, a node is a localized swelling (a "knot") or a point of intersection (a Vertex (graph theory), vertex). Node may refer to: In mathematics *Vertex (graph theory), a vertex in a mathematical graph *Node (autonomous system), behaviour fo ...
, superiors are called
parents A parent is a caregiver of the offspring in their own species. In humans, a parent is the caretaker of a child (where "child" refers to offspring, not necessarily age). A ''biological parent'' is a person whose gamete resulted in a child, a male ...
and subordinates are called
children Biologically, a child (plural children) is a human Humans (''Homo sapiens'') are the most abundant and widespread species In biology Biology is the natural science that studies life and living organisms, including their anat ...
. In a business setting, a superior is a supervisor/boss and a peer is a
colleague Collegiality is the relationship between ''colleagues''. Colleague is taken to mean a fellow member of the same profession, a group of colleagues united in a common purpose, and used in proper names, such as Electoral college, Electoral College, Co ...
.

## Degree of branching

Degree Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
of branching refers to the number of direct
subordinates A hierarchy (from the Ancient Greek, Greek: , from , 'president of sacred rites') is an arrangement of items (objects, names, values, categories, etc.) in which the items are represented as being "above", "below", or "at the same level as" one ano ...
or children an object has (in graph theory, equivalent to the number of other vertices connected to via outgoing arcs, in a directed graph) a node has. Hierarchies can be categorized based on the "maximum degree", the highest degree present in the system as a whole. Categorization in this way yields two broad classes: ''linear'' and ''branching''. In a linear hierarchy, the maximum degree is 1. In other words, all of the objects can be visualized in a line-up, and each object (excluding the top and bottom ones) has exactly one direct subordinate and one direct superior. Note that this is referring to the ''objects'' and not the ''levels''; every hierarchy has this property with respect to levels, but normally each level can have an infinite number of objects. An example of a linear hierarchy is the hierarchy of life. In a branching hierarchy, one or more objects has a degree of 2 or more (and therefore the minimum degree is 2 or higher). For many people, the word "hierarchy" automatically evokes an image of a branching hierarchy. Branching hierarchies are present within numerous systems, including
organization An organization, or organisation (Commonwealth English The use of the English language English is a of the , originally spoken by the inhabitants of . It is named after the , one of the ancient that migrated from , a peninsu ...

s and
classification scheme In information science and ontology (information science), ontology, a classification scheme is the product of arranging things into kinds of things (classes) or into ''groups'' of classes; this bears similarity to categorization, but with perhaps ...
s. The broad category of branching hierarchies can be further subdivided based on the degree. A flat hierarchy (also known for companies as
flat organization A flat organization (also known as horizontal organization or flat hierarchy) has an organizational structure An organizational structure defines how activities such as task allocation Task may refer to: * Task (project management), an activity t ...
) is a branching hierarchy in which the maximum degree approaches infinity, i.e., that has a wide span. Most often, systems intuitively regarded as hierarchical have at most a moderate span. Therefore, a flat hierarchy is often not viewed as a hierarchy at all. For example,
diamond Diamond is a Allotropes of carbon, solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. At Standard conditions for temperature and pressure, room temperature and pressure, another solid form of ...

s and
graphite Graphite (), archaically referred to as plumbago, is a Crystallinity, crystalline form of the element carbon with its atoms arranged in a Hexagonal crystal system, hexagonal structure. It occurs naturally in this form and is the most stable for ...

are flat hierarchies of numerous
carbon Carbon (from la, carbo "coal") is a with the C and 6. It is lic and —making four s available to form s. It belongs to group 14 of the periodic table. Carbon makes up only about 0.025 percent of Earth's crust. Three occur naturally, ...

atoms that can be further decomposed into subatomic particles. An overlapping hierarchy is a branching hierarchy in which at least one object has two parent objects. For example, a
graduate student Student receives degree from the Monterrey Institute of Technology and Higher Education, Mexico City, 2013 A graduate school (sometimes shortened to grad school) is a school that awards advanced academic degrees (e.g., master's degree, master's ...
can have two co-supervisors to whom the student reports directly and equally, and who have the same level of authority within the
university A university () is an educational institution, institution of higher education, higher (or Tertiary education, tertiary) education and research which awards academic degrees in several Discipline (academia), academic disciplines. Universities typ ...

hierarchy (i.e., they have the same position or
tenure Tenure is a category of academic appointment existing in some countries. A tenured post is an indefinite academic appointment that can be terminated only for cause or under extraordinary circumstances, such as financial exigency or program disco ...

status).

# Etymology

Possibly the first use of the English word ''hierarchy'' cited by the ''
Oxford English Dictionary The ''Oxford English Dictionary'' (''OED'') is the principal historical dictionary A historical dictionary or dictionary on historical principles is a dictionary which deals not only with the latterday meanings of words but also the historica ...
'' was in 1881, when it was used in reference to the three orders of three angels as depicted by
Pseudo-Dionysius the Areopagite Pseudo-Dionysius the Areopagite (or Dionysius the Pseudo-Areopagite) was a Syrian Christian theologian Christian theology is the theology Theology is the systematic study of the nature of the Divinity, divine and, more broadly, of religi ...
(5th–6th centuries). Pseudo-Dionysius used the related
Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...
word (ἱεραρχία, ) both in reference to the
celestial hierarchy ''De Coelesti Hierarchia'' ( grc-gre, Περὶ τῆς Οὐρανίας Ἱεραρχίας, "On the Celestial Hierarchy") is a Pseudo-Dionysus, Pseudo-Dionysian work on angelology, written in Greek language, Greek and dated to ca. AD the 5th ce ...
and the
ecclesiastical hierarchy In Christian theology, ecclesiology is the study of the Church (congregation), Church, the origins of Christianity, its relationship to Jesus, its role in salvation, its ecclesiastical polity, polity, its Church discipline, discipline, its eschat ...
. The Greek term ''hierarchia'' means 'rule of a high priest', from (ἱεράρχης, 'president of sacred rites, high-priest') and that from ''hiereus'' (ἱερεύς, 'priest') and ''arche'' (ἀρχή, 'first place or power, rule'). Dionysius is credited with first use of it as an abstract noun. Since hierarchical churches, such as the
Roman Catholic Roman or Romans most often refers to: *Rome , established_title = Founded , established_date = 753 BC , founder = King Romulus , image_map = Map of comune of Rome (metropolitan city of Capital Rome, region Laz ...
(see
Catholic Church hierarchy The hierarchy of the Catholic Church consists of its bishop (Catholic Church), bishops, Priesthood (Catholic Church), priests, and deacons. In the Catholic ecclesiology, ecclesiological sense of the term, "hierarchy" strictly means the "holy or ...
) and
Eastern Orthodox The Eastern Orthodox Church, also called the Orthodox Church, is the second-largest Christian church, with approximately 220 million baptised members. It operates as a communion Communion may refer to: Religion * The Eucharist (also cal ...
churches, had tables of organization that were "hierarchical" in the modern sense of the word (traditionally with
God In monotheistic Monotheism is the belief A belief is an attitude Attitude may refer to: Philosophy and psychology * Attitude (psychology) In psychology Psychology is the science of mind and behavior. Psychology includes the ...
as the pinnacle or head of the hierarchy), the term came to refer to similar organizational methods in
secular Secularity, also the secular or secularness (from Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through t ...

settings.

# Representing hierarchies

A hierarchy is typically depicted as a
pyramid A pyramid (from el, πυραμίς ') is a structure A structure is an arrangement and organization of interrelated elements in a material object or system A system is a group of Interaction, interacting or interrelated elements that act ...

, where the height of a level represents that level's status and width of a level represents the quantity of items at that level relative to the whole. For example, the few Directors of a company could be at the
apex Apex may refer to: Arts and media Fictional entities * Apex (comics), a teenaged super villainess in the Marvel Universe * Ape-X, a super-intelligent ape in the Squadron Supreme universe *Apex, a genetically-engineered human population in the TV s ...
, and the
base Base or BASE may refer to: Brands and enterprises * Base (mobile telephony provider), a Belgian mobile telecommunications operator *Base CRM Base CRM (originally Future Simple or PipeJump) is an enterprise software company based in Mountain Vie ...
could be thousands of people who have no subordinates. These pyramids are often diagrammed with a
triangle A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The ...

diagram A diagram is a symbolic Depiction, representation of information using Visualization (graphics), visualization techniques. Diagrams have been used since prehistoric times on Cave painting, walls of caves, but became more prevalent during the Age ...

which serves to emphasize the size differences between the levels (but note that not all triangle/pyramid diagrams are hierarchical; for example, the 1992 USDA food guide pyramid). An example of a triangle diagram appears to the right. Another common representation of a hierarchical scheme is as a .
Phylogenetic trees In biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interactions, Physiology, physiological mechanisms, ...
,
charts A chart is a graphical representation Graphic communication as the name suggests is communication using graphic elements. These elements include symbols such as glyphs and icon (computing), icons, images such as drawings and photographs, and c ...
showing the structure of , and playoff brackets in sports are often illustrated this way. More recently, as computers have allowed the storage and navigation of ever larger data sets, various methods have been developed to represent hierarchies in a manner that makes more efficient use of the available space on a computer's screen. Examples include
fractal In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

# Visual hierarchy

In the design field, mainly graphic design, successful layouts and formatting of the content on documents are heavily dependent on the rules of
visual hierarchy Visual hierarchy, according to Gestalt psychology, is a pattern in the visual field wherein some elements tend to "stand out," or attract attention, more strongly than other elements, suggesting a hierarchy of importance. While it may occur natural ...
. Visual hierarchy is also important for proper organization of files on computers. An example of visually representing hierarchy is through nested clusters. Nested clusters represent hierarchical relationships using layers of information. The child element is within the parent element, such as in a
Venn diagram A Venn diagram is a widely used diagram A diagram is a symbolic representation Representation may refer to: Law and politics *Representation (politics) Political representation is the activity of making citizens "present" in public policy ...
. This structure is most effective in representing simple hierarchical relationships. For example, when directing someone to open a file on a computer desktop, one may first direct them towards the main folder, then the subfolders within the main folder. They will keep opening files within the folders until the designated file is located. For more complicated hierarchies, the stair structure represents hierarchical relationships through the use of visual stacking. Visually imagine the top of a downward staircase beginning at the left and descending on the right. Child elements are towards the bottom of the stairs and parent elements are at the top. This structure represents hierarchical relationships through the use of visual stacking.

# Informal representation

In plain English, a hierarchy can be thought of as a set in which: # No element is superior to itself, and # One element, the ''hierarch'', is superior to all of the other elements in the set. The first requirement is also interpreted to mean that a hierarchy can have no circular relationships; the association between two objects is always
transitive Transitivity or transitive may refer to: Grammar * Transitivity (grammar), a property of verbs that relates to whether a verb can take direct objects * Transitive verb, a verb which takes an object * Transitive case, a grammatical case to mark arg ...
. The second requirement asserts that a hierarchy must have a leader or
root In vascular plant Vascular plants (from Latin ''vasculum'': duct), also known as Tracheophyta (the tracheophytes , from Greek τραχεῖα ἀρτηρία ''trācheia artēria'' 'windpipe' + φυτά ''phutá'' 'plants'), form a large grou ...
that is common to all of the objects.

# Mathematical representation

Mathematically, in its most general form, a hierarchy is a
partially ordered set upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparable, while e.g. \ and \ are not. In mathem ...
or ''poset''. The
system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purp ...
in this case is the entire poset, which is constituted of elements. Within this system, each element shares a particular unambiguous property. Objects with the same property value are grouped together, and each of those resulting levels is referred to as a
class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differently f ...
. "Hierarchy" is particularly used to refer to a poset in which the classes are organized in terms of increasing complexity. Operations such as addition, subtraction, multiplication and division are often performed in a certain sequence or order. Usually, addition and subtraction are performed after multiplication and division has already been applied to a problem. The use of parentheses is also a representation of hierarchy, for they show which operation is to be done prior to the following ones. For example: (2 + 5) × (7 - 4). In this problem, typically one would multiply 5 by 7 first, based on the rules of mathematical hierarchy. But when the parentheses are placed, one will know to do the operations within the parentheses first before continuing on with the problem. These rules are largely dominant in algebraic problems, ones that include several steps to solve. The use of hierarchy in mathematics is beneficial to quickly and efficiently solve a problem without having to go through the process of slowly dissecting the problem. Most of these rules are now known as the proper way into solving certain equations.

# Subtypes

## Nested hierarchy

A nested hierarchy or ''inclusion hierarchy'' is a hierarchical ordering of
nested set In a naive set theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematic ...
s. The concept of nesting is exemplified in Russian
matryoshka doll Matryoshka dolls ( rus, матрёшка, p=mɐˈtrʲɵʂkə, a=Ru-матрёшка.ogg; also known as babushka dolls, stacking dolls, nesting dolls, Russian tea dolls, or Russian dolls) are a set of wooden dolls of decreasing size placed one ...
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: : $\text \subset \text \subset \text \subset \text \,$ 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 A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The ...

,
hexagon In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

, etc. Nested hierarchies are the organizational schemes behind taxonomies and systematic classifications. For example, using the original
Linnaean taxonomy Linnaean taxonomy can mean either of two related concepts: # the particular form of biological classification In biology Biology is the natural science that studies life and living organisms, including their anatomy, physical stru ...
(the version he laid out in the 10th edition of ''
Systema Naturae ' (originally in Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to be in ...
''), a human can be formulated as: : $\text \subset \text \subset \text \subset \text \subset \text$ 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 __NOTOC__ has an infinitely repeating self-similarity when it is magnified. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra ...
and
recursion Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics Linguistics is the scientific study of language, meaning tha ...

. 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 subset, strict"—no two sets can be identical. The shapes example above can be modified to demonstrate this: : $\text \subsetneq \text \subsetneq \text \subsetneq \text \,$ 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 inheritance (computer science), 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.

## 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". 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 part 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 Concepts are defined as abstract ideas A mental representation (or cognitive representation), in philosophy of mind Philosophy of mind is a branch of philosophy that studies the ontology and nature of the mind and its relationship with the bo ...

s". 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. 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 organ (anatomy), organs, which are composed of tissue (biology), tissues, which are composed of cells (biology), 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 #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 neurons 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. ''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 ''xs properties and interactions. Furthermore, the entities found at each level are ''autonomous''.

# Contexts and applications

Kulish (2002) suggests that almost every system of organization which humans apply to the world is arranged hierarchically. Some conventional definitions of the terms "nation" and "government" suggest that every nation has a government and that every government is hierarchical. Sociologists can analyse socioeconomic systems in terms of stratification into a social hierarchy (the social stratification of societies), and all systematic name, systematic classification schemes ( taxonomies) are hierarchical. Most organized religions, regardless of their internal governance structures, operate as a hierarchy under deities and priesthoods. Many Christian denominations have an autocephalous
ecclesiastical hierarchy In Christian theology, ecclesiology is the study of the Church (congregation), Church, the origins of Christianity, its relationship to Jesus, its role in salvation, its ecclesiastical polity, polity, its Church discipline, discipline, its eschat ...
of leadership. Families can be viewed as hierarchical structures in terms of cousinship (e.g., first cousin once removed, second cousin, etc.), ancestry (as depicted in a family tree) and inheritance (order of succession, succession and heirship). All the requisites of a well-rounded life and lifestyle (sociology), lifestyle can be organized using Maslow's hierarchy of needs, Maslow's hierarchy of human needs - according to Maslow's hierarchy of human needs. Learning steps often follow a hierarchical scheme—to master differential equations one must first learn calculus; to learn calculus one must first learn elementary algebra; and so on. Nature offers hierarchical structures, as numerous schemes such as
Linnaean taxonomy Linnaean taxonomy can mean either of two related concepts: # the particular form of biological classification In biology Biology is the natural science that studies life and living organisms, including their anatomy, physical stru ...
, the biological organisation, organization of life, and biomass pyramids attempt to document. Hierarchies are so infused into daily life that they are viewed as trivial. While the above examples are often clearly depicted in a hierarchical form and are classic examples, hierarchies exist in numerous systems where this branching structure is not immediately apparent. For example, most postal code , postal-code systems are hierarchical. Using the Postal codes in Canada, Canadian postal code system as an example, the top level's binding concept, the forward sortation area, "postal district", consists of 18 objects (letters). The next level down is the "zone", where the objects are the digits 0–9. This is an example of an #Degree_of_branching , overlapping hierarchy, because each of these 10 objects has 18 parents. The hierarchy continues downward to generate, in theory, 7,200,000 unique codes of the format ''A0A 0A0'' (the second and third letter positions allow 20 objects each). Most library classification systems are also hierarchical. The Dewey Decimal Classification , Dewey Decimal System is infinitely hierarchical because there is no finite bound on the number of digits can be used after the decimal point. See also Dewey Decimal Classification, Wikipedia article.

## Organizations

Organizations can be structured as a dominance hierarchy. In an organizational hierarchy, there is a single person or group with the most power (philosophy) , power or
authority In the fields of sociology Sociology is the study of society, human social behaviour, patterns of social relationships, social interaction, and culture that surrounds everyday life. It is a social science that uses various methods of Empiric ...

, and each subsequent level represents a lesser authority. Most organizations are structured in this manner, including Forms of government, governments, Company, companies, Military , armed forces, militia and organized religions. The units or persons within an organization may be depicted hierarchically in an organizational chart. In a reverse hierarchy, the conceptual pyramid (geometry) , pyramid of authority is turned upside-down, so that the apex is at the bottom and the base is at the top. This mode represents the idea that members of the higher rankings are responsible for the members of the lower rankings.

## Biology

Empirically, when we observe in nature a large proportion of the (complex) biological systems, they exhibit hierarchic structure. On theoretical grounds we could expect complex systems to be hierarchies in a world in which complexity had to evolve from simplicity. Systems theory, System hierarchies analysis performed in the 1950s, laid the empirical foundations for a Branches of science , field that would become, from the 1980s, hierarchical ecology. The theoretical foundations are summarized by thermodynamics. When biological systems are modeled as physical systems, in the most general abstraction, they are Thermodynamic system#Open system, thermodynamic open systems that exhibit self-organisation, self-organised behavior, and the Set theory, set/subset relations between dissipative structures can be characterized in a hierarchy. Other hierarchical representations related to biology include ecological pyramids which illustrate energy flow or trophic levels in ecosystems, and Taxonomy (general), taxonomic hierarchies, including the Linnean classification scheme and phylogenetic trees that reflect inferred patterns of evolutionary relationship among living and extinct species.

## Computer-graphic imaging

Computer-generated imagery, CGI and computer animation , computer-animation computer program, programs mostly use hierarchies for models. On a 3D computer graphics , 3D 3d modeling , model of a human for example, the chest is a parent of the upper left arm, which is a parent of the lower left arm, which is a parent of the hand. This pattern is used in 3D modeling, modeling and animation for almost everything built as a 3D Digital data, digital model.

## Linguistics

Many grammatical theories, such as phrase-structure grammar, involve hierarchy. Direct–inverse languages such as Cree language , Cree and Mapudungun language, Mapudungun distinguish subject and object on verbs not by different subject and object markers, but via a hierarchy of persons. In this system, the three (or four with Algonquian languages) persons occur in a hierarchy of salience (language) , salience. To distinguish which is subject and which object, ''inverse markers'' are used if the object outranks the subject. On the other hand, languages include a variety of phenomena that are not hierarchical. For example, the relationship between a pronoun and a prior noun-phrase to which it refers commonly crosses grammatical boundaries in non-hierarchical ways.

## Music

The structure of a musical composition is often understood hierarchically (for example by Heinrich Schenker (1768–1835, see Schenkerian analysis), and in the (1985) Generative theory of tonal music, Generative Theory of Tonal Music, by composer Fred Lerdahl and linguist Ray Jackendoff). The sum of all notes in a piece is understood to be an all-inclusive surface, which can be reduced to successively more sparse and more fundamental types of motion. The levels of structure that operate in Schenker's theory are the foreground, which is seen in all the details of the musical score; the middle ground, which is roughly a summary of an essential contrapuntal progression and voice-leading; and the background or Ursatz, which is one of only a few basic "long-range counterpoint" structures that are shared in the gamut of tonal music literature. The pitch (music), pitches and Musical form, form of Tonality , tonal music are organized hierarchically, all pitches deriving their importance from their relationship to a Tonic (music) , tonic Key signature , key, and secondary themes in other keys are brought back to the tonic in a recapitulation of the primary theme. Susan McClary connects this specifically in the sonata-allegro form to the feminist hierarchy of gender (see above) in her 2002 book ''Feminine Endings'', even pointing out that primary themes were often previously called "masculine" and secondary themes "feminine".

## Examples of other applications

### Information-based

* Library classification ** Dewey Decimal Classification

### City planning-based

* Green transport hierarchy * Hierarchy of roads, Roads ** Street hierarchy, Streets * Settlement hierarchy ** Settlement hierarchy#Example of a settlement hierarchy, As of 2010 ** Ekistic units, As of 2100 (estimate according to Doxiadis, 1968)

### Linguistics-oriented

* Tree model, Language family tree * Levels of adequacy, Levels of adequacy for evaluating grammars * Direct–inverse languages * Structural linguistics ** Parse tree ** Formal grammars ** Abstract syntax tree * Color terms#Basic color terms, Evolution of basic color terminology in languages

### Power- or authority-based

* Noble ranks, Aristocratic hierarchies ** In Royal and noble ranks#General chart of "translations" between languages, Europe ** In Chinese nobility#Princehood and peerage, China * Ecclesiastical hierarchy, Ecclesiastical hierarchies **
Catholic Church hierarchy The hierarchy of the Catholic Church consists of its bishop (Catholic Church), bishops, Priesthood (Catholic Church), priests, and deacons. In the Catholic ecclesiology, ecclesiological sense of the term, "hierarchy" strictly means the "holy or ...
** Priesthood (LDS Church), LDS Church hierarchy ** Kimbanguism#Hierarchy, Kimbanguist Church hierarchy ** Raëlism#Member hierarchy, Raëlism Church hierarchy ** see also autocephaly * Prussian three-class franchise * Political party hierarchies ** Ranks and insignia of the Nazi Party, Nazi Party (''pace'' overlapping fields Compare: ) *** SS Ranks#Final SS ranks 1934–1945, SS *** Glossary of Nazi Germany#G, Hierarchy of subdivisions within the Gau ** Communist Party of the Soviet Union#Structure, Communist Party of the Soviet Union ** Communist Party of China * Command hierarchy, Chain of command ** List of comparative military ranks, Military ranks ** Military organization#Hierarchy of modern armies, Military units ** Unified Combatant Command, U.S. Military Combatant Commands * Dominance hierarchy, Intraspecial dominance ** Pecking order * Social stratification, Social classes ** Caste system in India ** Hierarchical structure of Feudal Japan ** Master race#Hierarchy, White racist hierarchy ** Hierarchy of Exclusion (Ender's Game)

### Value-related

* Hierarchy of genres, Hierarchy of genres in art * Hierarchy of evidence, Evidence * Maslow's hierarchy of needs, Human needs * Hierarchy of precious substances, Precious substances * Hierarchy of values, Judicial hierarchy of social values

### Perception-based

* Color wheel ** Primary colors *** Secondary colors **** Tertiary colors

### History-oriented

* Three-age system * Comparative history, Cyclic theory of civilization ** Spengler's civilization model, Oswald Spengler ** A Study of History#List of civilizations, Arnold J. Toynbee * Spiral dynamics

### Science-focussed

* Earth's location in the universe#Earth in the universe, Hierarchy of organization within the Universe * Hierarchical ternary star system, Star systems * Biological classification * Biological organization * Phylogenetic tree * Timeline of evolution, Evolutionary development * Ecological land classification#Hierarchy of classification levels in ecology compared to other fields, Hierarchy of ecological georegions

### Technology-based

* Memory hierarchy ** Cache hierarchy * Hierarchical clustering, Clusters * Hierarchy (object-oriented programming), Class constructs * Hierarchical database model, Data organization ** Hierarchical query * Hierarchical Data Format, Data storage ** Hierarchical File System, Computer files (Macintosh) * Hierarchical control system, Devices * Classless Inter-Domain Routing, IP addresses * Memory hierarchy, Memory ** Hierarchical page tables, Virtual memory allocation * Hierarchical internetworking model, Networks * Hierarchical cell structure, Radio cells * Hierarchical state machine, States (configurations) * Hierarchical name space, Web addresses * Structure ** Data Structure * Inheritance (object-oriented programming)

### Religion-related

* Levels of consciousness **Chakra#The seven major chakras, Chakras **Great chain of being **Ray of Creation, G.I. Gurdjieff **Eight Circuit Model of Consciousness#The eight circuits, Timothy Leary * Levels of spiritual development ** In Four stages of enlightenment, Theravada Buddhism ** In Bhumi (Buddhism), Mahayana Buddhism ** In Initiation (Theosophy), Theosophy * Ages in the evolution of society ** In Astrological age#Past ages, Astrology ** In Ages of Man, Hellenism (the Ancient Greek Religion) **Dispensation (period)#Protestant dispensations, Dispensations in Protestantism **Dispensation (period)#Latter Day Saint dispensations, Dispensations in Mormonism *Hierarchical communion, Degrees of communion between various Christian churches *UFO religions **Master Jesus#Airborne Division of the Brotherhood of Light, Command hierarchy of the ''Ashtar Galactic Command'' flying saucer fleet * Deities ** In Japanese Buddhist pantheon#Hierarchical structure of the Buddhist pantheon, Japanese Buddhism ** In Spiritual Hierarchy#Levels of the spiritual hierarchy, Theosophy * Angels ** In Christian angelic hierarchy, Christianity ** In Islamic view of angels#Angel hierarchy, Islam ** In Jewish angelic hierarchy, Judaism ***Kabbalistic angelic hierarchy, Kabbalistic ** In Yazata, Zoroastrianism * Devils and Demons **Hierarchy of devils, Devils **Hierarchy of demons, Demons *Hells ** In Inferno (Dante), Catholicism (Nine Levels of Hell) ** In Naraka (Buddhism), Buddhism (Sixteen Levels of Hell) *Religious stratification, Religions in society * (organizational hierarchies are listed under )

# Criticisms

In the work of diverse theorists such as William James (1842–1910), Michel Foucault (1926–1984) and Hayden White, important critiques of hierarchical epistemology are advanced. James famously asserts in his work Radical Empiricism that clear distinctions of type and category are a constant but unwritten goal of scientific reasoning, so that when they are discovered, success is declared. But if aspects of the world are organized differently, involving inherent and intractable ambiguities, then scientific questions are often considered unresolved. Feminists, Marxists, anarchists, communists, critical theorists and others, all of whom have multiple interpretations, criticize the hierarchies commonly found within human society, especially in social relationships. Hierarchies are present in all parts of society: in businesses, schools, families, etc. These relationships are often viewed as necessary. Entities that stand in hierarchical arrangements are animals, humans, plants, etc.

## Ethics, behavioral psychology, philosophies of identity

In ethics, various virtues are enumerated and sometimes organized hierarchically according to certain brands of virtue theory. In some of these random examples, there is an asymmetry of 'compositional' significance between levels of structure, so that small parts of the whole hierarchical array depend, for their meaning, on their membership in larger parts. There is a hierarchy of activities in human life: productive activity serves or is guided by the moral life; the moral life is guided by practical reason; practical reason (used in moral and political life) serves contemplative reason (whereby we contemplate God). Practical reason sets aside time and resources for contemplative reason.

## Structure-related concepts

''(For example, in )'' * Is-a ** Hypernymy (and supertype) ** Hyponymy (and Subtyping, subtype) * Has-a ** Holonymy ** Meronymy