Hierarchy
Encyclopedia
A hierarchy 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. Abstractly, a hierarchy is simply an ordered set
or an acyclic directed graph.
A hierarchy (sometimes abbreviated HR) can link entities either directly or indirectly, and either vertically or horizontally. 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. Indirect hierarchical links can extend "vertically" upwards or downwards via multiple links in the same direction, following a path
. All parts of the hierarchy which 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-workers or colleagues; each reports to a common superior, but they have the same relative amount of authority. Organizational forms exist that are both alternative and complimentary to hierarchy. Heterarchy
(sometimes abbreviated HT) is one such form.
The generic hierarchy uses the following terms:
(N.B., while hierarchies are commonly studied using graph theory
, the general terminology used is different, and words such as "direct" may have different general meanings)
Most hierarchies use a more specific vocabulary pertaining to their subject, but the idea behind them is the same. For example, with data structure
s, objects are known as nodes
, superiors are called parents and subordinates are called children. In a business setting, a superior is a supervisor/boss
and a peer is a colleague.
of branching refers to the number of direct subordinates or children an object has (equivalent to the number of vertices
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 lineup, 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 maximum 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
s and classification scheme
s. The broad category of branching hierarchies can be further subdivided based on the degree.
A flat hierarchy is a branching hierarchy in which the maximum degree approaches infinity, i.e., with 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 at first blush. For example, diamond
s and graphite
is a flat hierarchy of numerous carbon
atoms which 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 can have two co-supervisors
to whom the student reports directly and equally, and who have the same level of authority within the university
hierarchy (i.e., they have the same position or tenure
status
was in 1880, when it was used in reference to the three orders of three angels as depicted by Pseudo-Dionysius the Areopagite
(5th–6th centuries). Pseudo-Dionysius used the related Latin
word (hierarchia) both in reference to the celestial hierarchy
and the ecclesiastical hierarchy. His term is derived from the Greek term "ἱεραρχία" meaning "rule by priests" (from "ἱεράρχης" – ierarches, meaning "president of sacred rites, high-priest" and that from "ἱερεύς" – iereus, "priest" + "ἀρχή" – arche, amongst others "first place or power, rule"), and Dionysius is credited with first use of it as an abstract noun. Since hierarchical churches, such as the Roman Catholic (see Catholic Church hierarchy
) and Eastern Orthodox churches, had tables of organization that were "hierarchical" in the modern sense of the word (traditionally with God
as the pinnacle or head of the hierarchy), the term came to refer to similar organizational methods in secular settings.
, 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
, and the base
could be thousands of people who have no subordinates.
These pyramids are typically diagrammed with a tree
or triangle
diagram
(but note that not all triangle/pyramid diagrams are hierarchical), both of which serve to emphasize the size differences between the levels. An example of a triangle diagram appears to the right. An organizational chart
is the diagram of a hierarchy within an organization
, and is depicted in tree form below.
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 maps, TreeMaps
and Radial Trees
.
The first requirement is also interpreted to mean that a hierarchy can have no circular relationships
; the association between two objects is always transitive
.
The second requirement asserts that a hierarchy must have a leader or root that is common to all of the objects.
or poset. The system 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
.
"Hierarchy" is particularly used to refer to a poset in which the classes are organized in terms of increasing complexity.
A nested hierarchy or inclusion hierarchy is a hierarchical ordering of nested sets. 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:
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:
Taxonomies may change frequently (as seen in biological taxonomy
), but the underlying concept of nested hierarchies is always the same.
The notation
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.
A subsumptive
containment hierarchy is a classification of objects from the general to the specific. Other names for this type of hierarchy are "compositional hierarchy", "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, as are all systematic naming
schemes. 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
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 conceptualization of an animal.
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 organs
, which are composed of tissues
, which are composed of 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 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 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 xs properties and interactions. Furthermore, the entities found at each level are autonomous.
has a government and every government is hierarchical. Socioeconomic systems are stratified into a social hierarchy (the social stratification
of societies), and all systematic classification schemes
(taxonomies) are hierarchical. Most organized religions, regardless of their internal governance structures, operate as a hierarchy under God
. Many Christian denomination
s have an autocephalous ecclesiastical hierarchy of leadership. Families are viewed as a hierarchical structure in terms of cousinship (e.g., first cousin once removed, second cousin, etc.), ancestry (as depicted in a family tree
) and inheritance
(succession
and heirship). All the requisites of a well-rounded life and lifestyle can be organized using Maslow's hierarchy of human needs
. Learning
must often follow a hierarchical scheme—to learn differential equation
s one must first learn calculus
; to learn calculus one must first learn elementary algebra
; and so on. Even nature
itself has its own hierarchies, as demonstrated in numerous schemes such as Linnaean taxonomy
, the organization of life, and biomass pyramids. 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, all postal code
systems are necessarily hierarchical. Using the Canadian postal code system, the top level's binding concept is the "postal district", and 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 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. Most library classification
systems are also hierarchical. The Dewey Decimal System is regarded as infinitely hierarchical because there is no finite bound on the number of digits can be used after the decimal point.
s can be structured using a hierarchy. In an organizational hierarchy, there is a single person or group with the most power and authority
, and each subsequent level represents a lesser authority. Most organizations are structured in this manner, including government
s, companies, militia
and organized religions. The units or persons within an organization are depicted hierarchically in an organizational chart
.
In a reverse hierarchy
, the conceptual pyramid
of authority is turned upside-down, so that the apex is at the bottom and the base is at the top. This model represents the idea that members of the higher rankings are responsible for the members of the lower rankings.
and computer animation
programs
is the use of hierarchies. On a 3D
model
of a human
, 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 is used in modeling
and animation
of almost everything built as a 3D digital
model.
and Mapudungun, subject and object on verb
s are distinguished not by different subject and object markers, but via a hierarchy of persons.
In this system, the three (or four with Algonquian languages
) persons are placed in a hierarchy of salience. To distinguish which is subject and which object, inverse markers are used if the object outranks the subject.
In music
, the structure of a composition is often understood hierarchically (for example by Heinrich Schenker
(1768–1835, see Schenkerian analysis
), and in the (1985) 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 pitches
and form
of tonal
music are organized hierarchically, all pitches deriving their importance from their relationship to a tonic
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 book Feminine Endings, even pointing out that primary themes were often previously called "masculine" and secondary themes "feminine."
, various virtues are enumerated and sometimes organized hierarchically according to certain brands of virtue theory.
In all 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.
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. However, feminists, Marxists, critical theorists and others analyze hierarchy in terms of the values and power that it arbitrarily assigns to one group over another.
Ordered set
In order theory in mathematics, a set with a binary relation R on its elements that is reflexive , antisymmetric and transitive is described as a partially ordered set or poset...
or an acyclic directed graph.
A hierarchy (sometimes abbreviated HR) can link entities either directly or indirectly, and either vertically or horizontally. 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. Indirect hierarchical links can extend "vertically" upwards or downwards via multiple links in the same direction, following a path
Path (graph theory)
In graph theory, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence. A path may be infinite, but a finite path always has a first vertex, called its start vertex, and a last vertex, called its end vertex. Both of them...
. All parts of the hierarchy which 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-workers or colleagues; each reports to a common superior, but they have the same relative amount of authority. Organizational forms exist that are both alternative and complimentary to hierarchy. Heterarchy
Heterarchy
A heterarchy is a system of organization replete with overlap, multiplicity, mixed ascendancy, and/or divergent-but-coexistent patterns of relation...
(sometimes abbreviated HT) is one such form.
Terminology
Hierarchies have their own special vocabulary. These terms are easiest to understand when a hierarchy is diagrammed (see below).The generic hierarchy uses the following terms:
- ObjectEntityAn entity is something that has a distinct, separate existence, although it need not be a material existence. In particular, abstractions and legal fictions are usually regarded as entities. In general, there is also no presumption that an entity is animate.An entity could be viewed as a set...
: one entity (e.g., a person, department or conceptConceptThe word concept is used in ordinary language as well as in almost all academic disciplines. Particularly in philosophy, psychology and cognitive sciences the term is much used and much discussed. WordNet defines concept: "conception, construct ". However, the meaning of the term concept is much...
) or element of arrangement or member of a set - SystemSystemSystem is a set of interacting or interdependent components forming an integrated whole....
: the entire set of objects that are being arranged hierarchically (e.g., an administration) - DimensionDimensionIn physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
: another word for "system" from on-line analytical processing (e.g. cubes) - MemberMemberA member is a person who belongs to a social group or an entity such as a company or nation. By extension it can refer to any part of a whole.Member may also refer to:* Elected official** Member of Parliament...
: an (element or object) in a (system or dimension) at any (level or rank) - RankRankingA ranking is a relationship between a set of items such that, for any two items, the first is either 'ranked higher than', 'ranked lower than' or 'ranked equal to' the second....
: the relative valueValue (ethics)In ethics, value is a property of objects, including physical objects as well as abstract objects , representing their degree of importance....
, worth, complexityComplexityIn general usage, complexity tends to be used to characterize something with many parts in intricate arrangement. The study of these complex linkages is the main goal of complex systems theory. In science there are at this time a number of approaches to characterizing complexity, many of which are...
, power, importance, authorityAuthorityThe word Authority is derived mainly from the Latin word auctoritas, meaning invention, advice, opinion, influence, or command. In English, the word 'authority' can be used to mean power given by the state or by academic knowledge of an area .-Authority in Philosophy:In...
, level etc. of an object - Level: a set of objects with the same rank OR importance
- OrderingOrder of precedenceAn order of precedence is a sequential hierarchy of nominal importance of items. Most often it is used in the context of people by many organizations and governments...
: the arrangement of the (ranks or levels) - Hierarchy: the arrangement of a particular set of (ranks or levels) i.e. multiple hierarchies are possible per (dimension or system)
- CollectionFamily of setsIn set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets...
: all of the objects at one level - SuperiorSuperior (hierarchy)In a hierarchy or tree structure of any kind, a superior is an individual or position at a higher level in the hierarchy than another , and thus closer to the apex. It is often used in business terminology to refer to people who are supervisors and in the military to people who are higher in the...
: a higher level or an object ranked at a higher level (parent or ancestor) - Subordinate: a lower level or an object ranked at a lower level (child or descendent)
- Hierarch, the top level of the hierarchy, usually consisting of one object or member of a dimension
- Peer: an object with the same rank (and therefore at the same level)
- Neighbour: the adjacent level/ranking (the immediate superior and immediate inferior)
- InteractionInterpersonal relationshipAn interpersonal relationship is an association between two or more people that may range from fleeting to enduring. This association may be based on limerence, love, solidarity, regular business interactions, or some other type of social commitment. Interpersonal relationships are formed in the...
: the relationship between an object and its direct superior or subordinate (i.e. a superior/inferior pair)- a direct interactionAdjacentAdjacent is an adjective meaning contiguous, adjoining or abuttingIn geometry, adjacent is when sides meet to make an angle.In graph theory adjacent nodes in a graph are linked by an edge....
occurs when one object is on a level exactly one higher or one lower than the other (i.e., on a treeTree (graph theory)In mathematics, more specifically graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one simple path. In other words, any connected graph without cycles is a tree...
, the two objects have a line between them)
- a direct interaction
- DistanceDistance (graph theory)In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path connecting them. This is also known as the geodesic distance...
: 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 pathPath (graph theory)In graph theory, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence. A path may be infinite, but a finite path always has a first vertex, called its start vertex, and a last vertex, called its end vertex. Both of them...
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
(N.B., while hierarchies are commonly studied using graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
, the general terminology used is different, and words such as "direct" may have different general meanings)
Most hierarchies use a more specific vocabulary pertaining to their subject, but the idea behind them is the same. For example, with data structure
Data structure
In computer science, a data structure is a particular way of storing and organizing data in a computer so that it can be used efficiently.Different kinds of data structures are suited to different kinds of applications, and some are highly specialized to specific tasks...
s, objects are known as nodes
Node (computer science)
A node is a record consisting of one or more fields that are links to other nodes, and a data field. The link and data fields are often implemented by pointers or references although it is also quite common for the data to be embedded directly in the node. Nodes are used to build linked, often...
, superiors are called parents and subordinates are called children. In a business setting, a superior is a supervisor/boss
Supervisor
A supervisor, foreperson, team leader, overseer, cell coach, facilitator, or area coordinator is a manager in a position of trust in business...
and a peer is a colleague.
Degree of branching
DegreeDegree (graph theory)
In graph theory, the degree of a vertex of a graph is the number of edges incident to the vertex, with loops counted twice. The degree of a vertex v is denoted \deg. The maximum degree of a graph G, denoted by Δ, and the minimum degree of a graph, denoted by δ, are the maximum and minimum degree...
of branching refers to the number of direct subordinates or children an object has (equivalent to the number of vertices
Vertex (graph theory)
In graph theory, a vertex or node is the fundamental unit out of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges , while a directed graph consists of a set of vertices and a set of arcs...
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 lineup, 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 maximum 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
Organization
An organization is a social group which distributes tasks for a collective goal. The word itself is derived from the Greek word organon, itself derived from the better-known word ergon - as we know `organ` - and it means a compartment for a particular job.There are a variety of legal types of...
s and classification scheme
Classification scheme
In metadata a classification scheme is a hierarchical arrangement of kinds of things or groups of kinds of things. Typically it is accompanied by descriptive information of the classes or groups. A classification scheme is intended to be used for an arrangement or division of individual objects...
s. The broad category of branching hierarchies can be further subdivided based on the degree.
A flat hierarchy is a branching hierarchy in which the maximum degree approaches infinity, i.e., with 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 at first blush. For example, diamond
Diamond
In mineralogy, diamond is an allotrope of carbon, where the carbon atoms are arranged in a variation of the face-centered cubic crystal structure called a diamond lattice. Diamond is less stable than graphite, but the conversion rate from diamond to graphite is negligible at ambient conditions...
s and graphite
Graphite
The mineral graphite is one of the allotropes of carbon. It was named by Abraham Gottlob Werner in 1789 from the Ancient Greek γράφω , "to draw/write", for its use in pencils, where it is commonly called lead . Unlike diamond , graphite is an electrical conductor, a semimetal...
is a flat hierarchy of numerous carbon
Carbon
Carbon is the chemical element with symbol C and atomic number 6. As a member of group 14 on the periodic table, it is nonmetallic and tetravalent—making four electrons available to form covalent chemical bonds...
atoms which 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 can have two co-supervisors
Research supervisor
A research supervisor is responsible for the general oversight of an academic research project.Research Associates liaise with their research supervisor on a regular basis to advise him or her on the research status and receive any guidelines.Research supervisor for post-doctoral research usually...
to whom the student reports directly and equally, and who have the same level of authority within the university
University
A university is an institution of higher education and research, which grants academic degrees in a variety of subjects. A university is an organisation that provides both undergraduate education and postgraduate education...
hierarchy (i.e., they have the same position or tenure
Tenure
Tenure commonly refers to life tenure in a job and specifically to a senior academic's contractual right not to have his or her position terminated without just cause.-19th century:...
status
History of the term
The first use of the English word "hierarchy" cited by the Oxford English DictionaryOxford English Dictionary
The Oxford English Dictionary , published by the Oxford University Press, is the self-styled premier dictionary of the English language. Two fully bound print editions of the OED have been published under its current name, in 1928 and 1989. The first edition was published in twelve volumes , and...
was in 1880, when it was used in reference to the three orders of three angels as depicted by Pseudo-Dionysius the Areopagite
Pseudo-Dionysius the Areopagite
Pseudo-Dionysius the Areopagite, also known as Pseudo-Denys, was a Christian theologian and philosopher of the late 5th to early 6th century, the author of the Corpus Areopagiticum . The author is identified as "Dionysos" in the corpus, which later incorrectly came to be attributed to Dionysius...
(5th–6th centuries). Pseudo-Dionysius used the related Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
word (hierarchia) both in reference to the celestial hierarchy
De Coelesti Hierarchia
De Coelesti Hierarchia is a Pseudo-Dionysian work on angelology which exerted great influence on scholasticism...
and the ecclesiastical hierarchy. His term is derived from the Greek term "ἱεραρχία" meaning "rule by priests" (from "ἱεράρχης" – ierarches, meaning "president of sacred rites, high-priest" and that from "ἱερεύς" – iereus, "priest" + "ἀρχή" – arche, amongst others "first place or power, rule"), and Dionysius is credited with first use of it as an abstract noun. Since hierarchical churches, such as the Roman Catholic (see Catholic Church hierarchy
Catholic Church hierarchy
The term Hierarchy in the Catholic Church has a variety of related usages. Literally, "holy government", the term is employed in different instances. There is a Hierarchy of Truths, which refers to the levels of solemnity of the official teaching of the faith...
) and Eastern Orthodox churches, had tables of organization that were "hierarchical" in the modern sense of the word (traditionally with God
God in Christianity
In Christianity, God is the eternal being that created and preserves the universe. God is believed by most Christians to be immanent , while others believe the plan of redemption show he will be immanent later...
as the pinnacle or head of the hierarchy), the term came to refer to similar organizational methods in secular settings.
Visualization
A hierarchy is typically depicted as a pyramidPyramid (geometry)
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base....
, 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
Board of directors
A board of directors is a body of elected or appointed members who jointly oversee the activities of a company or organization. Other names include board of governors, board of managers, board of regents, board of trustees, and board of visitors...
of a company could be at the apex
Apex (geometry)
In geometry, an apex is the vertex which is in some sense the highest of the figure to which it belongs.*In an isosceles triangle, the apex is the vertex where the two sides of equal length meet, opposite the unequal third side....
, and the base
Base (geometry)
In geometry, a base is a side of a plane figure or face of solid, particularly one perpendicular to the direction height is measured or on what is considered to the bottom. This usage can be applied to a triangle, parallelogram, trapezoids, cylinder, pyramid, parallelopiped or frustum. By...
could be thousands of people who have no subordinates.
These pyramids are typically diagrammed with a tree
Tree structure
A tree structure is a way of representing the hierarchical nature of a structure in a graphical form. It is named a "tree structure" because the classic representation resembles a tree, even though the chart is generally upside down compared to an actual tree, with the "root" at the top and the...
or triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
diagram
Diagram
A diagram is a two-dimensional geometric symbolic representation of information according to some visualization technique. Sometimes, the technique uses a three-dimensional visualization which is then projected onto the two-dimensional surface...
(but note that not all triangle/pyramid diagrams are hierarchical), both of which serve to emphasize the size differences between the levels. An example of a triangle diagram appears to the right. An organizational chart
Organizational chart
An organizational chart is a diagram that shows the structure of an organization and the relationships and relative ranks of its parts and positions/jobs...
is the diagram of a hierarchy within an organization
Organization
An organization is a social group which distributes tasks for a collective goal. The word itself is derived from the Greek word organon, itself derived from the better-known word ergon - as we know `organ` - and it means a compartment for a particular job.There are a variety of legal types of...
, and is depicted in tree form below.
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 maps, TreeMaps
Treemapping
In information visualization and computing, treemapping is a method for displaying hierarchical data by using nested rectangles.- Main idea :...
and Radial Trees
Radial tree
A radial tree, or "radial map", is a method of displaying a tree structure in a way that expands outwards, radially. It is one of many ways to visually display a tree., with examples extending back to the early 20th century...
.
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
Cycle (graph theory)
In graph theory, the term cycle may refer to a closed path. If repeated vertices are allowed, it is more often called a closed walk. If the path is a simple path, with no repeated vertices or edges other than the starting and ending vertices, it may also be called a simple cycle, circuit, circle,...
; the association between two objects is always transitive
Transitive relation
In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
.
The second requirement asserts that a hierarchy must have a leader or root that is common to all of the objects.
Mathematical representation
Mathematically, in its most general form, a hierarchy is a partially ordered setPartially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
or poset. The system 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 (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...
.
"Hierarchy" is particularly used to refer to a poset in which the classes are organized in terms of increasing complexity.
Nested hierarchy
Matryoshka doll
A matryoshka doll is a Russian nesting doll which is a set of wooden dolls of decreasing size placed one inside the other. The first Russian nested doll set was carved in 1890 by Vasily Zvyozdochkin from a design by Sergey Malyutin, who was a folk crafts painter at Abramtsevo...
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:
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
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
, hexagon, etc.
Nested hierarchies are the organizational schemes behind taxonomies and systematic classifications. For example, using the original Linnaean taxonomy
Linnaean taxonomy
Linnaean taxonomy can mean either of two related concepts:# the particular form of biological classification set up by Carl Linnaeus, as set forth in his Systema Naturæ and subsequent works...
(the version he laid out in the 10th edition of Systema Naturae
Systema Naturae
The book was one of the major works of the Swedish botanist, zoologist and physician Carolus Linnaeus. The first edition was published in 1735...
), a human can be formulated as:
Taxonomies may change frequently (as seen in biological taxonomy
Biological classification
Biological classification, or scientific classification in biology, is a method to group and categorize organisms by biological type, such as genus or species. Biological classification is part of scientific 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:
The notation
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
Inheritance (computer science)
In object-oriented programming , inheritance is a way to reuse code of existing objects, establish a subtype from an existing object, or both, depending upon programming language support...
in object-oriented programming
Object-oriented programming
Object-oriented programming is a programming paradigm using "objects" – data structures consisting of data fields and methods together with their interactions – to design applications and computer programs. Programming techniques may include features such as data abstraction,...
.
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
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
containment hierarchy is a classification of objects from the general to the specific. Other names for this type of hierarchy are "compositional hierarchy", "taxonomic hierarchy" and "IS-A
Is-a
In knowledge representation, object-oriented programming and design, is-a or is_a or is a is a relationship where one class D is a subclass of another class B ....
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, as are all systematic naming
Systematic name
A systematic name is a name given in a systematic way to one unique group, organism, object or chemical substance, out of a specific population or collection...
schemes. 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
Concept
The word concept is used in ordinary language as well as in almost all academic disciplines. Particularly in philosophy, psychology and cognitive sciences the term is much used and much discussed. WordNet defines concept: "conception, construct ". However, the meaning of the term concept is much...
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 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 organs
Organ (anatomy)
In biology, an organ is a collection of tissues joined in structural unit to serve a common function. Usually there is a main tissue and sporadic tissues . The main tissue is the one that is unique for the specific organ. For example, main tissue in the heart is the myocardium, while sporadic are...
, which are composed of tissues
Tissue (biology)
Tissue is a cellular organizational level intermediate between cells and a complete organism. A tissue is an ensemble of cells, not necessarily identical, but from the same origin, that together carry out a specific function. These are called tissues because of their identical functioning...
, which are composed of cells, which are composed of molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
s, which are composed of atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
s. In fact, the last two levels apply to all matter
Matter
Matter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...
, 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
Cognition
In science, cognition refers to mental processes. These processes include attention, remembering, producing and understanding language, solving problems, and making decisions. Cognition is studied in various disciplines such as psychology, philosophy, linguistics, and computer science...
is not a property of neuron
Neuron
A neuron is an electrically excitable cell that processes and transmits information by electrical and chemical signaling. Chemical signaling occurs via synapses, specialized connections with other cells. Neurons connect to each other to form networks. Neurons are the core components of the nervous...
s but is of the brain
Human brain
The human brain has the same general structure as the brains of other mammals, but is over three times larger than the brain of a typical mammal with an equivalent body size. Estimates for the number of neurons in the human brain range from 80 to 120 billion...
)—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
Causality
Causality is the relationship between an event and a second event , where the second event is understood as a consequence of the first....
. 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
Almost every system within the world is arranged hierarchically. By their common definitions, every nationNation
A nation may refer to a community of people who share a common language, culture, ethnicity, descent, and/or history. In this definition, a nation has no physical borders. However, it can also refer to people who share a common territory and government irrespective of their ethnic make-up...
has a government and every government is hierarchical. Socioeconomic systems are stratified into a social hierarchy (the social stratification
Social stratification
In sociology the social stratification is a concept of class, involving the "classification of persons into groups based on shared socio-economic conditions ... a relational set of inequalities with economic, social, political and ideological dimensions."...
of societies), and all systematic classification schemes
Systematic name
A systematic name is a name given in a systematic way to one unique group, organism, object or chemical substance, out of a specific population or collection...
(taxonomies) are hierarchical. Most organized religions, regardless of their internal governance structures, operate as a hierarchy under God
God
God is the English name given to a singular being in theistic and deistic religions who is either the sole deity in monotheism, or a single deity in polytheism....
. Many Christian denomination
Christian denomination
A Christian denomination is an identifiable religious body under a common name, structure, and doctrine within Christianity. In the Orthodox tradition, Churches are divided often along ethnic and linguistic lines, into separate churches and traditions. Technically, divisions between one group and...
s have an autocephalous ecclesiastical hierarchy of leadership. Families are viewed as a hierarchical structure in terms of cousinship (e.g., first cousin once removed, second cousin, etc.), ancestry (as depicted in a family tree
Family tree
A family tree, or pedigree chart, is a chart representing family relationships in a conventional tree structure. The more detailed family trees used in medicine, genealogy, and social work are known as genograms.-Family tree representations:...
) and inheritance
Inheritance
Inheritance is the practice of passing on property, titles, debts, rights and obligations upon the death of an individual. It has long played an important role in human societies...
(succession
Order of succession
An order of succession is a formula or algorithm that determines who inherits an office upon the death, resignation, or removal of its current occupant.-Monarchies and nobility:...
and heirship). All the requisites of a well-rounded life and lifestyle can be organized using Maslow's hierarchy of human needs
Maslow's hierarchy of needs
Maslow's hierarchy of needs is a theory in psychology, proposed by Abraham Maslow in his 1943 paper A Theory of Human Motivation. Maslow subsequently extended the idea to include his observations of humans' innate curiosity...
. Learning
Learning
Learning is acquiring new or modifying existing knowledge, behaviors, skills, values, or preferences and may involve synthesizing different types of information. The ability to learn is possessed by humans, animals and some machines. Progress over time tends to follow learning curves.Human learning...
must often follow a hierarchical scheme—to learn differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s one must first learn calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
; to learn calculus one must first learn elementary algebra
Elementary algebra
Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. It is typically taught in secondary school under the term algebra. The major difference between algebra and...
; and so on. Even nature
Nature
Nature, in the broadest sense, is equivalent to the natural world, physical world, or material world. "Nature" refers to the phenomena of the physical world, and also to life in general...
itself has its own hierarchies, as demonstrated in numerous schemes such as Linnaean taxonomy
Linnaean taxonomy
Linnaean taxonomy can mean either of two related concepts:# the particular form of biological classification set up by Carl Linnaeus, as set forth in his Systema Naturæ and subsequent works...
, the organization of life, and biomass pyramids. 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, all postal code
Postal code
A postal code is a series of letters and/or digits appended to a postal address for the purpose of sorting mail. Once postal codes were introduced, other applications became possible.In February 2005, 117 of the 190 member countries of the Universal Postal Union had postal code systems...
systems are necessarily hierarchical. Using the Canadian postal code system, the top level's binding concept is the "postal district", and 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 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. Most library classification
Library classification
A library classification is a system of coding and organizing documents or library materials according to their subject and allocating a call number to that information resource...
systems are also hierarchical. The Dewey Decimal System is regarded as infinitely hierarchical because there is no finite bound on the number of digits can be used after the decimal point.
Organizations
OrganizationOrganization
An organization is a social group which distributes tasks for a collective goal. The word itself is derived from the Greek word organon, itself derived from the better-known word ergon - as we know `organ` - and it means a compartment for a particular job.There are a variety of legal types of...
s can be structured using a hierarchy. In an organizational hierarchy, there is a single person or group with the most power and authority
Authority
The word Authority is derived mainly from the Latin word auctoritas, meaning invention, advice, opinion, influence, or command. In English, the word 'authority' can be used to mean power given by the state or by academic knowledge of an area .-Authority in Philosophy:In...
, and each subsequent level represents a lesser authority. Most organizations are structured in this manner, including government
Government
Government refers to the legislators, administrators, and arbitrators in the administrative bureaucracy who control a state at a given time, and to the system of government by which they are organized...
s, companies, militia
Militia
The term militia is commonly used today to refer to a military force composed of ordinary citizens to provide defense, emergency law enforcement, or paramilitary service, in times of emergency without being paid a regular salary or committed to a fixed term of service. It is a polyseme with...
and organized religions. The units or persons within an organization are depicted hierarchically in an organizational chart
Organizational chart
An organizational chart is a diagram that shows the structure of an organization and the relationships and relative ranks of its parts and positions/jobs...
.
In a reverse hierarchy
Reverse hierarchy
A reverse hierarchy is a conceptual organizational structure that attempts to "invert" the classical pyramid of hierarchical organisations. The concept was pioneered by the total quality management movement....
, the conceptual pyramid
Pyramid (geometry)
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base....
of authority is turned upside-down, so that the apex is at the bottom and the base is at the top. This model represents the idea that members of the higher rankings are responsible for the members of the lower rankings.
Computer graphic imaging (CGI)
Within most CGIComputer-generated imagery
Computer-generated imagery is the application of the field of computer graphics or, more specifically, 3D computer graphics to special effects in art, video games, films, television programs, commercials, simulators and simulation generally, and printed media...
and computer animation
Computer animation
Computer animation is the process used for generating animated images by using computer graphics. The more general term computer generated imagery encompasses both static scenes and dynamic images, while computer animation only refers to moving images....
programs
Computer program
A computer program is a sequence of instructions written to perform a specified task with a computer. A computer requires programs to function, typically executing the program's instructions in a central processor. The program has an executable form that the computer can use directly to execute...
is the use of hierarchies. On a 3D
3D computer graphics
3D computer graphics are graphics that use a three-dimensional representation of geometric data that is stored in the computer for the purposes of performing calculations and rendering 2D images...
model
3D modeling
In 3D computer graphics, 3D modeling is the process of developing a mathematical representation of any three-dimensional surface of object via specialized software. The product is called a 3D model...
of a human
Human
Humans are the only living species in the Homo genus...
, the chest
Chest
The chest is a part of the anatomy of humans and various other animals. It is sometimes referred to as the thorax or the bosom.-Chest anatomy - Humans and other hominids:...
is a parent
Parent
A parent is a caretaker of the offspring in their own species. In humans, a parent is of a child . Children can have one or more parents, but they must have two biological parents. Biological parents consist of the male who sired the child and the female who gave birth to the child...
of the upper left arm, which is a parent of the lower left arm, which is a parent of the hand
Hand
A hand is a prehensile, multi-fingered extremity located at the end of an arm or forelimb of primates such as humans, chimpanzees, monkeys, and lemurs...
. This is used in modeling
3D modeling
In 3D computer graphics, 3D modeling is the process of developing a mathematical representation of any three-dimensional surface of object via specialized software. The product is called a 3D model...
and animation
Animation
Animation is the rapid display of a sequence of images of 2-D or 3-D artwork or model positions in order to create an illusion of movement. The effect is an optical illusion of motion due to the phenomenon of persistence of vision, and can be created and demonstrated in several ways...
of almost everything built as a 3D digital
Digital
A digital system is a data technology that uses discrete values. By contrast, non-digital systems use a continuous range of values to represent information...
model.
Hierarchical verbal alignment
In some languages, such as CreeCree language
Cree is an Algonquian language spoken by approximately 117,000 people across Canada, from the Northwest Territories and Alberta to Labrador, making it the aboriginal language with the highest number of speakers in Canada. It is also spoken in the U.S. state of Montana...
and Mapudungun, subject and object on verb
Verb
A verb, from the Latin verbum meaning word, is a word that in syntax conveys an action , or a state of being . In the usual description of English, the basic form, with or without the particle to, is the infinitive...
s are distinguished not by different subject and object markers, but via a hierarchy of persons.
In this system, the three (or four with Algonquian languages
Algonquian languages
The Algonquian languages also Algonkian) are a subfamily of Native American languages which includes most of the languages in the Algic language family. The name of the Algonquian language family is distinguished from the orthographically similar Algonquin dialect of the Ojibwe language, which is a...
) persons are placed in a hierarchy of salience. To distinguish which is subject and which object, inverse markers are used if the object outranks the subject.
In music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
, the structure of a composition is often understood hierarchically (for example by Heinrich Schenker
Heinrich Schenker
Heinrich Schenker was a music theorist, best known for his approach to musical analysis, now usually called Schenkerian analysis....
(1768–1835, see Schenkerian analysis
Schenkerian analysis
Schenkerian analysis is a method of musical analysis of tonal music based on the theories of Heinrich Schenker. The goal of a Schenkerian analysis is to interpret the underlying structure of a tonal work. The theory's basic tenets can be viewed as a way of defining tonality in music...
), and in the (1985) Generative Theory of Tonal Music, by composer Fred Lerdahl
Fred Lerdahl
Alfred Whitford Lerdahl is the Fritz Reiner Professor of Musical Composition at Columbia University, and a composer and music theorist best known for his work on pitch space and cognitive constraints on compositional systems or "musical grammar[s]." He has written many orchestral and chamber...
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 pitches
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...
and form
Musical form
The term musical form refers to the overall structure or plan of a piece of music, and it describes the layout of a composition as divided into sections...
of tonal
Tonality
Tonality is a system of music in which specific hierarchical pitch relationships are based on a key "center", or tonic. The term tonalité originated with Alexandre-Étienne Choron and was borrowed by François-Joseph Fétis in 1840...
music are organized hierarchically, all pitches deriving their importance from their relationship to a tonic
Tonic (music)
In music, the tonic is the first scale degree of the diatonic scale and the tonal center or final resolution tone. The triad formed on the tonic note, the tonic chord, is thus the most significant chord...
key
Key signature
In musical notation, a key signature is a series of sharp or flat symbols placed on the staff, designating notes that are to be consistently played one semitone higher or lower than the equivalent natural notes unless otherwise altered with an accidental...
, and secondary themes in other keys are brought back to the tonic in a recapitulation of the primary theme. Susan McClary
Susan McClary
Susan McClary is a musicologist associated with the "New Musicology". Noted for her work combining musicology and a feminist music criticism, McClary is Professor of Musicology at Case Western Reserve University.-Biography:...
connects this specifically in the sonata-allegro form to the feminist hierarchy of gender (see above) in her book Feminine Endings, even pointing out that primary themes were often previously called "masculine" and secondary themes "feminine."
Ethics, behavioral psychology, philosophies of identity
In ethicsEthics
Ethics, also known as moral philosophy, is a branch of philosophy that addresses questions about morality—that is, concepts such as good and evil, right and wrong, virtue and vice, justice and crime, etc.Major branches of ethics include:...
, various virtues are enumerated and sometimes organized hierarchically according to certain brands of virtue theory.
In all 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.
In the work of diverse theorists such as William James
William James
William James was a pioneering American psychologist and philosopher who was trained as a physician. He wrote influential books on the young science of psychology, educational psychology, psychology of religious experience and mysticism, and on the philosophy of pragmatism...
(1842–1910), Michel Foucault
Michel Foucault
Michel Foucault , born Paul-Michel Foucault , was a French philosopher, social theorist and historian of ideas...
(1926–1984) and Hayden White
Hayden White
Hayden White is a historian in the tradition of literary criticism, perhaps most famous for his work Metahistory: The Historical Imagination in Nineteenth-Century Europe...
, 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. However, feminists, Marxists, critical theorists and others analyze hierarchy in terms of the values and power that it arbitrarily assigns to one group over another.
City planning-based
- RoadsHierarchy of roadsThe hierarchy of roads categorizes roads according to their functions and capacities. While sources differ on the exact nomenclature, the basic hierarchy comprises freeways, arterials, collectors, and local roads....
- StreetsStreet hierarchyThe street hierarchy is an urban design technique for laying out road networks that exclude automobile through-traffic from developed areas. It is conceived as a hierarchy of roads that embeds the link importance of each road type in the network topology...
- Streets
- Settlement hierarchySettlement HierarchyA settlement is a general term used in archaeology, geography, landscape history and other subjects for a permanent or temporary community in which people live, without being specific as to size, population or importance. A settlement can therefore range in size from a small number of dwellings...
- As of 2010
- As of 2100 (estimate according to Doxiadis, 1968)
Linguistics-based
- Language family treeTree modelA language tree, or family tree with languages substituted for real family members, has the form of a node-link diagram of a logical tree structure. Additional linguistics terminology derives from it. Such a diagram contains branch points, or nodes, from which the daughter languages descend by...
- Levels of adequacy for evaluating grammarsLevels of adequacyIn his seminal work Aspects of the Theory of Syntax , Noam Chomsky introduces a hierarchy of Levels of Adequacy for evaluating grammars and metagrammars ....
- Direct–inverse languages
- Structural linguisticsStructural LinguisticsStructural linguistics is an approach to linguistics originating from the work of Swiss linguist Ferdinand de Saussure. De Saussure's Course in General Linguistics, published posthumously in 1916, stressed examining language as a static system of interconnected units...
- Parse treeParse treeA concrete syntax tree or parse tree or parsing treeis an ordered, rooted tree that represents the syntactic structure of a string according to some formal grammar. In a parse tree, the interior nodes are labeled by non-terminals of the grammar, while the leaf nodes are labeled by terminals of the...
- Formal grammarFormal grammarA formal grammar is a set of formation rules for strings in a formal language. The rules describe how to form strings from the language's alphabet that are valid according to the language's syntax...
s - Abstract syntax treeAbstract syntax treeIn computer science, an abstract syntax tree , or just syntax tree, is a tree representation of the abstract syntactic structure of source code written in a programming language. Each node of the tree denotes a construct occurring in the source code. The syntax is 'abstract' in the sense that it...
- Parse tree
- Evolution of basic color terminology in languages
Power- or authority-based
- Aristocratic hierarchies
- In Europe
- In China
- Ecclesiastical hierarchies
- Catholic Church hierarchyCatholic Church hierarchyThe term Hierarchy in the Catholic Church has a variety of related usages. Literally, "holy government", the term is employed in different instances. There is a Hierarchy of Truths, which refers to the levels of solemnity of the official teaching of the faith...
- LDS Church hierarchyPriesthood (LDS Church)In The Church of Jesus Christ of Latter-day Saints , the priesthood is the power and authority to act in the name of God for the salvation of humankind...
- Kimbanguist Church hierarchy
- Raelian Church hierarchy
- see also autocephalyAutocephalyAutocephaly , in hierarchical Christian churches and especially Eastern Orthodox and Oriental Orthodox churches, is the status of a hierarchical church whose head bishop does not report to any higher-ranking bishop...
- Catholic Church hierarchy
- Administrative branch of government geographical hierarchies
- Administrative divisions of China
- Administrative divisions of India
- Administrative divisions of the United States
- Administrative divisions of Russia
- Political party hierarchies
- Nazi PartyRanks and insignia of the Nazi PartyRanks and insignia of the Nazi Party were paramilitary titles used by the National Socialist German Workers Party between approximately 1928 and the fall of Nazi Germany in 1945...
- SS
- Hierarchy of subdivisions within the Gau
- Communist Party of the Soviet Union
- Communist Party of ChinaCommunist Party of ChinaThe Communist Party of China , also known as the Chinese Communist Party , is the founding and ruling political party of the People's Republic of China...
- Nazi Party
- Chain of commandCommand hierarchyA command hierarchy is a group of people committed to carrying out orders "from the top", that is, of authority. It is part of a power structure: usually seen as the most vulnerable and also the most powerful part of it.-Sociology:...
- Military ranks
- Military units
- U.S. Military Combatant CommandsUnified Combatant CommandA Unified Combatant Command is a United States Department of Defense command that is composed of forces from at least two Military Departments and has a broad and continuing mission. These commands are established to provide effective command and control of U.S. military forces, regardless of...
- Intraspecial dominanceDominance hierarchyA dominance hierarchy is the organization of individuals in a group that occurs when competition for resources leads to aggression...
- Pecking orderPecking orderPecking order or just peck order is the colloquial term for a hierarchical system of social organization in chickens. It was first described from the behaviour of poultry by Thorleif Schjelderup-Ebbe in 1921 under the German terms Hackordnung or Hackliste' ...
- Pecking order
- Social classesSocial stratificationIn sociology the social stratification is a concept of class, involving the "classification of persons into groups based on shared socio-economic conditions ... a relational set of inequalities with economic, social, political and ideological dimensions."...
- Caste system in IndiaCaste system in IndiaThe Indian caste system is a system of social stratification and social restriction in India in which communities are defined by thousands of endogamous hereditary groups called Jātis....
- Hierarchical structure of Feudal Japan
- White racist hierarchy
- Hierarchy of Exclusion (Ender's Game)
- Caste system in India
Value-based
- Art genresHierarchy of genresA hierarchy of genres is any formalization which ranks different genres in an art form in terms of their prestige and cultural value....
- EvidenceHierarchy of evidenceEvidence hierarchies reflect the relative authority of various types of biomedical research. Although there is no single, universally-accepted hierarchy of evidence, there is broad agreement on the relative strength of the principal types of research...
- Human needsMaslow's hierarchy of needsMaslow's hierarchy of needs is a theory in psychology, proposed by Abraham Maslow in his 1943 paper A Theory of Human Motivation. Maslow subsequently extended the idea to include his observations of humans' innate curiosity...
- Precious substancesHierarchy of precious substancesIn popular culture, sets of precious substances may form hierarchies which express conventional perceived relative value or merit. Precious metals appear prominently in such hierarchies, but as they grow, gems and semi-precious materials may be introduced as part of the system...
- Judicial hierarchy of social valuesHierarchy of valuesHierarchy of values is a concept in US legal analysis that Yale Law School Professor Myres MacDougal popularized. It refers to an ordered list of social values that influence judicial decision-making. Different jurists or legal analysts may order values in different hierarchies, which leads them to...
Perception-based
- Color wheelColor wheelA color wheel or color circle is an abstract illustrative organization of color hues around a circle that shows relationships between primary colors, secondary colors, complementary colors, etc....
- Primary colorsPrimary ColorsPrimary Colors: A Novel of Politics is a roman à clef, a work of fiction that purports to describe real life characters and events — namely, Bill Clinton's first presidential campaign in 1992...
- Secondary colors
- Tertiary colors
- Secondary colors
- Primary colors
History-based
- Three-age systemThree-age systemThe three-age system in archaeology and physical anthropology is the periodization of human prehistory into three consecutive time periods, named for their respective tool-making technologies:* The Stone Age* The Bronze Age* The Iron Age-Origin:...
- Cyclic theory of civilizationComparative historyComparative history is the comparison of different societies which existed during the same time period or shared similar cultural conditions. The comparative history of societies emerged as an important specialty among intellectuals in the Enlightenment in the 18th century, as typified by...
- Oswald SpenglerSpengler's civilization modelOswald Spengler's civilization model appears as three tables, each in a three-page long folded sheet, inserted between pages 68 and 69 of the first volume of his Der Untergang des Abendlandes, in the definitive edition published under the author's care by C. H. Beck in Munich, in 1931...
- Arnold J. Toynbee
- Oswald Spengler
- Spiral dynamicsSpiral dynamicsSpiral Dynamics is a theory of human development introduced in the 1996 book Spiral Dynamics by Don Beck and Chris Cowan. The book was based on the theory of psychology professor Clare W. Graves...
Science-based
- Hierarchy of organization within the Omniverse
- Hierarchy of organization within the Multiverse
- Hierarchy of organization within the Universe
- Orders of magnitude
- Energy qualityEnergy qualityEnergy quality is the contrast between different forms of energy, the different trophic levels in ecological systems and the propensity of energy to convert from one form to another. The concept refers to the empirical experience of the characteristics, or qualia, of different energy forms as they...
- Scientific knowledgeHierarchy of KnowledgeThe Hierarchy of knowledge is the idea that the basic fields of the sciences and mathematics can be organized linearly from its most basic pure form to its most derivative manifestation, at least as a convenience of discussion....
- Star systems
- Biological classificationBiological classificationBiological classification, or scientific classification in biology, is a method to group and categorize organisms by biological type, such as genus or species. Biological classification is part of scientific taxonomy....
- Biological organization
- Phylogenetic treePhylogenetic treeA phylogenetic tree or evolutionary tree is a branching diagram or "tree" showing the inferred evolutionary relationships among various biological species or other entities based upon similarities and differences in their physical and/or genetic characteristics...
- Evolutionary developmentTimeline of evolutionThis timeline of evolution of life outlines the major events in the development of life on planet Earth since it first originated until the present day. In biology, evolution is any change across successive generations in the heritable characteristics of biological populations...
- Hierarchy of ecological georegions
Technology-based
- ClustersHierarchical clusteringIn statistics, hierarchical clustering is a method of cluster analysis which seeks to build a hierarchy of clusters. Strategies for hierarchical clustering generally fall into two types:...
- Class constructs
- Data organization
- Hierarchical queryHierarchical queryA hierarchical query is a type of SQL query that handles hierarchical model data.Standard SQL specifies hierarchical queries by way of recursive common table expressions...
- Hierarchical query
- Data storageHierarchical Data FormatHierarchical Data Format is the name of a set of file formats and libraries designed to store and organize large amounts of numerical data...
- Computer files (Macintosh)Hierarchical File SystemHierarchical File System is a file system developed by Apple Inc. for use in computer systems running Mac OS. Originally designed for use on floppy and hard disks, it can also be found on read-only media such as CD-ROMs...
- Computer files (Macintosh)
- DevicesHierarchical control systemA Hierarchical control system is a form of Control System in which a set of devices and governing software is arranged in a hierarchical tree. When the links in the tree are implemented by a computer network, then that hierarchical control system is also a form of Networked control system.-...
- IP addressesClassless Inter-Domain RoutingClassless Inter-Domain Routing is a method for allocating IP addresses and routing Internet Protocol packets. The Internet Engineering Task Force introduced CIDR in 1993 to replace the previous addressing architecture of classful network design in the Internet...
- MemoryMemory hierarchyThe term memory hierarchy is used in the theory of computation when discussing performance issues in computer architectural design, algorithm predictions, and the lower level programming constructs such as involving locality of reference. A 'memory hierarchy' in computer storage distinguishes each...
- Virtual memory allocation
- NetworksHierarchical internetworking modelThe Hierarchical internetworking model, or three-layer model, is a network design model first proposed by Cisco. Thethree-layer model divides enterprise networks into three layers: core, distribution, and access layer. Each layer provides different...
- Radio cells
- States (configurations)
- Web addressesHierarchical name spaceThe DNS Hierarchical Name Space is a map of how DNS servers determine what IP address to connect to given a URL-Registrars:Registrars, such as VeriSign, the operator of the .com and .net top-level domains, let any company, organization, or private individual register a TLD for any number of years...
- StructureStructureStructure is a fundamental, tangible or intangible notion referring to the recognition, observation, nature, and permanence of patterns and relationships of entities. This notion may itself be an object, such as a built structure, or an attribute, such as the structure of society...
- Data StructureData structureIn computer science, a data structure is a particular way of storing and organizing data in a computer so that it can be used efficiently.Different kinds of data structures are suited to different kinds of applications, and some are highly specialized to specific tasks...
- Data Structure
Religion- and mythology-based
- Levels of consciousness
- Chakras
- Great chain of beingGreat chain of beingThe great chain of being , is a Christian concept detailing a strict, religious hierarchical structure of all matter and life, believed to have been decreed by the Christian God.-Divisions:...
- G.I. GurdjieffRay of CreationThe Ray of Creation is an esoteric cosmology which was taught by G. I. Gurdjieff. It is a diagram which better represents the place which Earth occupies in the Universe...
- Timothy Leary
- Levels of spiritual development
- In Therevada BuddhismFour stages of enlightenmentThe four stages of enlightenment in Buddhism are the four progressive stages culminating in full enlightenment as an Arahat, which an average, instructed person can attain in this life...
- In Mahayana BuddhismBhumi (Buddhism)The bodhisattva's path of awakening in the Mahayana tradition progresses through ten hierarchically arranged stages, referred to as the "bodhisattva bhūmis"...
- In TheosophyInitiation (Theosophy)Initiation is a concept in Theosophy that there are nine levels of spiritual development that beings who live on Earth can progress upward through. Within these levels, there are four basic levels of spiritual development that human beings on Earth progress through as they reincarnate, although...
- In Therevada Buddhism
- Ages in the evolution of society
- In Astrology
- In Hellenism (the Ancient Greek Religion)Ages of ManThe Ages of Man are the stages of human existence on the Earth according to Greek mythology. Two classical authors in particular offer accounts of the successive ages of mankind, which tend to progress from an original, long-gone age in which humans enjoyed a nearly divine existence to the current...
- Dispensations in Protestantism
- Dispensations in Mormonism
- Degrees of communion between various Christian churches
- UFO religionUFO religionUFO religion is an informal term used to describe a religion that equates UFO occupants with gods or other semi-divine beings. Typically, the UFO occupants are held to be extraterrestials and that humanity either currently is, or eventually will become, part of a preexisting extraterrestrial...
s- Command hierarchy of the Ashtar Galactic Command flying saucer fleet
- Deities
- In Japanese Buddhism
- In Theosophy
- Angels
- In Christianity
- In JudaismJewish angelic hierarchyMaimonides, in his Yad ha-Chazakah: Yesodei ha-Torah, counts ten ranks of angels in the Jewish angelic hierarchy, beginning from the highest:- Kabbalistic angelic hierarchy :...
- Kabbalistic
- In Islam
- In ZoroastrianismYazataYazata is the Avestan language word for a Zoroastrian concept. The word has a wide range of meanings but generally signifies a divinity...
- Devils and Demons
- Devils
- Demons
- HellHellIn many religious traditions, a hell is a place of suffering and punishment in the afterlife. Religions with a linear divine history often depict hells as endless. Religions with a cyclic history often depict a hell as an intermediary period between incarnations...
s- In Catholicism (Nine Levels of Hell)Inferno (Dante)Inferno is the first part of Dante Alighieri's 14th-century epic poem Divine Comedy. It is followed by Purgatorio and Paradiso. It is an allegory telling of the journey of Dante through what is largely the medieval concept of Hell, guided by the Roman poet Virgil. In the poem, Hell is depicted as...
- In Buddhism (Sixteen Levels of Hell)Naraka (Buddhism)Naraka नरक or Niraya निरय is the name given to one of the worlds of greatest suffering in Buddhist cosmology.Naraka is usually translated into English as "hell", "hell realm", or "purgatory"...
- In Catholicism (Nine Levels of Hell)
- Religions in societyReligious stratificationReligious stratification is the division of a society into hierarchical layers on the premise of religious beliefs, affiliation, or faith practices.According to Kingsley Davis and Wilbert E...
- (organizational hierarchies are listed under )
Methods using the hierarchical model
See also
External links
- Principles and annotated bibliography of hierarchy theory
- Summary of the Principles of Hierarchy Theory — S.N. Salthe
- Everything is Hierarchical - Think in a different way.