A hierarchy (from
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
: , 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
architecture
Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and constructing buildings ...
,
philosophy
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. ...
,
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, product, or process. The verb ''to design' ...
,
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
,
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
,
organizational theory
Organizational theory refers to the set of interrelated concepts that involve the sociological study of the structures and operations of formal social organizations. Organizational theory also attempts to explain how interrelated units of organiz ...
,
systems theory
Systems theory is the interdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or human-made. Every system has causal boundaries, is influenced by its context, defined by its structu ...
,
systematic biology
Biological systematics is the study of the diversification of living forms, both past and present, and the relationships among living things through time. Relationships are visualized as evolutionary trees (synonyms: cladograms, phylogenetic tr ...
, and the
social sciences
Social science is one of the branches of science, devoted to the study of societies and the relationships among individuals within those societies. The term was formerly used to refer to the field of sociology, the original "science of so ...
(especially
political philosophy
Political philosophy or political theory is the philosophical study of government, addressing questions about the nature, scope, and legitimacy of public agents and institutions and the relationships between them. Its topics include politics, ...
).
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
subordinate
A hierarchy (from Greek: , 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 ...
s, 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 a relationship between two parties regulating the provision of paid labour services. Usually based on a contract, one party, the employer, which might be a corporation, a not-for-profit organization, a co-operative, or any o ...
s or
colleague
Collegiality is the relationship between colleagues. A colleague is a fellow member of the same profession.
Colleagues are those explicitly united in a common purpose and respect each other's abilities to work toward that purpose. A colleague is ...
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) or where they possess the potential to be ranked a number of different ways. Definitions of the term vary among the disciplines: in soci ...
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
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
).
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
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ...
: one entity (e.g., a person, department or
concept
Concepts are defined as abstract ideas. They are understood to be the fundamental building blocks of the concept behind principles, thoughts and beliefs.
They play an important role in all aspects of cognition. As such, concepts are studied by ...
or element of arrangement or member of a set)
*
System
A system is a group of 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 purpose and express ...
: the entire set of objects that are being arranged hierarchically (e.g., an administration)
*
Dimension
In physics and mathematics, the dimension of a mathematical 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 (1D) because only one coord ...
: another word for "system" from on-line analytical processing (e.g. cubes)
*
Member
Member may refer to:
* Military jury, referred to as "Members" in military jargon
* Element (mathematics), an object that belongs to a mathematical set
* In object-oriented programming, a member of a class
** Field (computer science), entries in ...
: 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, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
: the relative
value
Value or values may refer to:
Ethics and social
* Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them
** Values (Western philosophy) expands the notion of value beyo ...
, worth,
complexity
Complexity characterises the behaviour of a system or model whose components interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and emergence.
The term is generally used to ch ...
,
power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may a ...
, importance,
authority
In the fields of sociology and political science, authority is the legitimate power of a person or group over other people. In a civil state, ''authority'' is practiced in ways such a judicial branch or an executive branch of government.''T ...
, 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
*Spirit level, an instrument designed to indicate whether a surface is horizontal or vertical
* Canal pound or level
*Reg ...
or Tier: a set of objects with the same rank OR importance
**
Ordering
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
: 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 ''ordinarius'') is an officer of a church or civic authority who by reason of office has ordinary power to execute laws.
Such officers are found in hierarchically organised churches of Western Christianity which have an ...
, the apex of the hierarchy, consisting of one single orphan (object or member) in the top level of a dimension. The root of an
inverted-tree structure
**
Member
Member may refer to:
* Military jury, referred to as "Members" in military jargon
* Element (mathematics), an object that belongs to a mathematical set
* In object-oriented programming, a member of a class
** Field (computer science), entries in ...
, 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.
In common usage, only a child who has lost both parents due to death is called an orphan. When referring to animals, only the mother's condition is usuall ...
, 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 ( : leaves) is any of the principal appendages of a vascular plant stem, usually borne laterally aboveground and specialized for photosynthesis. Leaves are collectively called foliage, as in "autumn foliage", while the leaves, ...
, 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
* The subject of the Great Commandment
Films
* ''Neighbors'' (1920 film), a short film starring Buster Keaton
* ''Neighbours'' (1952 film), a ...
: a member adjacent to another member in the same (level or rank). Always a peer.
**
Superior
Superior may refer to:
*Superior (hierarchy), something which is higher in a hierarchical structure of any kind
Places
*Superior (proposed U.S. state), an unsuccessful proposal for the Upper Peninsula of Michigan to form a separate state
*Lake ...
: a higher level or an object ranked at a higher level (A parent or an ancestor)
**
Subordinate
A hierarchy (from Greek: , 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 ...
: a lower level or an object ranked at a lower level (A child or a descendant)
**
Collection: 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; related to the term "peer of the realm"
Computing
* Peer, one of several functional units in the same layer of a ne ...
: an object with the same rank (and therefore at the same level)
**
Interaction
Interaction is action that occurs between two or more objects, with broad use in philosophy and the sciences. It may refer to:
Science
* Interaction hypothesis, a theory of second language acquisition
* Interaction (statistics)
* Interaction ...
: 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 stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...
, the two objects have a line between them)
**
Distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
: 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
Span may refer to:
Science, technology and engineering
* Span (unit), the width of a human hand
* Span (engineering), a section between two intermediate supports
* Wingspan, the distance between the wingtips of a bird or aircraft
* Sorbitan ester ...
: 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, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
), 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, a data structure is a data organization, management, and storage format that is usually chosen for efficient access to data. More precisely, a data structure is a collection of data values, the relationships among them, ...
s, objects are known as
nodes, superiors are called
parents and subordinates are called
children
A child ( : children) is a human being between the stages of birth and puberty, or between the developmental period of infancy and puberty. The legal definition of ''child'' generally refers to a minor, otherwise known as a person younger ...
. In a business setting, a superior is a
supervisor/boss and a peer is a
colleague
Collegiality is the relationship between colleagues. A colleague is a fellow member of the same profession.
Colleagues are those explicitly united in a common purpose and respect each other's abilities to work toward that purpose. A colleague is ...
.
Degree of branching
Degree of
branching refers to the number of direct
subordinates 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; see spelling differences), is an entity—such as a company, an institution, or an association—comprising one or more people and having a particular purpose.
The word is derived f ...
s and
classification scheme
In information science and 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 a more theoretical bent, as cla ...
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) 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 solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Another solid form of carbon known as graphite is the chemically stable form of carbon at room temperature and pressure, b ...
s and
graphite
Graphite () is a crystalline form of the element carbon. It consists of stacked layers of graphene. Graphite occurs naturally and is the most stable form of carbon under standard conditions. Synthetic and natural graphite are consumed on la ...
are flat hierarchies of numerous
carbon
Carbon () is a chemical element with the symbol C and atomic number 6. It is nonmetallic and tetravalent—its atom making four electrons available to form covalent chemical bonds. It belongs to group 14 of the periodic table. Carbon ma ...
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
Postgraduate or graduate education refers to academic or professional degrees, certificates, diplomas, or other qualifications pursued by post-secondary students who have earned an undergraduate (bachelor's) degree.
The organization and 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 institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United Stat ...
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 first and foundational historical dictionary of the English language, published by Oxford University Press (OUP). It traces the historical development of the English language, providing a c ...
'' 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 Greek author, Christian theologian and Neoplatonic philosopher of the late 5th to early 6th century, who wrote a set of works known as the ''Corpus Areopagiticum'' ...
(5th–6th centuries). Pseudo-Dionysius used the related
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
word (ἱεραρχία, ) both in reference to the
celestial hierarchy and the
ecclesiastical hierarchy
The hierarchy of the Catholic Church consists of its bishops, priests, and deacons. In the ecclesiological sense of the term, "hierarchy" strictly means the "holy ordering" of the Church, the Body of Christ, so to respect the diversity of gif ...
. 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, the capital city of Italy
*Ancient Rome, Roman civilization from 8th century BC to 5th century AD
* Roman people, the people of ancient Rome
*'' Epistle to the Romans'', shortened to ''Romans'', a let ...
(see
Catholic Church hierarchy
The hierarchy of the Catholic Church consists of its bishops, priests, and deacons. In the ecclesiological sense of the term, "hierarchy" strictly means the "holy ordering" of the Church, the Body of Christ, so to respect the diversity of gifts ...
) and
Eastern Orthodox
Eastern Orthodoxy, also known as Eastern Orthodox Christianity, is one of the three main branches of Chalcedonian Christianity, alongside Catholicism and Protestantism.
Like the Pentarchy of the first millennium, the mainstream (or " canonical ...
churches, had tables of organization that were "hierarchical" in the modern sense of the word (traditionally with
God
In monotheistic thought, God is usually viewed as the supreme being, creator, and principal object of faith. Swinburne, R.G. "God" in Honderich, Ted. (ed)''The Oxford Companion to Philosophy'', Oxford University Press, 1995. God is typically ...
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 ''saeculum'', "worldly" or "of a generation"), is the state of being unrelated or neutral in regards to religion. Anything that does not have an explicit reference to religion, either negativ ...
settings.
Representing hierarchies
![Maslow's hierarchy of needs](https://upload.wikimedia.org/wikipedia/commons/c/c3/Maslow%27s_hierarchy_of_needs.png)
A hierarchy is typically depicted as a
pyramid
A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilate ...
, 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
Director may refer to:
Literature
* ''Director'' (magazine), a British magazine
* ''The Director'' (novel), a 1971 novel by Henry Denker
* ''The Director'' (play), a 2000 play by Nancy Hasty
Music
* Director (band), an Irish rock band
* ''D ...
of a company could be at the
apex
The apex is the highest point of something. The word may also 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, ...
, and the
base could be thousands of people who have no subordinates.
These pyramids are often diagrammed with a
triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
diagram
A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three ...
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
tree diagram.
Phylogenetic trees
A phylogenetic tree (also phylogeny or evolutionary tree Felsenstein J. (2004). ''Inferring Phylogenies'' Sinauer Associates: Sunderland, MA.) is a branching diagram or a tree showing the evolutionary relationships among various biological spec ...
,
charts
A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent tabu ...
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, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as ill ...
maps,
TreeMaps and
Radial Trees.
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 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 style that shows the logical relation between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationship ...
. 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
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
in which:
# No element is superior to itself, and
# One element, the (''apex'' or ''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.
The second requirement asserts that a hierarchy must have a leader or
root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
that is common to all of the objects.
Mathematical representation
Mathematically, in its most general form, a hierarchy is a
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) 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 ...
or ''poset''.
The
system
A system is a group of 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 purpose and express ...
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 ...
.
"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
![Russian-Matroshka no bg](https://upload.wikimedia.org/wikipedia/commons/d/d2/Russian-Matroshka_no_bg.jpg)
A nested hierarchy or ''inclusion hierarchy'' is a hierarchical ordering of
nested set
In a naive set theory, a nested set is a set containing a chain of subsets, forming a hierarchical structure, like Russian dolls.
It is used as reference-concept in all scientific hierarchy definitions, and many technical approaches, like the ...
s.
The concept of nesting is exemplified in Russian
matryoshka doll
Matryoshka dolls ( ; rus, матрёшка, p=mɐˈtrʲɵʂkə, a=Ru-матрёшка.ogg), also known as stacking dolls, nesting dolls, Russian tea dolls, or Russian dolls, are a set of wooden dolls of decreasing size placed one inside ano ...
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
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
,
hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A '' regular hexagon'' has ...
, 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 (taxonomy) set up by Carl Linnaeus, as set forth in his ''Systema Naturae'' (1735) and subsequent works. In the taxonomy of Linnaeus t ...
(the version he laid out in the 10th edition of ''
Systema Naturae
' (originally in Latin written ' with the ligature æ) is one of the major works of the Swedish botanist, zoologist and physician Carl Linnaeus (1707–1778) and introduced the Linnaean taxonomy. Although the system, now known as binomial ...
''), a human can be formulated as:
:
Taxonomies may change frequently (as seen in
biological taxonomy
In biology, taxonomy () is the scientific study of naming, defining ( circumscribing) and classifying groups of biological organisms based on shared characteristics. Organisms are grouped into taxa (singular: taxon) and these groups are given ...
), 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__
In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...
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 to logic. The most common application of recursion is in mathematic ...
. 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
In mathematical writing, the term strict refers to the property of excluding equality and equivalence and often occurs in the context of inequality and monotonic functions. It is often attached to a technical term to indicate that the exclusive ...
"—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 in
object-oriented programming
Object-oriented programming (OOP) is a programming paradigm based on the concept of "objects", which can contain data and code. The data is in the form of fields (often known as attributes or ''properties''), and the code is in the form of ...
.
Two types of containment hierarchies are the ''subsumptive'' containment hierarchy and the ''compositional'' containment hierarchy. A subsumptive hierarchy "
subsumes" its children, and a compositional hierarchy is "
composed" of its children. A hierarchy can also be both subsumptive ''and'' compositional.
Subsumptive containment hierarchy
A ''
subsumptive'' containment hierarchy is a classification of object classes from the general to the specific. Other names for this type of hierarchy are "taxonomic hierarchy" and "
IS-A
In knowledge representation, object-oriented programming and design (see object-oriented program architecture), is-a (is_a or is a) is a subsumption relationship between abstractions (e.g. types, classes), wherein one class ''A'' is a subclass ...
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. They are understood to be the fundamental building blocks of the concept behind principles, thoughts and beliefs.
They play an important role in all aspects of cognition. As such, concepts are studied by ...
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
In information science a conceptualization is an abstract simplified view of some selected part of the world, containing the objects, concepts, and other entities that are presumed of interest for some particular purpose and the relationships betw ...
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 system
An organ system is a biological system consisting of a group of organs that work together to perform one or more functions. Each organ has a specialized role in a plant or animal body, and is made up of distinct tissues.
Plants
Plants have ...
s, which are composed of
organs, which are composed of
tissues, which are composed of
cells
Cell most often refers to:
* Cell (biology), the functional basic unit of life
Cell may also refer to:
Locations
* Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
, which are composed of
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and b ...
s, which are composed of
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, a ...
s. In fact, the last two levels apply to all
matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic part ...
, at least at the
macroscopic scale
The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopic.
Overview
When applied to physical phenomena a ...
. Moreover, each of these levels inherit all the properties of their
children
A child ( : children) is a human being between the stages of birth and puberty, or between the developmental period of infancy and puberty. The legal definition of ''child'' generally refers to a minor, otherwise known as a person younger ...
.
In this particular example, there are also ''
emergent properties''—functions that are not seen at the lower level (e.g.,
cognition
Cognition refers to "the mental action or process of acquiring knowledge and understanding through thought, experience, and the senses". It encompasses all aspects of intellectual functions and processes such as: perception, attention, though ...
is not a property of
neuron
A neuron, neurone, or nerve cell is an electrically excitable cell that communicates with other cells via specialized connections called synapses. The neuron is the main component of nervous tissue in all animals except sponges and placozoa ...
s but is of the
brain
A brain is an organ (biology), organ that serves as the center of the nervous system in all vertebrate and most invertebrate animals. It is located in the head, usually close to the sensory organs for senses such as Visual perception, vision. I ...
)—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 In philosophy, downward causation is a causal relationship from higher levels of a system to lower-level parts of that system: for example, mental events acting to cause physical events. The term was originally coined in 1974 by the philosopher and ...
'' 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
In developmental psychology and moral, political, and bioethical philosophy, autonomy, from , ''autonomos'', from αὐτο- ''auto-'' "self" and νόμος ''nomos'', "law", hence when combined understood to mean "one who gives oneself one's ow ...
''.
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
A nation is a community of people formed on the basis of a combination of shared features such as language, history, ethnicity, culture and/or society. A nation is thus the collective identity of a group of people understood as defined by th ...
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
Social stratification refers to a society's categorization of its people into groups based on socioeconomic factors like wealth, income, race, education, ethnicity, gender, occupation, social status, or derived power (social and politi ...
of societies), and all
systematic classification schemes (
taxonomies) are hierarchical. Most
organized religion
Organized religion, also known as institutional religion, is religion in which belief systems and rituals are systematically arranged and formally established. Organized religion is typically characterized by an official doctrine (or dogma), ...
s, regardless of their internal governance structures, operate as a hierarchy under
deities
A deity or god is a supernatural being who is considered divine or sacred. The ''Oxford Dictionary of English'' defines deity as a god or goddess, or anything revered as divine. C. Scott Littleton defines a deity as "a being with powers greater ...
and
priesthoods. Many
Christian denomination
A Christian denomination is a distinct Religion, religious body within Christianity that comprises all Church (congregation), church congregations of the same kind, identifiable by traits such as a name, particular history, organization, leadersh ...
s have an
autocephalous
Autocephaly (; from el, αὐτοκεφαλία, meaning "property of being self-headed") is the status of a hierarchical Christian church whose head bishop does not report to any higher-ranking bishop. The term is primarily used in Eastern O ...
ecclesiastical hierarchy
The hierarchy of the Catholic Church consists of its bishops, priests, and deacons. In the ecclesiological sense of the term, "hierarchy" strictly means the "holy ordering" of the Church, the Body of Christ, so to respect the diversity of gif ...
of
leadership
Leadership, both as a research area and as a practical skill, encompasses the ability of an individual, group or organization to "lead", influence or guide other individuals, teams, or entire organizations. The word "leadership" often gets v ...
. Families can be viewed as hierarchical structures in terms of
cousinship
Most generally, in the lineal kinship system used in the English-speaking world, a cousin is a type of familial relationship in which two relatives are two or more familial generations away from their most recent common ancestor. Commonly, ...
(e.g., first cousin once removed, second cousin, etc.),
ancestry
An ancestor, also known as a forefather, fore-elder or a forebear, is a parent or ( recursively) the parent of an antecedent (i.e., a grandparent, great-grandparent, great-great-grandparent and so forth). ''Ancestor'' is "any person from w ...
(as depicted in a
family tree
A family tree, also called a genealogy or a pedigree chart, is a chart representing family relationships in a conventional tree structure. More detailed family trees, used in medicine and social work, are known as genograms.
Representations ...
) and
inheritance
Inheritance is the practice of receiving private property, titles, debts, entitlements, privileges, rights, and obligations upon the death of an individual. The rules of inheritance differ among societies and have changed over time. Of ...
(
succession
Succession is the act or process of following in order or sequence.
Governance and politics
*Order of succession, in politics, the ascension to power by one ruler, official, or monarch after the death, resignation, or removal from office of ...
and
heir
Inheritance is the practice of receiving private property, titles, debts, entitlements, privileges, rights, and obligations upon the death of an individual. The rules of inheritance differ among societies and have changed over time. Offic ...
ship). All the requisites of a well-rounded life and
lifestyle
Lifestyle often refers to:
* Lifestyle (sociology), the way a person lives
* ''Otium'', ancient Roman concept of a lifestyle
* Style of life (german: Lebensstil, link=no), dealing with the dynamics of personality
Lifestyle may also refer to:
Bu ...
can be organized using
Maslow's hierarchy of human needs - according to Maslow's hierarchy of human needs.
Learning
Learning is the process of acquiring new understanding, knowledge, behaviors, skills, values, attitudes, and preferences. The ability to learn is possessed by humans, animals, and some machines; there is also evidence for some kind of lea ...
steps often follow a hierarchical scheme—to master
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s one must first learn
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
; to learn calculus one must first learn
elementary algebra
Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values).
This use of variables entail ...
; and so on.
Nature
Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are ...
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 (taxonomy) set up by Carl Linnaeus, as set forth in his ''Systema Naturae'' (1735) and subsequent works. In the taxonomy of Linnaeus t ...
, the
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 systems are hierarchical. Using the
Canadian postal code system as an example, the top level's binding concept, the
"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
overlapping hierarchy
A hierarchy (from Greek: , 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 ...
, 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
A library classification is a system of organization of knowledge by which library resources are arranged and ordered systematically. Library classifications are a notational system that represents the order of topics in the classification and al ...
systems are also hierarchical. The
Dewey Decimal System
The Dewey Decimal Classification (DDC), colloquially known as the Dewey Decimal System, is a proprietary library classification system which allows new books to be added to a library in their appropriate location based on subject.
Section 4.1 ...
is infinitely hierarchical because there is no finite bound on the number of digits can be used after the decimal point.
[ See also Wikipedia article.]
Organizations
Organization
An organization or organisation (Commonwealth English; see spelling differences), is an entity—such as a company, an institution, or an association—comprising one or more people and having a particular purpose.
The word is derived f ...
s can be structured as a
dominance hierarchy
In biology, a dominance hierarchy (formerly and colloquially called a pecking order) is a type of social hierarchy that arises when members of animal social groups interact, creating a ranking system. A dominant higher-ranking individual is so ...
. In an organizational hierarchy, there is a single person or group with the most
power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may a ...
or
authority
In the fields of sociology and political science, authority is the legitimate power of a person or group over other people. In a civil state, ''authority'' is practiced in ways such a judicial branch or an executive branch of government.''T ...
, and each subsequent level represents a lesser authority. Most organizations are structured in this manner, including
governments
A government is the system or group of people governing an organized community, generally a state.
In the case of its broad associative definition, government normally consists of legislature, executive, and judiciary. Government is a ...
,
companies
A company, abbreviated as co., is a legal entity representing an association of people, whether natural, legal or a mixture of both, with a specific objective. Company members share a common purpose and unite to achieve specific, declared go ...
,
armed forces
A military, also known collectively as armed forces, is a heavily armed, highly organized force primarily intended for warfare. It is typically authorized and maintained by a sovereign state, with its members identifiable by their distinct ...
,
militia
A militia () is generally an army or some other fighting organization of non-professional soldiers, citizens of a country, or subjects of a state, who may perform military service during a time of need, as opposed to a professional force of r ...
and
organized religion
Organized religion, also known as institutional religion, is religion in which belief systems and rituals are systematically arranged and formally established. Organized religion is typically characterized by an official doctrine (or dogma), ...
s. The units or persons within an organization may be depicted hierarchically in an
organizational chart
An organizational chart, also called organigram, organogram, or organizational breakdown structure (OBS) is a diagram that shows the structure of an organization and the relationships and relative ranks of its parts and positions/jobs. The term ...
.
In a
reverse hierarchy, the conceptual
pyramid
A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilate ...
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.
System
A system is a group of 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 purpose and express ...
hierarchies analysis performed in the 1950s, laid the empirical foundations for a
field that would become, from the 1980s, hierarchical ecology.
The theoretical foundations are summarized by
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...
.
When
biological systems
A biological system is a complex network which connects several biologically relevant entities. Biological organization spans several scales and are determined based different structures depending on what the system is. Examples of biological syst ...
are modeled as
physical system
A physical system is a collection of physical objects.
In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment. The environment is ignored except for its effects on the ...
s, in the most general abstraction, they are
thermodynamic open systems that exhibit
self-organised behavior, and the
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
An ecosystem (or ecological system) consists of all the organisms and the physical environment with which they interact. These biotic and abiotic components are linked together through nutrient cycles and energy flows. Energy enters the syst ...
, and
taxonomic hierarchies, including the
Linnean classification
Linnaean taxonomy can mean either of two related concepts:
# The particular form of biological classification (taxonomy) set up by Carl Linnaeus, as set forth in his '' Systema Naturae'' (1735) and subsequent works. In the taxonomy of Linnaeus ...
scheme and
phylogenetic trees
A phylogenetic tree (also phylogeny or evolutionary tree Felsenstein J. (2004). ''Inferring Phylogenies'' Sinauer Associates: Sunderland, MA.) is a branching diagram or a tree showing the evolutionary relationships among various biological spec ...
that reflect inferred patterns of evolutionary relationship among living and extinct species.
Computer-graphic imaging
CGI and
computer-animation programs
Program, programme, programmer, or programming may refer to:
Business and management
* Program management, the process of managing several related projects
* Time management
* Program, a part of planning
Arts and entertainment Audio
* Progra ...
mostly use hierarchies for models. On a
3D model
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models c ...
of a
human
Humans (''Homo sapiens'') are the most abundant and widespread species of primate, characterized by bipedalism and exceptional cognitive skills due to a large and complex brain. This has enabled the development of advanced tools, cultu ...
for example, the
chest
The thorax or chest is a part of the anatomy of humans, mammals, and other tetrapod animals located between the neck and the abdomen. In insects, crustaceans, and the extinct trilobites, the thorax is one of the three main divisions of the crea ...
is a
parent
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 t ...
of the upper left arm, which is a parent of the lower left arm, which is a parent of the
hand
A hand is a prehensile, multi-fingered appendage located at the end of the forearm or forelimb of primates such as humans, chimpanzees, monkeys, and lemurs. A few other vertebrates such as the koala (which has two opposable thumbs on each "h ...
. This pattern is used in
modeling and
animation
Animation is a method by which still figures are manipulated to appear as moving images. In traditional animation, images are drawn or painted by hand on transparent celluloid sheets to be photographed and exhibited on film. Today, most ani ...
for almost everything built as a 3D
digital
Digital usually refers to something using discrete digits, often binary digits.
Technology and computing Hardware
*Digital electronics, electronic circuits which operate using digital signals
** Digital camera, which captures and stores digital ...
model.
Linguistics
Many grammatical theories, such as
phrase-structure grammar, involve hierarchy.
Direct–inverse languages such as
Cree
The Cree ( cr, néhinaw, script=Latn, , etc.; french: link=no, Cri) are a North American Indigenous people. They live primarily in Canada, where they form one of the country's largest First Nations.
In Canada, over 350,000 people are Cree o ...
and
Mapudungun
Mapuche (, Mapuche & Spanish: , or Mapudungun; from ' 'land' and ' 'speak, speech') is an Araucanian language related to Huilliche spoken in south-central Chile and west-central Argentina by the Mapuche people (from ''mapu'' 'land' and ''che ...
distinguish subject and object on
verb
A verb () is a word ( part of speech) that in syntax generally conveys an action (''bring'', ''read'', ''walk'', ''run'', ''learn''), an occurrence (''happen'', ''become''), or a state of being (''be'', ''exist'', ''stand''). In the usual descr ...
s not by different subject and object markers, but via a hierarchy of persons.
In this system, the three (or four with
Algonquian languages
The Algonquian languages ( or ; also Algonkian) are a subfamily of indigenous American languages that include most languages in the Algic language family. The name of the Algonquian language family is distinguished from the orthographically simi ...
) persons occur in a hierarchy of
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 Schenkerian analysis is a method of analyzing tonal music based on the theories of Heinrich Schenker (1868–1935). The goal is to demonstrate the organic coherence of the work by showing how it relates to an abstracted deep structure, the ''Urs ...
), and in the (1985)
Generative Theory of Tonal Music, by composer
Fred Lerdahl
Alfred Whitford (Fred) Lerdahl (born March 10, 1943, in Madison, Wisconsin) is the Fritz Reiner Professor Emeritus of Musical Composition at Columbia University, and a composer and music theorist best known for his work on musical grammar an ...
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
In Schenkerian analysis, the fundamental structure (german: Ursatz) describes the structure of a tonal work as it occurs at the most remote (or " background") level and in the most abstract form. A basic elaboration of the tonic triad, it consi ...
, 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
Form is the shape, visual appearance, or configuration of an object. In a wider sense, the form is the way something happens.
Form also refers to:
*Form (document), a document (printed or electronic) with spaces in which to write or enter data
* ...
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.
Examples of other applications
Information-based
*
Library classification
A library classification is a system of organization of knowledge by which library resources are arranged and ordered systematically. Library classifications are a notational system that represents the order of topics in the classification and al ...
**
Dewey Decimal Classification
The Dewey Decimal Classification (DDC), colloquially known as the Dewey Decimal System, is a proprietary library classification system which allows new books to be added to a library in their appropriate location based on subject.
Section 4.1 ...
City planning-based
*
Green transport hierarchy
*
Roads
A road is a linear way for the conveyance of traffic that mostly has an improved surface for use by vehicles (motorized and non-motorized) and pedestrians. Unlike streets, the main function of roads is transportation.
There are many types of ...
**
Streets
Streets is the plural of street, a type of road.
Streets or The Streets may also refer to: Music
* Streets (band), a rock band fronted by Kansas vocalist Steve Walsh
* ''Streets'' (punk album), a 1977 compilation album of various early UK punk ba ...
*
Settlement hierarchy
**
As of 2010
**
As of 2100 (estimate according to Doxiadis, 1968)
Linguistics-oriented
*
Language family tree
*
Levels of adequacy for evaluating grammars
*
Direct–inverse languages
*
Structural linguistics
Structural linguistics, or structuralism, in linguistics, denotes schools or theories in which language is conceived as a self-contained, self-regulating semiotic system whose elements are defined by their relationship to other elements within t ...
**
Parse tree
A parse tree or parsing tree or derivation tree or concrete syntax tree is an ordered, rooted tree that represents the syntactic structure of a string according to some context-free grammar. The term ''parse tree'' itself is used primarily in co ...
**
Formal grammar
In formal language theory, a grammar (when the context is not given, often called a formal grammar for clarity) describes how to form strings from a language's alphabet that are valid according to the language's syntax. A grammar does not describe ...
s
**
Abstract syntax tree
In computer science, an abstract syntax tree (AST), or just syntax tree, is a tree representation of the abstract syntactic structure of text (often source code) written in a formal language. Each node of the tree denotes a construct occurr ...
*
Evolution of basic color terminology in languages
Power- or authority-based
*
Aristocratic hierarchies
** In
Europe
Europe is a large peninsula conventionally considered a continent in its own right because of its great physical size and the weight of its history and traditions. Europe is also considered a Continent#Subcontinents, subcontinent of Eurasia ...
** In
China
China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's List of countries and dependencies by population, most populous country, with a Population of China, population exceeding 1.4 billion, slig ...
*
Ecclesiastical hierarchies
**
Catholic Church hierarchy
The hierarchy of the Catholic Church consists of its bishops, priests, and deacons. In the ecclesiological sense of the term, "hierarchy" strictly means the "holy ordering" of the Church, the Body of Christ, so to respect the diversity of gifts ...
**
LDS Church hierarchy
**
Kimbanguist Church hierarchy
**
Raëlism Church hierarchy
** see also
autocephaly
Autocephaly (; from el, αὐτοκεφαλία, meaning "property of being self-headed") is the status of a hierarchical Christian church whose head bishop does not report to any higher-ranking bishop. The term is primarily used in Eastern Or ...
*
Prussian three-class franchise
The Prussian three-class franchise (German: ''Preußisches Dreiklassenwahlrecht'') was an indirect electoral system used from 1848 until 1918 in the Kingdom of Prussia and for shorter periods in other German states. Voters were grouped by distric ...
* Political party hierarchies
**
Nazi Party
The Nazi Party, officially the National Socialist German Workers' Party (german: Nationalsozialistische Deutsche Arbeiterpartei or NSDAP), was a far-right politics, far-right political party in Germany active between 1920 and 1945 that crea ...
(''pace'' overlapping fields
[
Compare:
])
***
SS
***
Hierarchy of subdivisions within the Gau
**
Communist Party of the Soviet Union
"Hymn of the Bolshevik Party"
, headquarters = 4 Staraya Square, Moscow
, general_secretary = Vladimir Lenin (first) Mikhail Gorbachev (last)
, founded =
, banned =
, founder = Vladimir Lenin
, newspaper ...
**
Chinese Communist Party
The Chinese Communist Party (CCP), officially the Communist Party of China (CPC), is the founding and One-party state, sole ruling party of the China, People's Republic of China (PRC). Under the leadership of Mao Zedong, the CCP emerged victoriou ...
*
Chain of command
A command hierarchy is a group of people who carry out orders based on others' authority within the group. It can be viewed as part of a power structure, in which it is usually seen as the most vulnerable and also the most powerful part.
Milit ...
**
Military ranks
Military ranks are a system of hierarchy, hierarchical relationships, within armed forces, police, intelligence agencies or other institutions organized along military lines. The military rank system defines dominance, authority, and responsibi ...
**
Military units
Military organization or military organisation is the structuring of the armed forces of a state so as to offer such military capability as a national defense policy may require. In some countries paramilitary forces are included in a nation' ...
**
U.S. Military Combatant Commands
A unified combatant command (CCMD), also referred to as a combatant command, is a joint military command of the United States Department of Defense that is composed of units from two or more service branches of the United States Armed Forces, an ...
*
Intraspecial dominance
**
Pecking order
In biology, a dominance hierarchy (formerly and colloquially called a pecking order) is a type of social hierarchy that arises when members of animal social animal , social groups interact, creating a ranking system. A dominant higher-ranking i ...
*
Social classes
A social class is a grouping of people into a set of hierarchical social categories, the most common being the upper, middle and lower classes. Membership in a social class can for example be dependent on education, wealth, occupation, inco ...
**
Caste system in India
The caste system in India is the paradigmatic ethnographic example of classification of castes. It has its origins in Outline of ancient India, ancient India, and was transformed by various ruling elites in medieval, early-modern, and modern ...
**
Hierarchical structure of Feudal Japan
Edo society refers to the society of Japan under the rule of the Tokugawa Shogunate during the Edo period from 1603 to 1868.
Edo society was a feudal society with strict social stratification, customs, and regulations intended to promote p ...
**
White racist hierarchy
**
Hierarchy of Exclusion
The ''Ender's Game'' series (often referred to as the ''Ender'' saga and also the Enderverse) is a series of science fiction books written by American author Orson Scott Card. The series started with the novelette ''Ender's Game'', which was la ...
(Ender's Game)
Value-related
*
Hierarchy of genres in art
*
Evidence
Evidence for a proposition is what supports this proposition. It is usually understood as an indication that the supported proposition is true. What role evidence plays and how it is conceived varies from field to field.
In epistemology, evidenc ...
*
Human needs
*
Precious substances
*
Judicial hierarchy of social values
Perception-based
*
Color wheel
A color wheel or color circle is an abstract illustrative organization of color hues around a circle, which shows the relationships between primary colors, secondary colors, tertiary colors etc.
Some sources use the terms ''color wheel'' & ' ...
**
Primary colors
A set of primary colors or primary colours (see spelling differences) consists of colorants or colored lights that can be mixed in varying amounts to produce a gamut of colors. This is the essential method used to create the perception of a bro ...
***
Secondary colors
****
Tertiary colors
A tertiary color or intermediate color is a color made by mixing full saturation of one primary color with half saturation of another primary color and none of a third primary color, in a given color space such as RGB, CMYK (more modern) or RYB ...
History-oriented
*
Three-age system
The three-age system is the periodization of human pre-history (with some overlap into the historical periods in a few regions) into three time-periods: the Stone Age, the Bronze Age, and the Iron Age; although the concept may also refer to o ...
*
Cyclic theory of civilization
**
Oswald Spengler
Oswald Arnold Gottfried Spengler (; 29 May 1880 – 8 May 1936) was a German historian and philosopher of history whose interests included mathematics, science, and art, as well as their relation to his organic theory of history. He is best known ...
**
Arnold J. Toynbee
Arnold Joseph Toynbee (; 14 April 1889 – 22 October 1975) was an English historian, a philosopher of history, an author of numerous books and a research professor of international history at the London School of Economics and King's Colleg ...
*
Spiral dynamics
Science-focussed
*
Hierarchy of organization within the Universe
*
Star systems
*
Biological classification
In biology, taxonomy () is the scientific study of naming, defining ( circumscribing) and classifying groups of biological organisms based on shared characteristics. Organisms are grouped into taxa (singular: taxon) and these groups are given ...
*
Biological organization
Biological organisation is the hierarchy of complex biological structures and systems that define life using a reductionistic approach. The traditional hierarchy, as detailed below, extends from atoms to biospheres. The higher levels of this s ...
*
Phylogenetic tree
A phylogenetic tree (also phylogeny or evolutionary tree Felsenstein J. (2004). ''Inferring Phylogenies'' Sinauer Associates: Sunderland, MA.) is a branching diagram or a tree showing the evolutionary relationships among various biological spec ...
*
Evolutionary development
*
Hierarchy of ecological georegions
Technology-based
*
Memory hierarchy
In computer architecture, the memory hierarchy separates computer storage into a hierarchy based on response time. Since response time, complexity, and capacity are related, the levels may also be distinguished by their performance and controlli ...
**
Cache hierarchy
Cache hierarchy, or multi-level caches, refers to a memory architecture that uses a hierarchy of memory stores based on varying access speeds to cache data. Highly requested data is cached in high-speed access memory stores, allowing swifter access ...
*
Clusters
*
Class constructs
*
Data organization
**
Hierarchical query
A hierarchical query is a type of SQL query that handles hierarchical model data. They are special cases of more general recursive fixpoint queries, which compute transitive closures.
In standard SQL:1999 hierarchical queries are implemented b ...
*
Data storage
Data storage is the recording (storing) of information (data) in a storage medium. Handwriting, phonographic recording, magnetic tape, and optical discs are all examples of storage media. Biological molecules such as RNA and DNA are conside ...
**
Computer files (Macintosh)
*
Devices
*
IP addresses
An Internet Protocol address (IP address) is a numerical label such as that is connected to a computer network that uses the Internet Protocol for communication.. Updated by . An IP address serves two main functions: network interface ident ...
*
Memory
Memory is the faculty of the mind by which data or information is encoded, stored, and retrieved when needed. It is the retention of information over time for the purpose of influencing future action. If past events could not be remembered, ...
**
Virtual memory allocation
*
Networks
Network, networking and networked may refer to:
Science and technology
* Network theory, the study of graphs as a representation of relations between discrete objects
* Network science, an academic field that studies complex networks
Mathematics
...
*
Radio cells
*
States (configurations)
*
Web addresses
*
Structure
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
**
Data Structure
In computer science, a data structure is a data organization, management, and storage format that is usually chosen for efficient access to data. More precisely, a data structure is a collection of data values, the relationships among them, a ...
*
Inheritance (object-oriented programming)
In object-oriented programming, inheritance is the mechanism of basing an object or class upon another object ( prototype-based inheritance) or class ( class-based inheritance), retaining similar implementation. Also defined as deriving new classe ...
Religion-related
* Levels of consciousness
**
Chakras
Chakras (, ; sa , text=चक्र , translit=cakra , translit-std=IAST , lit=wheel, circle; pi, cakka) are various focal points used in a variety of ancient meditation practices, collectively denominated as Tantra, or the esoteric or ...
**
Great chain of being
The great chain of being is a hierarchical structure of all matter and life, thought by medieval Christianity to have been decreed by God. The chain begins with God and descends through angels, humans, animals and plants to minerals.
The great ...
**
G.I. Gurdjieff
George Ivanovich Gurdjieff (; rus, Гео́ргий Ива́нович Гурджи́ев, r=Geórgy Ivánovich Gurdzhíev, p=ɡʲɪˈorɡʲɪj ɪˈvanəvʲɪd͡ʑ ɡʊrd͡ʐˈʐɨ(j)ɪf; hy, Գեորգի Իվանովիչ Գյուրջիև; c. 1 ...
**
Timothy Leary
Timothy Francis Leary (October 22, 1920 – May 31, 1996) was an American psychologist and author known for his strong advocacy of psychedelic drugs. Evaluations of Leary are polarized, ranging from bold oracle to publicity hound. He was "a her ...
* Levels of spiritual development
** In
Theravada Buddhism
''Theravāda'' () ( si, ථේරවාදය, my, ထေရဝါဒ, th, เถรวาท, km, ថេរវាទ, lo, ເຖຣະວາດ, pi, , ) is the most commonly accepted name of Buddhism's oldest existing school. The school' ...
** In
Mahayana Buddhism
''Mahāyāna'' (; "Great Vehicle") is a term for a broad group of Buddhist traditions, texts, philosophies, and practices. Mahāyāna Buddhism developed in India (c. 1st century BCE onwards) and is considered one of the three main existing bra ...
** In
Theosophy
Theosophy is a religion established in the United States during the late 19th century. It was founded primarily by the Russian Helena Blavatsky and draws its teachings predominantly from Blavatsky's writings. Categorized by scholars of religion a ...
* Ages in the evolution of society
** In
Astrology
Astrology is a range of Divination, divinatory practices, recognized as pseudoscientific since the 18th century, that claim to discern information about human affairs and terrestrial events by studying the apparent positions of Celestial o ...
** In
Hellenism (the Ancient Greek Religion)
**
Dispensations in Protestantism
**
Dispensations in Mormonism
*
Degrees of communion between various Christian churches
*
UFO religion
A UFO religion is any religion in which the existence of extraterrestrial (ET) entities operating unidentified flying objects (UFOs) is an element of belief. Typically, adherents of such religions believe the ETs to be interested in the welfar ...
s
**
Command hierarchy of the ''Ashtar Galactic Command'' flying saucer fleet
* Deities
** In
Japanese Buddhism
Buddhism has been practiced in Japan since about the 6th century CE. Japanese Buddhism () created many new Buddhist schools, and some schools are original to Japan and some are derived from Chinese Buddhist schools. Japanese Buddhism has had a ...
** In
Theosophy
Theosophy is a religion established in the United States during the late 19th century. It was founded primarily by the Russian Helena Blavatsky and draws its teachings predominantly from Blavatsky's writings. Categorized by scholars of religion a ...
* Angels
** In
Christianity
Christianity is an Abrahamic monotheistic religion based on the life and teachings of Jesus of Nazareth. It is the world's largest and most widespread religion with roughly 2.38 billion followers representing one-third of the global pop ...
** In
Islam
Islam (; ar, ۘالِإسلَام, , ) is an Abrahamic religions, Abrahamic Monotheism#Islam, monotheistic religion centred primarily around the Quran, a religious text considered by Muslims to be the direct word of God in Islam, God (or ...
** In
Judaism
Judaism ( he, ''Yahăḏūṯ'') is an Abrahamic, monotheistic, and ethnic religion comprising the collective religious, cultural, and legal tradition and civilization of the Jewish people. It has its roots as an organized religion in the ...
***
Kabbalistic
Kabbalah ( he, קַבָּלָה ''Qabbālā'', literally "reception, tradition") is an esoteric method, discipline and Jewish theology, school of thought in Jewish mysticism. A traditional Kabbalist is called a Mekubbal ( ''Məqūbbāl'' "rece ...
** In
Zoroastrianism
Zoroastrianism is an Iranian religions, Iranian religion and one of the world's History of religion, oldest organized faiths, based on the teachings of the Iranian peoples, Iranian-speaking prophet Zoroaster. It has a Dualism in cosmology, du ...
* Devils and Demons
**
Devils
A devil is the personification of evil as it is conceived in many and various cultures and religious traditions.
Devil or Devils may also refer to:
* Satan
* Devil in Christianity
* Demon
* Folk devil
Art, entertainment, and media
Film an ...
**
Demons
A demon is a malevolent supernatural entity. Historically, belief in demons, or stories about demons, occurs in religion, occultism, literature, fiction, mythology, and folklore; as well as in media such as comics, video games, movies, anime, ...
*
Hell
In religion and folklore, hell is a location in the afterlife in which evil souls are subjected to punitive suffering, most often through torture, as eternal punishment after death. Religions with a linear divine history often depict hell ...
s
** In
Catholicism (Nine Levels of Hell)
** In
Buddhism (Sixteen Levels of Hell)
*
Religions in society
* (organizational hierarchies are listed under )
Methods using hierarchy
Criticisms
In the work of diverse theorists such as
William James
William James (January 11, 1842 – August 26, 1910) was an American philosopher, historian, and psychologist, and the first educator to offer a psychology course in the United States.
James is considered to be a leading thinker of the lat ...
(1842 to 1910),
Michel Foucault
Paul-Michel Foucault (, ; ; 15 October 192625 June 1984) was a French philosopher, historian of ideas, writer, political activist, and literary critic. Foucault's theories primarily address the relationship between power and knowledge, and how ...
(1926 to 1984) and
Hayden White
Hayden V. White (July 12, 1928 – March 5, 2018) was an American historian in the tradition of literary criticism, perhaps most famous for his work '' Metahistory: The Historical Imagination in Nineteenth-Century Europe'' (1973/2014).
Career
W ...
(1928 to 2018), important critiques of hierarchical
epistemology
Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics.
Episte ...
are advanced. James famously asserts in his work
Radical Empiricism
Radical empiricism is a philosophical doctrine put forth by William James. It asserts that experience includes both particulars and relations between those particulars, and that therefore both deserve a place in our explanations. In concrete terms: ...
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
Feminism is a range of socio-political movements and ideologies that aim to define and establish the political, economic, personal, and social equality of the sexes. Feminism incorporates the position that society prioritizes the male poi ...
,
Marxists
Marxism is a left-wing to far-left method of socioeconomic analysis that uses a materialist interpretation of historical development, better known as historical materialism, to understand class relations and social conflict and a dialectic ...
,
anarchists
Anarchism is a political philosophy and movement that is skeptical of all justifications for authority and seeks to abolish the institutions it claims maintain unnecessary coercion and hierarchy, typically including, though not necessari ...
,
communists
Communism (from Latin la, communis, lit=common, universal, label=none) is a far-left sociopolitical, philosophical, and economic ideology and current within the socialist movement whose goal is the establishment of a communist society, a so ...
,
critical theorists
A critical theory is any approach to social philosophy that focuses on society and culture to reveal, critique and challenge power structures. With roots in sociology and literary criticism, it argues that social problems stem more from socia ...
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
![Hierarchy Of Purposes](https://upload.wikimedia.org/wikipedia/commons/0/0e/Hierarchy_Of_Purposes.jpg)
In
ethics
Ethics or moral philosophy is a branch of philosophy that "involves systematizing, defending, and recommending concepts of right and wrong behavior".''Internet Encyclopedia of Philosophy'' The field of ethics, along with aesthetics, concerns m ...
, various
virtues
Virtue ( la, virtus) is moral excellence. A virtue is a trait or quality that is deemed to be morally good and thus is valued as a foundation of principle and good moral being. In other words, it is a behavior that shows high moral standard ...
are enumerated and sometimes organized hierarchically according to certain brands of
virtue theory
Virtue ethics (also aretaic ethics, from Greek ἀρετή arete_(moral_virtue).html"_;"title="'arete_(moral_virtue)">aretḗ''_is_an_approach_to_ethics_that_treats_the_concept_of_virtue.html" ;"title="arete_(moral_virtue)">aretḗ''.html" ;" ...
.
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.
See also
Structure-related concepts
''(For example, in )''
*
Is-a
In knowledge representation, object-oriented programming and design (see object-oriented program architecture), is-a (is_a or is a) is a subsumption relationship between abstractions (e.g. types, classes), wherein one class ''A'' is a subclass ...
**
Hypernymy
In linguistics, semantics, general semantics, and ontologies, hyponymy () is a semantic relation between a hyponym denoting a subtype and a hypernym or hyperonym (sometimes called umbrella term or blanket term) denoting a supertype. In other wor ...
(and
supertype)
**
Hyponymy
In linguistics, semantics, general semantics, and ontologies, hyponymy () is a semantic relation between a hyponym denoting a subtype and a hypernym or hyperonym (sometimes called umbrella term or blanket term) denoting a supertype. In other wor ...
(and
subtype
Subtype may refer to:
* Viral subtypes, such as Subtypes of HIV
* Subtyping
In programming language theory, subtyping (also subtype polymorphism or inclusion polymorphism) is a form of type polymorphism in which a subtype is a datatype that is ...
)
*
Has-a
In database design, object-oriented programming and design (see object oriented program architecture), has-a (has_a or has a) is a composition relationship where one object (often called the constituted object, or part/constituent/member object) ...
**
Holonymy
In linguistics, meronymy () is a semantic relation between a meronym denoting a part and a holonym denoting a whole. In simpler terms, a meronym is in a ''part-of'' relationship with its holonym. For example, ''finger'' is a meronym of ''hand' ...
**
Meronymy
In linguistics, meronymy () is a semantic relation between a meronym denoting a part and a holonym denoting a whole. In simpler terms, a meronym is in a ''part-of'' relationship with its holonym. For example, ''finger'' is a meronym of ''hand' ...
Footnotes
Further reading
*
*
*
*
*
*
*
*
*
** Also includes full copies of:
**
**
External links
*
Principles and annotated bibliography of hierarchy theorySummary of the Principles of Hierarchy Theory— S.N. Salthe
{{Authority control
Patterns
Structure
Political culture