About: Hierarchy (mathematics)

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In mathematics, a hierarchy is a set-theoretical object, consisting of a preorder defined on a set. This is often referred to as an ordered set, though that is an ambiguous term that many authors reserve for partially ordered sets or totally ordered sets. The term pre-ordered set is unambiguous, and is always synonymous with a mathematical hierarchy. The term hierarchy is used to stress a hierarchical relation among the elements.

Attributes	Values
rdfs:label	Hierarchy (mathematics) (en)
rdfs:comment	In mathematics, a hierarchy is a set-theoretical object, consisting of a preorder defined on a set. This is often referred to as an ordered set, though that is an ambiguous term that many authors reserve for partially ordered sets or totally ordered sets. The term pre-ordered set is unambiguous, and is always synonymous with a mathematical hierarchy. The term hierarchy is used to stress a hierarchical relation among the elements. (en)
sameAs	Hierarchy (mathematics)
dbp:wikiPageUsesTemplate	dbt:Citation_needed dbt:Div_col dbt:Div_col_end dbt:More_citations_needed dbt:Other_uses dbt:Math-stub
Subject	Hierarchy
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Hierarchy_(mathematics)?oldid=1052143142&ns=0
Wikipage page ID	3612259 (xsd:integer)
page length (characters) of wiki page	3459 (xsd:nonNegativeInteger)
Wikipage revision ID	1052143142 (xsd:integer)
Link from a Wikipage to another Wikipage	Analytical hierarchy Arithmetical hierarchy Borel hierarchy Monoid Category (mathematics) Category of sets Hierarchy Difference hierarchy Leibniz operator Set (mathematics) Set theory Chomsky hierarchy Class (computer programming) Decision problem Hyperarithmetical theory Preorder Time hierarchy theorem Complex network Complexity Computer science Hierarchy Order theory Partially ordered set Wadge hierarchy Tree (data structure) Tree (descriptive set theory) Tree (graph theory) Tree (set theory) Tree network Tree structure Theoretical computer science Total order Lattice Mathematics History of numbers Nested set Polynomial hierarchy
has abstract	In mathematics, a hierarchy is a set-theoretical object, consisting of a preorder defined on a set. This is often referred to as an ordered set, though that is an ambiguous term that many authors reserve for partially ordered sets or totally ordered sets. The term pre-ordered set is unambiguous, and is always synonymous with a mathematical hierarchy. The term hierarchy is used to stress a hierarchical relation among the elements. Sometimes, a set comes equipped with a natural hierarchical structure. For example, the set of natural numbers N is equipped with a natural pre-order structure, where whenever we can find some other number so that . That is, is bigger than only because we can get to from using . This idea can be applied to any commutative monoid. On the other hand, the set of integers Z requires a more sophisticated argument for its hierarchical structure, since we can always solve the equation by writing . A mathematical hierarchy (a pre-ordered set) should not be confused with the more general concept of a hierarchy in the social realm, particularly when one is constructing computational models that are used to describe real-world social, economic or political systems. These hierarchies, or complex networks, are much too rich to be described in the category Set of sets. This is not just a pedantic claim; there are also mathematical hierarchies, in the general sense, that are not describable using set theory. Other natural hierarchies arise in computer science, where the word refers to partially ordered sets whose elements are classes of objects of increasing complexity. In that case, the preorder defining the hierarchy is the class-containment relation. Containment hierarchies are thus special cases of hierarchies. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Hierarchy_(mathematics)
is Link from a Wikipage to another Wikipage of	Abstract algebraic logic Arithmetical hierarchy Boolean hierarchy Difference hierarchy Hierarchy (disambiguation) Neighbor-net Nested set
is foaf:primaryTopic of	http://en.wikipedia.org/wiki/Hierarchy_(mathematics)

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