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| {{Other uses|hierarchy (disambiguation)}}
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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, which many authors reserve for [[partially ordered set | partially ordered sets]] or [[totally ordered set | 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.
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| 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 <math>n \le n'</math> whenever we can find some other number <math>m</math> so that <math>n + m = n'</math>. That is, <math>n'</math> is bigger than <math>n</math> only because we can get to <math>n'</math> from <math>n</math> ''using'' <math>m</math>. This is true for 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 <math>n + m = n'</math> by writing <math>m = (n'-n)</math>.{{Citation needed|date=January 2013 | text="The argument should be provided here in the text."}}
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| 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 which 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 [http://ncatlab.org/nlab/show/Set Set] of sets.<ref>We may need a bigger [[topos]].</ref> This is not just a pedantic claim; there are also mathematical hierarchies which are not describable using set theory.{{Citation needed|date=January 2013}}
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| Another natural hierarchy arises in [[computer science]], where the word refers to [[poset |partially ordered sets]]s whose elements are [[class (set theory)|classes]] of objects of increasing [[complexity]]. In that case, the preorder defining the hierarchy is the class-containment relation. [[Containment hierarchy|Containment hierarchies]] are thus special cases of hierarchies.
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| ==Related terminology==
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| Individual elements of a hierarchy are often called '''levels''' and a hierarchy is said to be infinite if it has infinitely many distinct levels but said to '''collapse''' if it has only finitely many distinct levels.
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| ==Example==
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| In [[theoretical computer science]], the [[time hierarchy]] is a classification of [[decision problem]]s according to the amount of time required to solve them.
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| ==See also==
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| *[[Order theory]]
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| *[[Tree structure]]
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| *[[Lattice (mathematics)|Lattice]]{{dn|date=May 2012}}
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| *[[Polynomial hierarchy]]
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| *[[Chomsky hierarchy]]
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| *[[Analytical hierarchy]]
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| *[[Arithmetical hierarchy]]
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| *[[Hyperarithmetical hierarchy]]
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| * [[Abstract algebraic hierarchy]]
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| *[[Borel hierarchy]]
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| *[[Wadge hierarchy]]
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| *[[Difference hierarchy]]
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| * [[Tree (data structure)]]
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| * [[Tree (graph theory)]]
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| * [[Tree network]]
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| * [[Tree (descriptive set theory)]]
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| * [[Tree (set theory)]]
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| {{col-end}}
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| == References ==
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| <references />
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| [[Category:Hierarchy]]
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| {{math-stub}}
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