In mathematics, a hierarchy is a set-theoretical object, consisting of a preorder defined on a set.
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.
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
[citation needed] 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.
[1] This is not just a pedantic claim; there are also mathematical hierarchies, in the general sense, that are not describable using set theory.
[citation needed] 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.
In theoretical computer science, the time hierarchy is a classification of decision problems according to the amount of time required to solve them.
Tree related topics: Effective complexity hierarchies: Ineffective complexity hierarchies: In set theory or logic: