Hierarchy (mathematics)

For other uses, see hierarchy (disambiguation).

In mathematics, a hierarchy is a preorder, i.e. an ordered set. The term is used to stress a natural hierarchical relation among the elements. In particular, it is the preferred terminology for posets 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.

Related terminology

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.

Example

In theoretical computer science, the time hierarchy is a classification of decision problems according to the amount of time required to solve them.

See also

• Order theory
• Tree structure
• Lattice
• Polynomial hierarchy
• Chomsky hierarchy
• Analytical hierarchy
• Arithmetical hierarchy
• Hyperarithmetical hierarchy

• Abstract algebraic hierarchy
• Borel hierarchy
• Wadge hierarchy
• Difference hierarchy
• Tree (data structure)
• Tree (graph theory)
• Tree network
• Tree (descriptive set theory)
• Tree (set theory)