Hierarchy (mathematics)

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 ${\displaystyle n\leq n'}$ whenever we can find some other number ${\displaystyle m}$ so that ${\displaystyle n+m=n'}$. That is, ${\displaystyle n'}$ is bigger than ${\displaystyle n}$ only because we can get to ${\displaystyle n'}$ from ${\displaystyle n}$ using ${\displaystyle m}$. 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 ${\displaystyle n+m=n'}$ by writing ${\displaystyle m=(n'-n)}$.

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