# Prewellordering

In set theory, a prewellordering on a set ${\displaystyle X}$ is a preorder ${\displaystyle \leq }$ on ${\displaystyle X}$ (a transitive and strongly connected relation on ${\displaystyle X}$) that is wellfounded in the sense that the relation ${\displaystyle x\leq y\land y\nleq x}$ is wellfounded. If ${\displaystyle \leq }$ is a prewellordering on ${\displaystyle X}$, then the relation ${\displaystyle \sim }$ defined by

${\displaystyle x\sim y\iff x\leq y\land y\leq x}$

is an equivalence relation on ${\displaystyle X}$, and ${\displaystyle \leq }$ induces a wellordering on the quotient ${\displaystyle X/\sim }$. The order-type of this induced wellordering is an ordinal, referred to as the length of the prewellordering.

A norm on a set ${\displaystyle X}$ is a map from ${\displaystyle X}$ into the ordinals. Every norm induces a prewellordering; if ${\displaystyle \phi :X\to Ord}$ is a norm, the associated prewellordering is given by

${\displaystyle x\leq y\iff \phi (x)\leq \phi (y)}$

Conversely, every prewellordering is induced by a unique regular norm (a norm ${\displaystyle \phi :X\to Ord}$ is regular if, for any ${\displaystyle x\in X}$ and any ${\displaystyle \alpha <\phi (x)}$, there is ${\displaystyle y\in X}$ such that ${\displaystyle \phi (y)=\alpha }$).