Prewellordering (original) (raw)

In set theory, a prewellordering on a set is a preorder on (a transitive and strongly connected relation on ) that is wellfounded in the sense that the relation is wellfounded. If is a prewellordering on then the relation defined by is an equivalence relation on and induces a wellordering on the quotient The order-type of this induced wellordering is an ordinal, referred to as the length of the prewellordering. A norm on a set is a map from into the ordinals. Every norm induces a prewellordering; if is a norm, the associated prewellordering is given by