Prewellordering

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

Conversely, every prewellordering is induced by a unique regular norm (a norm is regular if, for any and any , there is such that ).


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