Prewellordering

rdf:langString Prewellordering

rdf:langString 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 ).