What is "prewellordering"

n. (context math set theory English) A binary relation that is transitive, total, and well-founded.

In set theory, a prewellordering is a binary relation  ≤  that is transitive, total, and wellfounded (more precisely, the relation $x\le y\land y\nleq x$ is wellfounded). In other words, if  ≤  is a prewellordering on a set X, and if we define  ∼  by

x ∼ y ⇔ x ≤ y ∧ y ≤ x
then  ∼  is an equivalence relation on X, and  ≤  induces a wellordering on the quotient X/ ∼ . The order-type of this induced wellordering is an ordinal, referred to as the length of the prewellordering.

A norm on a set X is a map from X into the ordinals. Every norm induces a prewellordering; if ϕ : X → Ord is a norm, the associated prewellordering is given by

x ≤ y ⇔ ϕ(x) ≤ ϕ(y)
Conversely, every prewellordering is induced by a unique regular norm (a norm ϕ : X → Ord is regular if, for any x ∈ X and any α < ϕ(x), there is y ∈ X such that ϕ(y) = α).