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prewe
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There are 5 letters in PREWE ( E1P3R1W4 )
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Definitions of prewe in various dictionaries:
PREWE - In set theory, a prewellordering is a binary relation ≤ {\displaystyle \leq } that is transitive, total,...
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In set theory, a prewellordering is a binary relation * * * * ≤ * * * {\displaystyle \leq } * that is transitive, total, and wellfounded (more precisely, the relation * * * * x * ≤ * y * ∧ * y * ≰ * x * * * {\displaystyle x\leq y\land y\nleq x} * is wellfounded). In other words, if * * * * ≤ * * * {\displaystyle \leq } * is a prewellordering on a set * * * * X * * * {\displaystyle X} * , and if we define * * * * ∼ * * * {\displaystyle \sim } * by* * * * x * ∼ * y * * ⟺ * * x * ≤ * y * ∧ * y * ≤ * x * * * {\displaystyle x\sim y\iff x\leq y\land y\leq x} * then * * * * ∼ * * * {\displaystyle \sim } * is an equivalence relation on * * * * X * * * {\displaystyle X} * , and * * * * ≤ * * * {\displaystyle \leq } * induces a wellordering on the quotient * * * * X * * / * * ∼ * * * {\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 * * * * X * * * {\displaystyle X} * is a map from * * * * X * * * {\displaystyle X} * into the ordinals. Every norm induces a prewellordering; if * * * * ϕ * : * X * → * O * r * d * * * {\displaystyle \phi :X\to Ord} * is a norm, the associated prewellordering is given by * * * * * x * ≤ * y * * ⟺ * * ϕ * ( * x * ) * ≤ * ϕ * ( * y * ) * * * {\displaystyle x\leq y\iff \phi (x)\leq \phi (y)} * Conversely, every prewellordering is induced by a unique regular norm (a norm * * * * ϕ * : * X * → * O * r * d * * * {\displaystyle \phi :X\to Ord} * is regular if, for any * * * * x * ∈ * X * * * {\displaystyle x\in X} * and any * * * * α * < * ϕ * ( * x * ) * * * {\displaystyle \alpha <\phi (x)} * , there is * * * * y * ∈ * X * * * {\displaystyle y\in X} * such that * * * * ϕ * ( * y * ) * = * α * * * {\displaystyle \phi (y)=\alpha } * ). |