An extension is a mapping from an incompletely ordered set to a completely ordered set.

for "=>" a preorder "=>'" extends "=>" if forall x,y in X x"=>"y => x"=>'"y

example:

X={x,y,z} with x>y,x>z has extensions: y>z, z<y, z~y

Lemma:

> a strict partial order on a set X => there exists a strict order >'
which extends >

source

jl@crush.caltech.edu index

representation_theorem