A weak order satisfies:

completeness /\ transitivity.

A weak order is commonly denoted as "=>".

(=> also satisfies reflexivity)

From a weak order you can build a strong order and equivalence as follows:
x~y <=> x=>y /\ y=>x

x>y <=> x=>y /\ y!=>x

There is a theorem which states that the above definitions lead to each other.

X/~ == {{x}: x in X, forall y in {x} y~x} is a partition of the set into equivalence classes.

source

jl@crush.caltech.edu index

representation_theorem

strict_weak_order

weighted_utility

strict_order

utility

choice

order

chain

QP1

AA1

Ps

P1

J1