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.