Weak orders

Weak orders are isomorphic to ordered partitions:

(#{0 1} #{2 3} #{4 5 6})
(#{0 1 2} #{3 4 5} #{6 7 8})

Total orders are weak orders with singleton equivalence classes:

(#{0} #{1} #{2} #{3})

Reductions are built out of weak orders with a single well defined maximal element:

(#{0 1 2} #{3})
(#{0 1 2 3} #{4})

We can weakly order the vertices in a binary relation by connectivity, then refine that with degree characteristics, and other vertex invariants, and so on to canonize the relation.