Partitions of a poset

Given a partitioned partial ordering relation we can produce a new partial order on the parts of the partition defined the condition that part one is less the part two if for every element of part one there exists some element of part two such that these two elements are less then one another in the underlying partial order.

We can use this notion of partitioned posets to partially order isomorphism types including cardinal numbers and the canonical labelings of relations. This allows us to maintain the use of partial orders even when we aren't dealing with structured sets which demonstrates how fundamental order theory really is.