I'm not sure of the etiquette here and whether this should be in a separate
topic, or if anything Chapter 1-related goes here. I have a question on the
order of partitions.
Eq. (1.2) on page 5 shows an order by coarseness of the partition. In other
words, we say that \\(A ≤ B\\) if, whenever \\(x\\) is connected to \\(y\\) in
\\(A\\), then \\(x\\) is connected to \\(y\\) in \\(B\\).
Then on the bottom of page 12, having introduced the notion that partitions on \\(A\\) can be
thought of as surjective functions out of \\(A\\), we can come up with a
slightly more formal definition of that same notion of order. We say that \\(f:
A ↠ P\\) is finer than \\(g: A ↠ Q\\) if there is a function \\(h: P → Q\\) such
that \\(f.h = g\\).
I'm a bit puzzled by this. I can see from the basic left-totality requirement of
a function that you cannot map a partition of lower cardinality to a partition
of higher cardinality. That will allow the arrows in Eq. (1.2) and disallow the
reverse of those arrows. My problem is that it would seem to me that you can
easily map partitions of the same cardinality to each other. So I can define a
function that maps the partitions in the middle row to each other. By the above
definition, that means that those partitions are all mutually finer than each
other. But we have already learnt on page 11 that the middle row are
non-comparable. So does the definition on the bottom of page introduce
unwanted additional comparisons over the earlier definition?