The division lattice as a category: is 0 prime?

[Vaughan Pratt <pratt@cs.stanford.edu>]

Has the division lattice been organized as a category somewhere in the
literature in a way that accounts somehow for 0?

A simplistic construction is [FinSet^N] denoting the finitary
finite-set-valued functors from the discrete category N.  ("Finitary" in
the sense of being zero almost everywhere.)

Interpreting N as indexing the primes and coproduct as numeric
multiplication, product becomes gcd and the pushout of (a,b) over its
gcd is its lcm.

It is simplistic by virtue of omitting (numeric) 0, which is standardly
placed at the top of the division lattice.  Unlike the rest of the
lattice 0 is not generated from the primes by finite coproducts,
suggesting it needs to feature in some sort of basis for the complete
lattice.

The natural thing would be to remove all but the primes and 0 from the
division lattice and then try to put them back finitarily.  This makes
the starting point the inverted flat CPO N^* where N consists of the
primes and "bottom" * is now at the top, denoting 0.  (That convention
makes N_* the usual flat CPO of natural numbers.)

If I'm not mistaken the completion of N^* under finite coproducts has as
objects two copies of the natural numbers.  Below * is [FinSet^N]
understood as previously.  Above * is FinSet, which was created from *
by completion under coproducts.  This amounts to FinSet + [FinSet^N]
joined at the hip with a shared initial object (numeric 1) and a shared
final object (numeric 0, or *).

 From the Yoneda standpoint the objects are functors from N_* (the usual
CPO) to FinSet.  The ones below are the functors that are 0 at * and
cofinitely many elements of N.  The ones above are 0 at N, with *
unconstrained.

Yoneda's hands are a bit tied here because we are only taking
coproducts.  Closing under finite colimits presumably frees up Yoneda to
produce FinSet x [FinSet^N]  = [FinSet^{N_*}].  This might come in handy
when one wants a system of pairs of numbers (m,n) for which
(m,n)+(m',n') = (m+m',nxn').

Is there some abstract nonsense reason why coproducts produced the sum
(actually pushout over the initial and final objects) while colimits
produced the product?

I arrived at all this after Steve Vickers mentioned on the univalg
mailing list that ring theorists define 0 to be a prime number because
then they could define n to be prime just when the ring Z/nZ extends to
a field.  This got me to wondering how this could be reflected in the
division lattice, which has 0 at the top without however being
considered a prime.  I personally am too old to believe that 0 is a
prime, but I can see where a younger generation could be hoodwinked.
Even with the above understanding however I don't see how 0 can be
understood as just another ordinary prime, any more than bottom is just
another ordinary number in N_*.

Vaughan Pratt