The division lattice as a category: is 0 prime?

The division lattice as a category: is 0 prime?

[Vaughan Pratt]

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

Re: The division lattice as a category: is 0 prime?

[Bill Lawvere]

Vaughn remarks
...........I would have thought intersecting them could only get you the
square-free ideals........

Indeed, as Jeff points out, we learned from Kummer and Dedekind to replace
elements by ideals, but we categorists have been late in providing a clear
account of this transition and, in particular, of the reason why the
result is  not primarily a lattice, but a monoidal closed category with
colimits.

Below I will elaborate on the following three points:

(1) The actual "ideal number" functor itself is clear enough (though never
made explicit), but why should it exist?

(2) The standard account of "why" is very categorical, but does not
directly address the algebraic category of rings nor the geometry of
intersection theory.

(3) The universal algebraists have developed a tool that might be applied
to the "why", but for some reason the universality is not often applied to
algebra or to geometry.

In more detail:

(1) The Kummer functor I goes from rigs (or K-rigs, where K is given, e.g.
Z or Q) to 2-rigs, where 2 is the 2-element rig in which 1+1=1. (Yes, the
rig that launched ring theory is not itself a ring). The functorality, as
well as the multiplication itself, depends on the set-theoretic operation
of image. The principal ideal concept is a natural transformation from M
to MI where M is the underlying multiplication.

(2) A rig can serve (not only as functions on a scheme but) as an abstract
general whose semantically corresponding concrete general is its category
of modules, which is a monoidal closed category with colimits. This
2-functor can be composed with the functor to posets that extracts from
the big category of modules just the submodules of the unit object. Again,
the image operation must in general be applied to the result of tensoring
two submodules (because of the lack of flatness). A monoidal poset with
colimits is also a 2-rig.

(3) Intersecting closed subspaces of a space may give only a shadow of a
description of their clash (e.g. the clash of Africa & Europe produces a
bulge i.e. the Alps). Although geometric figures are in general singular,
a notion of closed subspace which refines the notion of mere subset
provides a useful partial record.
In terms of the rigs of variable quantity on the spaces there is a
corresponding refinement:
The distributive lattice of radical ideals is refined to the monoidal
poset of all ideals. The ideal product under discussion is a key
ingredient in a construction of unions of subspaces that takes into
account the clashes. As it would be desirable geometrically to see even
nonsingular figures as images of maps in the category of spaces itself, it
would dually be desirable to see R/ab as the result of a construction on
R/a and R/b within the category of K-rigs itself, without the detour (2)
through modules. At least the case where K is a ring is indeed covered in
principle by a construction called the "commutator" (a misleadingly
particular "general terminology", ....groups are apparently not a typical
algebraic category). That this construction does reduce to a certain
concatenation of limits and colimits has been shown by categorists in
terms of congruence relations.

But the application to rings and ideals still remains to be done.

Bill


************************************************************
F. William Lawvere, Professor emeritus
Mathematics Department, State University of New York
244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA
Tel. 716-645-6284
HOMEPAGE:  http://www.acsu.buffalo.edu/~wlawvere
************************************************************



On Thu, 27 Sep 2007, Vaughan Pratt wrote:

> Jeff Egger wrote:
>> And I thought that every generation since Dedekind, Krull and Noether
>> knew that divisibility lattices are (in the general case) a red herring
>> and that it is the lattice of ideals of a ring (or its opposite, if you
>> prefer) which is really important.
>> ...
>> although I don't really understand your motivation.
>
> Right, I should have been clearer about the motivation.  I wanted to
> construct the division lattice abstractly from the primes in some
> finitary way, analogously to how one can construct the power set 2^X as
> the free upper semilattice generated by the singletons of X.  Putting
> that in terms of ideals, I'd like to be able to form all the ideals of Z
> from just the prime ideals.  I don't know much about ring theory so I
> could be confused about this, but I would have thought intersecting them
> could only get you the square-free ideals.  Starting from the prime
> power ideals takes care of that but what's the trick for getting all the
> ideals from just the prime ideals?  The category Div was my suggestion
> for that, but if there's a more standard approach in ring theory I'd be
> happy to use that instead (or at least be aware of it---Div is starting
> to grow on me).
>
> Now that I think of it, I suppose the standard completion must be the
> formation of finite subdirect products (aka sums?) of the quotients Z_p
> = Z/pZ over the prime ideals pZ.  By including Z along with the Z's,
> that way you reconstruct Div with the lower part consisting of Z_n =
> Z/nZ and the upper part n.Z (if I understand the notation).  That puts
> the ring structure of Z back into play however, which doesn't feel quite
> as "pure" as simply closing a flat inverted CPO under finite coproducts.
>
>>  Perhaps the answer to your original
>> question is to take (finite-valued) sheaves on this space of primes,
>
> Right, that (by Yoneda) was the completion under finite colimits
> approach at the end of my 10:40 am message this morning, which didn't
> "work" in the sense of not being the minimal solution and not having an
> obviously pleasing structure either.  Completion under finite coproducts
> was as small as I could make it, and initially I was miffed that there
> was still this junk above the division lattice that I was hoping would
> go away.  But then I decided that rather than complicate the completion
> process to prevent 0 from sprouting sow's ears above it, I'd try to make
> a silk purse out of the ears.  This ended up being the two-part
> Fundamental Theorem of Arithmetic via the single construction.
>
> With coproducts instead of colimits it's still sheaves but with the
> condition that if the stalk at * is nonempty then all the other stalks
> must be empty.  I don't know what the abstract-nonsense name for that is.
>
> Vaughan
>
>
>
>

Re: The division lattice as a category: is 0 prime?

[Vaughan Pratt]

Jeff Egger wrote:
> And I thought that every generation since Dedekind, Krull and Noether
> knew that divisibility lattices are (in the general case) a red herring
> and that it is the lattice of ideals of a ring (or its opposite, if you
> prefer) which is really important.
 > ...
 > although I don't really understand your motivation.

Right, I should have been clearer about the motivation.  I wanted to
construct the division lattice abstractly from the primes in some
finitary way, analogously to how one can construct the power set 2^X as
the free upper semilattice generated by the singletons of X.  Putting
that in terms of ideals, I'd like to be able to form all the ideals of Z
from just the prime ideals.  I don't know much about ring theory so I
could be confused about this, but I would have thought intersecting them
could only get you the square-free ideals.  Starting from the prime
power ideals takes care of that but what's the trick for getting all the
ideals from just the prime ideals?  The category Div was my suggestion
for that, but if there's a more standard approach in ring theory I'd be
happy to use that instead (or at least be aware of it---Div is starting
to grow on me).

Now that I think of it, I suppose the standard completion must be the
formation of finite subdirect products (aka sums?) of the quotients Z_p
= Z/pZ over the prime ideals pZ.  By including Z along with the Z's,
that way you reconstruct Div with the lower part consisting of Z_n =
Z/nZ and the upper part n.Z (if I understand the notation).  That puts
the ring structure of Z back into play however, which doesn't feel quite
as "pure" as simply closing a flat inverted CPO under finite coproducts.

>  Perhaps the answer to your original
> question is to take (finite-valued) sheaves on this space of primes,

Right, that (by Yoneda) was the completion under finite colimits
approach at the end of my 10:40 am message this morning, which didn't
"work" in the sense of not being the minimal solution and not having an
obviously pleasing structure either.  Completion under finite coproducts
was as small as I could make it, and initially I was miffed that there
was still this junk above the division lattice that I was hoping would
go away.  But then I decided that rather than complicate the completion
process to prevent 0 from sprouting sow's ears above it, I'd try to make
a silk purse out of the ears.  This ended up being the two-part
Fundamental Theorem of Arithmetic via the single construction.

With coproducts instead of colimits it's still sheaves but with the
condition that if the stalk at * is nonempty then all the other stalks
must be empty.  I don't know what the abstract-nonsense name for that is.

Vaughan

Re: The division lattice as a category: is 0 prime?

[Jeff Egger]

--- Vaughan Pratt <pratt@cs.stanford.edu> wrote:

> 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.  

Um, well, for arbitrary ideals I in a commutative ring R, R/I "extends to 
a field" (or, in more common parlance, "is an integral domain") if and only 
if I is a prime ideal; hence the previous assertion can be simplified to 

  ring theorists define 0 to be a prime number because
  then they could define n to be prime just when nZ is a prime ideal. 

which doesn't seem so unreasonable.  

> 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.

And I thought that every generation since Dedekind, Krull and Noether
knew that divisibility lattices are (in the general case) a red herring 
and that it is the lattice of ideals of a ring (or its opposite, if you 
prefer) which is really important.  Surely, it makes sense to fix 
terminology according to what does work in the general case.  

> 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_*.

Although 0 can be a prime (depending on the ring under consideration), 
it is plainly never "just another ordinary prime": there is a well-known
topology on the set of prime ideals of a commutative ring which clearly
distinguishes 0 from its fellows.  Perhaps the answer to your original 
question is to take (finite-valued) sheaves on this space of primes, 
although I don't really understand your motivation.  

Cheers,
Jeff Egger.




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Re: The division lattice as a category: is 0 prime?

[Vaughan Pratt]

Mike Barr's response to my

> 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_*.

was "I don't see your problem."  And now that I reflect on that comment,
I don't see it myself.

With the benefit of sleeping on the problem combined with Mike's prod,
the moral for me is clear.  The Fundamental Theorem of Arithmetic is
incomplete as stated.  It should read as follows.

   Every natural number partitions uniquely as a sum of 1's, and every
   positive integer factors uniquely as a product of primes.

The constructive proof of the theorem exhibits this ostensibly two-part
structure uniformly as the completion under coproducts of the inverted
flat CPO N^*.  This coproduct-complete category is naturally analyzed
into two components, additive upstairs and multiplicative downstairs.
The components share the initial and final objects of the category, with
the former manifesting as 0 in the additive component and 1 in the
multiplicative, and conversely for the latter.

And that's why 0 is at the top of the division lattice.  The reason it
(qua 1) is at the bottom of the additive component and not the top (its
default location in an arbitrary category with 1) is because it
generates Set (or in this case FinSet).

It makes no sense to consider 0 as a prime because there is no way to
define things such that 0 factors uniquely.  The role of the morphisms
in N^* is to prevent 0 from being an atom in the completion, instead
making it final in the inverted CPO, which coproducts preserve and
colimits do not.

Had we completed under colimits, by Yoneda the final object would have
been the constantly 1 functor.  The top * of N^* would then no longer be
the final object of the completion, being the unit functor for *, namely
1 at * and 0 elsewhere.  With either completion the primes are the other
unit functors, but only with the coproduct completion is the final
object a unit functor.

Vaughan