Of chickens and eggs [was: is 0 prime?]

Of chickens and eggs [was: is 0 prime?]

[Jeff Egger]

--- Vaughan Pratt <pratt@cs.stanford.edu> wrote:
> Presumably by Sup you mean what Peter Johnstone calls CSLat,
> complete semilattices, which is a lovely self-dual category.  

Yes, indeed I did provide an equivalent definition:

> > [Sup denotes the category of complete lattices and sup-homomorphisms.] 

> According it the status of "the most awesome" however is a symptom of
> not yet having come to grips with the joy of Chu, 

At the risk of appearing pretentious, I'd like to quote Chekhov: de gustibus,
aut bene aut nihil.  ;)

[Incidentally, I do like Chu categories, but I will play devil's advocate
here.]

> a more awesome
> self-dual category (fully) embedding CSLat in a duality-preserving and
> concrete-preserving way 

...but not tensor-preserving?  I could just as easily say that Chu(Set,2) 
is (equivalent to) a lluf subcategory of Rel^2 (2 here denoting the arrow
category), which is in turn (equivalent to) a full subcategory of Sup^2; 
the latter carries a fascinating *-autonomous structure derived from those 
of Sup and 2, and the composite embedding is duality-preserving (though only 
the first part is "concrete-preserving").  

> [...] which is more awesome but still not awesome to the max.  

Word.  

> If going up only reduces the awe, then one should instead go down from
> CSLat for greater awe.  

The trouble with (Dedekind-)infinite things is that one can argue about 
which way is up and which way is down.  For example, both the forgetful
functor Sup ---> Pos, and its left adjoint can be regarded as "embeddings"
---thus one could perversely regard complete (semi)lattices as more, not
less, general than arbitrary posets.  

> Not only am I not a ring theorist but it's never occurred to me even to
> play one on the Internet. 

I hope no-one would accuse me of "playing the ring theorist" on the 
Internet or elsewhere, merely as a result of quoting some of the subject's
most celebrated theorems.  [I was glad to learn that I have forgotten a 
smaller chunk of my undergraduate education than I would have suspected.]  

Cheers,
Jeff.

P.S. It has been pointed out to me, by a reader of this list, that the 
"conventional wisdom" I quoted in re the history of ideal theory is 
flawed (as I suspected, for no deeper reason than a profound mistrust 
of conventional wisdom).

> > [...] is commonly cited as Dedekind's original motivation 
> > for defining ideals.
> 
> Hi Jeff,
>   in fact Kummer defined ideal numbers and proved the Fermat
> conjecture for regular primes before Lame' presented the
> fallacious argument (by some years, I think, but I can't recall
> just how many).  There's a lot of information about this in
> the Edwards book named after the conjecture (and some more in
> his recent book on constructive algebra).

Re: Of chickens and eggs [was: is 0 prime?]

[Vaughan Pratt]

Jeff Egger wrote:

>
> ...but not tensor-preserving?

Tensor-preserving is the exception, the rule is a tensorial strength, as
in this case.

Vaughan