Re: The Category of Semimodules over Semirings

["Stephen Lack" <S.Lack@uws.edu.au>]

As Fred says, the semimodules over a given semiring are 
determined by operations and equations, and so are complete
and cocomplete. In terms of exactness properties they are
also 

(i) locally finitely presentable (so that finite limits commute
with filtered colimits, and 
(ii) Barr-exact (so that there is an equivalence between quotients
and congruences)

If we restrict to the cancellative case, we still have completeness
and  cocompleteness and (i), but (ii) fails.  

Steve Lack.

> -----Original Message-----
> From: cat-dist@mta.ca [mailto:cat-dist@mta.ca] On Behalf Of 
> Fred E.J. Linton
> Sent: Monday, March 17, 2008 4:43 AM
> To: categories@mta.ca
> Subject: categories: Re: The Category of Semimodules over Semirings
> 
> On Sun, 16 Mar 2008 09:26:35 AM EDT Jawad Abuhlail 
> <abuhlail@kfupm.edu.sa> wrote, in part, on the Subject: The 
> Category of Semimodules over Semirings,
> 
> > ...  The category of semimodules had products, equalizers 
> and products 
> > (however not necessarily coequalizers).
> 
> I must be missing something here. Don't the (say, left-) 
> semimodules (over a given semiring) constitute an 
> equationally definable class of algebras? That is, aren't 
> they determined entirely by operations and equations?
> 
> If they DO, that is, if they ARE, then the category of them 
> all (together with their homomorphisms) must, like all such 
> "varietal categories," have all (small) limits and colimits, 
> and, in particular, all coequalizers.
> 
> Alas, I have little else to offer. Cheers, and Happy St. Paddy's Day,
> 
> -- Fred
> 
> 
> 
> 
>