RE: The Category of Semimodules over Semirings

RE: The Category of Semimodules over Semirings

[Katsov, Yefim]

Dear Jawad,

Here are some suggestions that hopefully may help you:

1) In regard of literature on the subject, I'd suggest to look at the book, "A Guide to the Literature on Semirings and their Applications in Mathematics and Information Sciences (With Complete Bibliography)," by Kazimierz Glazek, Kluwer Academic Publishers, 2002.

2) Concerning categorical aspects of semimodules categories, I'd suggest to check publications of Ildar S. Safuanov whose Ph.D. dissertation, supervised by late Prof. L.A. Skornyakov (Moscow State U., MGU) in 80-th, was about categorical aspect of semimodules.

With my best wishes,

Yefim
____________________________________________________________________
Prof. Yefim Katsov
Department of Mathematics & CS
Hanover College
Hanover, IN 47243-0890, USA
telephones: office (812) 866-6119;
                 home (812) 866-4312;
                  fax   (812) 866-7229
-----Original Message-----
From: cat-dist@mta.ca [mailto:cat-dist@mta.ca] On Behalf Of Jawad Abuhlail
Sent: Saturday, March 15, 2008 9:33 PM
To: categories@mta.ca
Subject: categories: The Category of Semimodules over Semirings

Dear colleagues,

A semiring is roughly speaking a ring without subtraction, i.e. (R,+,0) is
an Abelian monoid & (R,*,1) is a semigroup with distribution of * over +
(e.g. the set of non-negative integers). A semimodule over a semiring is
roughly speaking a module without subtraction, i.e. (M,+,0) is an Abelian
Monoid, and there is a scalar multiplication of the semiring on M with the
usual expected properties. A semimodule over a semiring is cancellative, if
the Abelian monoid (R,+) is cancellative.

The category of N0-Semimoduels is just the category of "Commutative
Monoids".

Indeed the category of left semimodules over an arbitrary semiring R? (A
special example would be the category of commutative monoids) is not
pre-additive. However, for any left semimodules M and N over a semiring R,
(Hom_R(M,N),+,0) is an Abelian monoid and it has kernels and cokernels.

A monomorphism of semimodules is injective, however only an "image-regular"
epimorphism is subjective. For a morphism of semimodules f: M --> N, the
sequence 0 ---> Coimage(f) --> Im(f) --> is exact, but the canonical map
Coimage(f) --> Im(f) is not an isomorphism (neither a bimorphism), unless f
is regular.  The category of semimodules had products, equalizers and
products (however not necessarily coequalizers). The category of
cancellative semimodules is complete and cocomplete. It has a generator,
namely the semiring itself and I also "expect" it to have exact colimits
(ANY REFERENCE?).

What kind of Categories is the category of (cancellative) semimodules over
semiring? Is there a notion of "almost Grothendieck categories" or
"Semi-Grothendieck Categories" to which categories of (cancellative)
semimodules fit? Unfortunately, I did not find a single book that clarifies
the categorical aspects of semimodules (The 3 books of Golan as well as the
books on the subject by others are devoted more to semirings and automata
and not have much on semimodules).

I appreciate very much your comments, suggestions for literature (e.g.
books, Ph.D. dissertations, articles) on the CATEGROY OF (Cancellative)
SEMIMODULES (other than the papers of Takahashi and Katsov, which I already
have).

With best regards,

Jawad

-----------------------------------------------------

Dr. Jawad Abuhlail

Dept. of Math. & Stat.

Box # 5046, KFUPM

31261 Dhahran (KSA)

http://faculty.kfupm.edu.sa/math/abuhlail

Re: The Category of Semimodules over Semirings

[Jawad Abuhlail]

Dear Prof. Linton,
Many thanks for your comments about the existence of coequalizers in
categories of semimodules.

What I mentioned (that the category of left semimodules over an arbitrary
semiring has in general no coequalizers) was due to CONFUSION caused by the
way some results in "M. Takahashi, Completeness and $C$-co completeness of
the category of semimodules. Math. Sem. Notes Kobe Univ. 10 (1982), no. 2,
551--562." are stated.

In that paper, Takahashi proved that the category of left semimodules over
an arbitrary semiring has $c$-coequalizers and is $c$-cocomplete (where $c :
R-smod ---> C-R-smod$ denote the functor that assigns to each semimodule its
associated cancellative semimodule).

Indeed his proof does not exclude that this category has coequalizers as I
(apparently incorrectly) stated. For your convenience, I summarize what
Takahashi did in the above mentioned paper:

Denote the category of left semimodules over a semirings $R$ by $R-smod$ and
its full subcategory of cancellative semimodules by $C-R-smod$. Then
$R-smod$ has products and equalizers, whence complete. Let $c : R-smod --->
C-R-smod$ denote the functor that assigns to each semimodule its associated
cancellative semimodule. This functor is left adjoint to the embedding
functor $U : C-R-smod ---> R-smod$. Then $R-smod$ has coproducts and
$c$-coequalizers, whence $c$-cocomplete.

The confusion is caused by his statement that "$c$-cocompleteness is a
relaxation of cocompleteness" and the last Corollary in the paper, in which
he deduced that the full subcategory $C-R-smod$ of CANCELLATIVE semimodules
has coequalizers and is cocomplete!!

Anyway, I am so grateful for your comments and would appreciate as well any
comment about exactness of colimits in $C-R-smod$ and $C-R-smod$ (in the
category of modules over rings, colimits are exact!!) will be highly
appreciated.

Wassalam,
Jawad


-----Original Message-----
From: Fred E.J. Linton [mailto:fejlinton@usa.net]
Sent: Sunday, March 16, 2008 8:43 PM
To: categories@mta.ca
Cc: Jawad Abuhlail
Subject: Re: categories: 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

Re: The Category of Semimodules over Semirings

[Stephen Lack]

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
> 
> 
> 
> 
> 

Re: The Category of Semimodules over Semirings

[Fred E.J. Linton]

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

The Category of Semimodules over Semirings

[Jawad Abuhlail]

Dear colleagues,

A semiring is roughly speaking a ring without subtraction, i.e. (R,+,0) is
an Abelian monoid & (R,*,1) is a semigroup with distribution of * over +
(e.g. the set of non-negative integers). A semimodule over a semiring is
roughly speaking a module without subtraction, i.e. (M,+,0) is an Abelian
Monoid, and there is a scalar multiplication of the semiring on M with the
usual expected properties. A semimodule over a semiring is cancellative, if
the Abelian monoid (R,+) is cancellative.

The category of N0-Semimoduels is just the category of "Commutative
Monoids".

Indeed the category of left semimodules over an arbitrary semiring R? (A
special example would be the category of commutative monoids) is not
pre-additive. However, for any left semimodules M and N over a semiring R,
(Hom_R(M,N),+,0) is an Abelian monoid and it has kernels and cokernels.

A monomorphism of semimodules is injective, however only an "image-regular"
epimorphism is subjective. For a morphism of semimodules f: M --> N, the
sequence 0 ---> Coimage(f) --> Im(f) --> is exact, but the canonical map
Coimage(f) --> Im(f) is not an isomorphism (neither a bimorphism), unless f
is regular.  The category of semimodules had products, equalizers and
products (however not necessarily coequalizers). The category of
cancellative semimodules is complete and cocomplete. It has a generator,
namely the semiring itself and I also "expect" it to have exact colimits
(ANY REFERENCE?).

What kind of Categories is the category of (cancellative) semimodules over
semiring? Is there a notion of "almost Grothendieck categories" or
"Semi-Grothendieck Categories" to which categories of (cancellative)
semimodules fit? Unfortunately, I did not find a single book that clarifies
the categorical aspects of semimodules (The 3 books of Golan as well as the
books on the subject by others are devoted more to semirings and automata
and not have much on semimodules).

I appreciate very much your comments, suggestions for literature (e.g.
books, Ph.D. dissertations, articles) on the CATEGROY OF (Cancellative)
SEMIMODULES (other than the papers of Takahashi and Katsov, which I already
have).

With best regards,

Jawad

-----------------------------------------------------

Dr. Jawad Abuhlail

Dept. of Math. & Stat.

Box # 5046, KFUPM

31261 Dhahran (KSA)

http://faculty.kfupm.edu.sa/math/abuhlail