The Category of Semimodules over Semirings

[Jawad Abuhlail <abuhlail@kfupm.edu.sa>]

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