re: Small semirings

[Marco Grandis <grandis@dima.unige.it>]

This site lists a lot of algebraic structures and often gives  
information on finite examples:

     http://math.chapman.edu/cgi-bin/structures

-----
Thus, for commutative rings (with 1) you have:

http://math.chapman.edu/structuresold/files/ 
Commutative_rings_with_identity.pdf

where you can find that there are:

- 1 structure with 1 element (or 2, 3, 5, 6 elements)
- 4 structures with 4 elements.

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The case of semirings is not (yet?) much developed: just a few  
results and trivial examples.
See:

http://math.chapman.edu/structuresold/files/ 
Semirings_with_identity_and_zero.pdf
http://math.chapman.edu/structuresold/files/Semirings_with_zero.pdf

-------
Commutative semirings are not in the list, I think.

Marco Grandis


On 3 Jan 2007, at 23:09, Andrej Bauer wrote:

> Dear categorists,
>
> I have no idea where to ask the following algebra question. Hoping  
> that
> some of you are algebraists, I am asking it here.
>
> I am looking for examples of small (finite and with few elements,  
> say up
> to 8) commutative semirings with unit, by which I mean an algebraic
> structure which has +, *, 0 and 1, both operations are commutative  
> and *
> distributes over +. The initial such structure are the natural  
> numbers.
>
> Here are the examples I know:
>
> 1) Modular arithmetic, i.e., (Z_n, +, *, 0, 1)
>
> 2) Distributive lattices with 0 and 1.
>
> 3) "Cut-off" semiring, in which we compute like with natural numbers,
> but if a value exceeds a given constant N, then we cut it off at N.  
> For
> example, if N = 7 then we would have 3 + 3 = 6, 3 + 6 = 7, 4 * 4 = 7,
> etc. Do such semirings have a name?
>
> There must be a census of small commutative rings, or even semirings.
> Does anyone know?
>
> Andrej
>
>
>