Small semirings

[Andrej Bauer <Andrej.Bauer@fmf.uni-lj.si>]

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