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. ----- 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