From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3567 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: re: Small semirings Date: Thu, 4 Jan 2007 17:52:53 +0100 Message-ID: <41321.9766378917$1241019382@news.gmane.org> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v752.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019381 9287 80.91.229.2 (29 Apr 2009 15:36:21 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:36:21 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Jan 4 12:53:12 2007 -0400 X-Keywords: X-UID: 59 Original-Lines: 66 Xref: news.gmane.org gmane.science.mathematics.categories:3567 Archived-At: 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 > > >