>> On 03/11/2007 03:25 PM, Ralf Hemmecke wrote: >>> Could someone tell me what the cycle type of \sigma \in S_0 is. I >>> somehow think that (), i.e. the empty tuple is a good candidate. >> What else. > > Hmmm, I wanted to say the sequence which consists of zeros only. > Well, that's about the same thing, I guess :-)
Also quite OK. At one point I thought that one should rather take the point of view that all permutations are automorphisms of N (natural numbers). But that doesn't quite fit to the fact that a species is a endo-functor of the category B of *finite* sets and bijections. >>> Take n=0 in Definition 2.2.4 of BLL. What happens? Let \sigma\in >>> S_0. > So, \sigma is the only permutation of the empty set. Something rather > imaginary... There is exactly one such permutation, which explains > 0!=1. Of course there must be one. \emptyset is an object in B and in a category there is always the 1 arrow from an object to itself. > Note that - since it is a function from the empty set to the empty > set, one cannot really say that it takes an argument :-) I don't say that. >>> What is G[\sigma]? Ah, sure, it is a permutation in some S_m. >>> First question: what is m? >> The only good choice is m=\card G[0]. And to be functorial >> G[\sigma] must be the identity permuation of S_m. > > I just wanted to look that up. OK, I did: one can say (although I > don't really have a good argument for that) that the sigma above is > the identity on the empty set... No. G[\emptyset] is not necessarily empty. So the G transports the arrow 1: \emptyset -> \emptyset to G[1]: G[\emptyset] -> G[\emptyset]. And the latter is most certainly the identity on G[\emptyset], because G is a functor. > Note that axiom says lcm(0,n)=0 for any n. I don't think that is important since you suggested to take n := lcm([i for i in 1..#p | p.i ~= 0]); and I would have to compute the lcm of the empty list. That is more interesting. :-) >> So I would be happy if someone could confirm my thinking. > > confirmed? Thank you? ;-) Ralf ------------------------------------------------------------------------- Take Surveys. Earn Cash. Influence the Future of IT Join SourceForge.net's Techsay panel and you'll get the chance to share your opinions on IT & business topics through brief surveys-and earn cash http://www.techsay.com/default.php?page=join.php&p=sourceforge&CID=DEVDEV _______________________________________________ Aldor-combinat-devel mailing list Aldor-combinat-devel@lists.sourceforge.net https://lists.sourceforge.net/lists/listinfo/aldor-combinat-devel
