On Apr 30, 2008, at 10:43 PM, David Harvey wrote: > Regarding ticket > > http://sagetrac.org/sage_trac/ticket/3045 > > can someone explain to me what the "gens" and "ngens" methods are > supposed to mean? There seems to be a lot of inconsistency. For > example: > > sage: ZZ.gens() > (1,) > > These are the additive generators. > > Ditto here: > > sage: GF(7).gens()(1,) > > Okay, what about this: > > sage: GF(49, "a").gens()(a,) > > That's not an additive generator. That's a generator as an algebra > over the base ring (GF(7) in this case). Similarly: > > sage: ZZ["x", "y"].gens()(x, y) > > So is this rule that > > (1) If R has a base ring distinct from R, then R.gens() returns the > generators over the base ring, where "generator" is interpreted > according to what category we're working in, and > (2) If the base ring of R is just R, then R.gens() returns the > additive generators? > > That's a bit weird. > > I hate to think what happens if we implement, e.g. the ring of > continuous functions on the interval [0, 1]. I suppose then "gens" > needs to return some kind of uncountable generator object > perhaps.... (excuse the pun)?
This is one of the things David Roe and I have been looking at as part of the coercion model. One change is that one will be able to specify the category for which the generators are given, and are also lifting the assumption of finiteness (and the indexing may be something other than the integers). I think the way to do things is (1). It is a little odd when the base of R is R, this happens for non-algebraic operations (e.g. completion of QQ to RR). If the R has a 1, then the 1 in R does generate over R and in some sense it's the best we can make sense of. One question is what does one want to be able to do with generators. Ideally, for example, one could define morphisms by giving the images of the generators. One could do very generic operations (e.g. differentiation, quotients, etc.) just by knowing how the operators act on generators. To do this one needs to be able to decompose any element as some kind of a "structure" (e.g. algebraic expression) involving only generators and elements of the base. It is unclear what the best way of going about this is. In any case, I'm curious what everyone else thinks as well. - Robert --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
