[Om3] OpenMath Symbols for Symbolic Computation

[Professor James Davenport]

On Fri, 12 Sep 2008, Sebastian Freundt wrote:
> Paul Libbrecht <paul at activemath.org> writes:
> > Le 11-sept.-08 à 18:39, Sebastian Freundt a écrit :
> >> for one, NONE of the provided matrix CDs actually define matrices
> >> (from an
> >> algebraic point of view), a matrix is just an element of a matrix
> >> algebra
> >> or, more generally, a specific representation of a linear mapping,
> >> so it's
> >> an element of Hom(R,S), R,S being rings, Hom being the space of
> >> mappings
> >> that preserve homomorphy.
> >
> > Honestly, this is a linear mapping, not a matrix, at least to what I
> > was educated in.
> > A matrix has a notion of being an array so associating the matrix to
> > an element of Hom(R,S) actually requires a basis.
> >
> > I would agree with an element Hom(R^k, S^k) maybe, then it is the same
> > notion.
> 
> Yeah, sorry, I had to write the mail about 10 times because of this
> gmane-gateway.  I meant R,S being modules of course.  And yes, you need a
> basis.  Admittedly, and that's exactly what I'm arguing for, additional
> information are mandatory.  Specifically, our CD is actually a compromise
I would disgaree here. A Matrix is precisely that, and needs no additional 
information. IF you wnat that matrix to represent a linear transformation 
on a space, THEN you need additional information.
> and we limit to the case where R,S are both M-modules, so you have to 
> (at least) specify the ground domain (as we call it) M.


> >>> From a computational POV, it is quite essential to know the parent
> >> structure (the space where the element lives) upfront, because
> >> usually you
> >> equip your parent structures with certain methods (comparison,
> >> addition,
> >> etc.) and not the elements themselves.
> >
> > But when you define a function only the very eager people define their
> > domain and range well!
> > (I agree it's best practice!).
> >
> >> Well, and then we were thinking it's better to start over with a
> >> fully-fledged CD instead of, um, `improving' the existing ones.
> >
> > Maybe you have a biassed interpretation?
> 
> Sure. We want to work with matrices (and other objects) in computer
> algebra systems, and as you can imagine it's hard to guess (from the CAS'
> POV) what the user is trying to tell us when even the most basic
> information is missing.
Right - rather like the polyd family. of CDs. Having such a set of CDs 
certainly makes sense.
I'd be happy to look at what you have: I have Peter Horn's matrix1, but 
don't quite see, for instance, what the point of entry_domain is?

James Davenport