Dear Florent,
I hope you don't feel offended by my questions. I knew the mathematics
you were writing about. I was just saying that either the function
basis() should be renamed to canonical_basis() or all what you wrote in
your mail should better go into the specification of basis().
I just love to see extensive documentation. The current documentation is
not as precise as a mathematician wants. If I try to understand a
function, I don't want to be forced to look at the code.
Translating programs into mathematics, I would say, the specification is
what goes int \begin{theorem} ... \end{theorem}, the code however, lives
in \begin{proof} ... \end{proof}. Hmm.. I don't know whether this is the
best comparison. But it is completely fine to use theorem without
knowing its proof.
Ralf
On 03/03/2009 01:56 PM, Florent Hivert wrote:
> Dear Ralf,
>
>> Interesting! You probably meant F.basis(). But anyway, how can you be
>> sure that b['a'] is the "same" as the 'a' in the free module?
>
> As usual in mathematics, we identify two different things, whereas when
> writing code we have to be very precise. The universal property in the
> category of R-module state that:
>
> For any ring R and set E, there exist an unique (upto isomorphism) R-module
> R(E), and a mapping i: E -> R(E) such that if f is an arbitrary mapping
> from E to some R-module M, then there exists a unique module homomorphism
> g: R(E) -> M such that f = g o i. The mapping i is called the canonical
> injection from E to R(E).
>
> Usually when speaking about R(E), we don't make any difference between any e
> in E, and the corresponding element i(e) in R(E). The same abuse of language
> is usually done in free group. For many reason, we don't want (cannot ?) do
> this abuse anymore in Sage. This being clear, in R(E) there is a canonical
> basis (i(e))_{e in E}. The basis Nicolas is speaking about is this basis. so
> that "b['a'] is the "same" as the 'a'" is "true" by construction. The rigorous
> statement is that b['a'] is the image of 'a' by the canonical injection i.
> Now the printing b['a'] is not very clear, and the user is supposed to provide
> it's own printing, usually by giving a name to the basis b.
>
> Note that the mathematical notation (i(e))_{e in E} is exactly an instance of
> the notion of family, as stated by Nicolas. This is a very simple and very
> natural notion in mathematics... It should be as natural in Sage... As an
> example: for any species F, we have a family (F_n)_{n in N} where F_n is
> defined as F[{1 ... n}]. And there is a canonical EnumerableSet associated to
> F that is Union((F_n)_{n in N}).
>
> This discussion should clearly end is the doc... Any help welcome.
>
> Cheers,
>
> Florent
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