On Sun, Mar 27, 2011 at 12:00:37PM -0700, Simon King wrote:
> Anyway, I agree with you that my question should be answered in a
> practical way and can not really have a "categorical" answer. That's
> why I formulated "are free algebras and polynomial rings too
> different?" in my original question.
>
> Since I tend to believe that free algebras and polynomial rings are
> not too different, I would say that it is a practical solution to
> identify a unigenerated free algebra with a univariate polynomial
> ring. But other people may have the opposite opinion - that's why I
> ask.
I don't have a good answer. However there is one obvious constraint:
``any code working with the free algebra on n generators should work
-- or at least degenerate sanely -- on one (and actually even zero)
generators''.
(or at least: it should be possible for the user to construct the free
algebra on n generators in such a way that the above holds)
By obvious, I mean that it's just the usual thing about base cases
which should be consistent with the generic cases. Also we encountered
several similar occasions in MuPAD where such consistency was not met,
and where that was concretely hurting us.
Now to an example of consequence: if, say, the basis of the free
algebra on n generators is indexed by words, then so should be the
basis of the free algebra on one generator. Or at least it should be
indexed by some objects which *behave* like words.
With tells us that we should start by answering the same question
about the free monoid on one versus n generators.
Cheers,
Nicolas
--
Nicolas M. Thiéry "Isil" <nthi...@users.sf.net>
http://Nicolas.Thiery.name/
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