Obviously I don't want to sound obstinate either, but I really cannot
imagine you find a majority on this list which prefers to write R.mul(a,
b) instead of a*b, whatever the argument for or against.
On 31.03.2012 16:05, Sergiu Ivanov wrote:
Summarising: when you work in concrete rings (integers, reals, etc.)
and focus on elements thereof, the "real" stuff is indeed in elements.
However, if the focus is on the rings themselves, the situation is
utterly different. Consider, for example, a quite real task of
deciding whether a ring with a given set of generators is commutative
or not. The focus is on rings.
Maybe this is the crux of the matter. Some questions inherently deal
with the *whole* algebraic structure (is this ring commutative? is it a
field?). The elements themselves don't really enter. Other questions
deal mostly with elements (is this polynomial irreducible? do these
polynomials generate a radical ideal?).
There are borderline questions, like "is this element invertible?" or
"is this relation trivial in the group we are studying?".
I guess the reason why I did not consider this dichotomy so far is that
groebner basis computations seem, to me, to very much fall in the second
category (being about elements, not the whole ring).
Actually, thinking about it, this discussion does not seem terribly
useful. Either approach can emulate the other easily (one can define
R.mul(a, b) to return a*b, and one can define a*b to return R.mul(a,b),
assuming elements know to which ring they belong). How can one approach
have a decisive advantage over the other if they are almost equivalent?
Obviously, both approaches discussed here are approaches which *work*.
However, we should also choose the approach which is correct and
corresponds to the entities in the situation we want to model.
I don't want to sound obstinate, really :-) It's just that I have yet
to see a solid argument which would prove my position wrong or
inefficient.
Sergiu
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