Hi all!
On 9 Feb., 15:57, luisfe <lftab...@yahoo.es> wrote:
> On Feb 9, 9:46 am, "D. S. McNeil" <dsm...@gmail.com> wrote:
>
> > >> (1) gcd is broken. http://trac.sagemath.org/sage_trac/ticket/10459
> > [..]
> > > I'm personally OK either way with this.
>
> > IMO a*b = gcd(a,b)*lcm(a,b) should be maintained wherever possible.
What does "Wherever possible" mean in an ordered ring, R? In
particular, is it possible for R=QQ? Or even for R=ZZ?
If I am not mistaken, lcm(x,y) could be defined as
inf {x*m | m in R, x*m>0, there is n in R with x*m==y*n} (1)
At least, that definition matches the situation in ZZ (note that
lcm(-2,-2) = 2).
Or is there any reason to have a minimum rather than an infimum?
On the other hand, what is the notion of a "multiple" of x? Is it
simply x*m for any m in R? Or is it rather x*m for any abs(m)>=1 in R?
In that case, definition (1) above became
inf {x*m | m in R, abs(m)>=1, x*m>0, there is n in R with abs(n)>=1
and x*m==y*n} (2)
Again, it's one definition that matches the situation in ZZ.
Now, consider R=QQ.
According to definition (1), the least common multiple of 1/1 and 1/1
is: Zero! 'Cause any positive rational is a multiple of 1/1, and the
infimum is zero.
But according to definition (2), the least common multiple of 1/1 and
1/1 is: 1/1.
Perhaps there are other possible ways to extend the notion of lcm from
ZZ to any ordered ring. I could imagine that different textbooks use
slightly different ways of defining lcm and gcd in ZZ, extending to QQ
differently.
So, is QQ reasonably covered by "Wherever possible"?? I doubt.
Note that currently we have
sage: gcd(-2,1)
1
sage: lcm(-2,1)
2
So, gcd(x,y)*lcm(x,y) == x*y doesn't even hold in ZZ. Why should it
hold in QQ?
Cheers,
Simon
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