Hi Simon,
Thank you! I'll have a look.
Nicolas, I wanted to add the patch trac7797-full_letterplace_wrapper.patch
to the sage-combinat queue to test it, but it does not commute with your patch
trac_10961-lie_bracket_in_rings-nt.patch. Would it be possible to rebase it?
Also, would it be possible to get the code
sage.misc.misc.fibers into sage? I need it quite often, but it comes late in the
sage-combinat queue.
Best,
Anne
On 3/24/11 9:36 AM, Simon King wrote:
Hi Anne,
On 24 Mrz., 10:53, Anne Schilling<a...@math.ucdavis.edu> wrote:
...
Ok, this should not be a problem since in the application I have in mind
the relations are homogeneous and hence only terms in homogeneous terms
will cancel.
Therefore, in my to-be-submitted-as-soon-as-the-damned-documentation-
correctly-builds patch, it is excluded to even *create* an
inhomogeneous element. Probably that will be allowed as soon as the
next Singular version is in Sage.
Ok, let me know when your patch is available! I would be super happy to try it
out.
The patch is submitted -- see #7797.
Here are some examples, also addressed to people who might be
interested in reviewing.
1. The arithmetic is fairly quick (but mind that it is restricted to
homogeneous elements, and so it can currently not be the default
implementation):
sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: F
Free Associative Unital Algebra on 3 generators ('x', 'y', 'z') over
Rational Field
sage: F_old.<a,b,c> = FreeAlgebra(QQ)
sage: timeit('t=(x+y)^15')
5 loops, best of 3: 27.7 ms per loop
sage: %time t=(a+b)^15
CPU times: user 4.51 s, sys: 0.09 s, total: 4.60 s
Wall time: 6.46 s
sage: 4510/27.7
162.815884476534
sage: timeit('t=(x+y)^15')
25 loops, best of 3: 19.7 ms per loop
sage: %time t=(a+b)^15
CPU times: user 2.70 s, sys: 0.02 s, total: 2.72 s
Wall time: 2.73 s
sage: 2700/19.7
137.055837563452
3. We have Groebner bases and normal forms wrt. twosided homogeneous
ideals, and we can even construct the quotient ring:
sage: I = F*[x*y+y*z,x^2+x*y-y*x-y^2]*F
sage: I.groebner_basis(degbound=3)
Twosided Ideal (y*y*y - y*y*z + y*z*y - y*z*z, y*y*x + y*y*z + y*z*x +
y*z*z, x*y + y*z, x*x - y*x - y*y - y*z) of Free Associative Unital
Algebra on 3 generators ('x', 'y', 'z') over Rational Field
sage: (x*y*z*y*x).normal_form(I)
y*z*z*y*z + y*z*z*z*x + y*z*z*z*z
sage: x*y*z*y*x - (x*y*z*y*x).normal_form(I) in I
True
sage: x*I.0-I.1*y+I.0*y in I
True
sage: 1 in I
False
sage: Q.<a,b,c> = F.quo(I); Q
Quotient of Free Associative Unital Algebra on 3 generators ('x', 'y',
'z') over Rational Field by the ideal (x*y + y*z, x*x + x*y - y*x -
y*y)
sage: a*b
-b*c
sage: a^3
-b*c*a - b*c*b - b*c*c
A quotient ring of a quotient ring works as well.
sage: J = Q*[a^3-b^3]*Q
sage: R.<i,j,k> = Q.quo(J); R
Quotient of Free Associative Unital Algebra on 3 generators ('x', 'y',
'z') over Rational Field by the ideal (-y*y*z - y*z*x - 2*y*z*z, x*y +
y*z, x*x + x*y - y*x - y*y)
sage: i^3
-j*k*i - j*k*j - j*k*k
sage: j^3
-j*k*i - j*k*j - j*k*k
Best regards,
Simon
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