On Jul 18, rrbold wrote:
The problem of thinking that graded Abelian groups and graded modules are not abelian groups, and similarly graded Algebras are not Algebras, in the context of Sage, is that the whole category framework is in place, and your classes will not be able to use this. A parent defined as above cannot be a parent in the category of graded modules. This means that you have to double whatever code is already in place in the category framework and this should lead to more programming errors than the ones that may be solved by having a zero on each degree instead of checking for zero before using degree(). R.Hi Christian and John,Christian, your first sentence puts the finger on the correct spot: I take the position that a graded abelian group is not an abelian group. It is a sequence of abelian groups. For any category C, one can consider Gr(C), the category of graded objects in C, which has objects the functions from your grading monoid, frequently the natural numbers, to Obj(C), and morphisms the sequences of morphisms of C. There is no need for C to have a direct sum or categorical coproduct which will allow you to combine these into a single object in C, in order to consider such things. Mathematically, consider singular n-cochains on a space X with values in a module M. These are functions from the set Top(\Delta_n,X) of continuous maps \Delta_n --> X into the module M, i.e., elements of Set(Top(\Delta_n,X),M), given the natural module structure inherited from M. If n \neq k, then there is no sensible relation between the zero function Top(\Delta_n, X) --> M and the zero function Top(\Delta_k,X). Only inductive generalization or habit would suggest that sending all the elements of Top(\Delta_n,X) to zero \in M means you should also do this to all the elements in the entirely different set Top(Delta_k,X). The fact that this causes difficulties in the programming is a hint that we make an error in thinking of graded objects as their direct sum. I think it is better to take the mathematically sensible solution, and accept that there is a different 0 in each degree of a graded module. Best, Bob (rrb - old)
On Saturday, July 18, 2020 at 5:57:21 PM UTC-4, Christian Nassau wrote:
Hi Sverre,
I don't think it's a good idea to have different zeroes in an algebraic
structure that is also categorized as an abelian group, unless you take the
point that a "graded abelian group" should not be an "abelian group".
But let me also point out that something similar to what you want already
exists: you can take a homogeneous component of the Steenrod algebra and
look at its zero:
sage: A=SteenrodAlgebra(2)
sage: A[18]
Vector space spanned by (Sq(0,1,0,1), Sq(3,0,0,1), Sq(1,1,2), Sq
(4,0,2), Sq(2,3,1), Sq(5,2,1), Sq(8,1,1), Sq(11,0,1), Sq(0,6), Sq(3,5),
Sq(6,4), Sq(9,3), Sq(12,2), Sq(15,1), Sq(18)) over Finite Field of size
2
sage: A[18].zero() == A.zero()
True
sage: A[18].zero() == A[17].zero()
False
This suggests that "A[18].zero().degree()" could give 18, and the fact that
it currently gives a ValueError might be considered a bug.
Best,
Christian
On 18.07.20 23:35, Sverre Lunøe-Nielsen wrote:
Hi,
Thank you for your comments so far. I feel I need to expand some more
on the issue of zero elements which is the central thing for the
problem we are adressing.
It is mathematically equivalent to think of a graded k-algebra A as
either
1) a direct sum A = \bigosum_i A_i, together with a graded k-linear map
from
the graded tensor product A\tensor_k A --> A,
or
2) a sequence of k-vectorspaces {A_i}_i, together with a set of
structure maps
\{ A_i \tensor_R A_j --> A_{i+j} \}_{i,j}.
(In both cases the structure maps should satisfy usual algebraic
conditions.)
Similar for graded A-modules.
The implementation of the SteenrodAlgebra package takes the approach of
1), and never speaks about the zero element z_i \in A_i for any i.
Rather, they are all identified in A via the canonical injection A_i
--> A. It is tradition not to worry too much about this since you can
"figure it out" if you have to, and know how you ended up with a zero.
However, it is arguably better, specially when writing software, to
avoid this simplifaction since it leads to a corner case which has to
be dealt with over and over again. A great share of the bugs I have
corrected in the package I have been editing have been caused by the
wrongful assumption that all elements have an integer degree. Having
not to worry about this would make our code cleaner, and so will all
future code building on it.
I was being rather vague about making proposals for change in the
SteenrodAlgebra package in my last post, so to be clear let me propose
a specific change and invite anyone to share their opinion on it:
Change SteenrodAlgebra such that _all_ homogeneous elements have a well
defined degree. For the user, this means in particular that when
constructing the zero element, its degree must be given:
sage: A = SteenrodAlgebra(p=2)
sage: z = A.zero(degree=2)
sage: Sq(1)*Sq(1) == z
True
sage: Sq(2)*Sq(1)*Sq(1) == z
False
This involves adding the degree as internal data to zero elements, and
change the behaviour of degree() such that it raises an exception only
for inhomogeneous elements.
I hope I have clearified that I am not seeking a strange new definition
of graded module or algebra, and that I am merely wanting to discuss
the possibility of changing the implementation of SteenrodAlgebra.
E.g. are there perhaps unwanted software ramifications that our
proposal would bring about?
Regards,
Sverre
On Saturday, July 18, 2020 at 11:31:43 PM UTC+2, John H Palmieri wrote:
On Saturday, July 18, 2020 at 2:31:01 AM UTC-7, Sverre
Lunøe-Nielsen wrote:
Dear list,
I have been involved in preparing a package by M. Catanzaro and
R. Bruner lately, which implements finitely presented modules
over the mod `p` Steenrod algebra.
We have encountered a conflict regarding how to present graded
objects, and I am writing to the list to get other people's
opinion on how to proceed on this matter.
Briefly, the issue is that the Steenrod algebra allows
inhomogeneous elements and our graded modules do not. Thus,
the Steenrod algebra has a single zero element with no well
defined degree, while our modules could potentially have one
zero element for each degree.
My wish is to allow degreewise zero elements in our graded
modules, so that x.degree() would return an integer for every
element x. But because the unique zero in the Steenrod algebra
has no well defined degree, I am forced to let degree() treat
all zero elements in our modules the same way and return ``None
``.
A more precise description of the issue is found in the Sphinx
note below.
My questions to the list are: Has similar issues been discussed
and/or resolved before? And more specificly: What acceptable
changes could be made to the Steenrod algebra package to
achieve what I want?
Regards,
Sverre Lunøe-Nielsen
.. NOTE::
Our implementation treats a graded module as the disjoint
union, rather than a
direct sum, of vectorspaces of homogeneous elements. Elements
are therefore
always homogeneous, which also implies that sums between
elements of different
degrees are not allowed. This also means that acting by an
inhomogeneous
element of the Steenrod algebra makes no sense.
In this setting there is no single zero element, but rather a
zero for every
degree. It follows that, in theory, all elements, including
the zero elements,
have a well defined degree.
This way of representing a graded object differs from the way
the graded
Steenrod algebra is represented by :class:`sage.algebras.
steenrod` where inhomogeneous
elements are allowed and there is only a single zero element.
Consequently,
this zero element has no well defined degree.
Thus, because of the module action, we are forced to follow the
same convention
when it comes to the degree of zero elements in a module: The
method
:meth:`sage.modules.finitely_presented_over_the_steenrod_
algebra.module.fp_element.FP_Element.degree'
returns the value ``None`` for zero elements.
An example which highlights this problem is the following::
sage: F = FPA_Module([0], SteenrodAlgebra(p=2)) # The
free module on a single generator in degree 0.
sage: g = F.generator(0)
sage: x1 = Sq(1)*g
sage: x2 = Sq(1)*x1
Clearly, the code implementing the module action has all the
information it needs
to conclude that the element ``x2`` is the zero element in the
second degree.
However, because of the module action, we cannot distinguish it
from the element::
sage: x2_ = (Sq(1) * Sq(1))*g
The latter is equal to the action of the zero element of the
Steenrod
algebra on `g`, but since the zero element has no degree in the
Steenrod algebra,
the module class cannot deduce what degree the zero element
`x2_` should belong
to.
In my experience, algebraic topologists often think of graded
objects as disjoint unions, and you can often get away with this,
but really they're not — they're direct sums. I think you should
use Sage's categories framework, graded modules with basis or
whatever, to set these up. In any case where the degree matters,
you should first test whether an element is zero (in which case it
won't have a degree) and then perhaps whether it is homogeneous. If
not, you can raise an error (to keep someone from multiplying a
module element by Sq(1) + Sq(2), for example). If it is
homogeneous, you can proceed the way you want.
--
John
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