#9280: Implement an example of a graded algebra with basis, and improve the 
later
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   Reporter:  jhpalmieri                        |       Owner:  nthiery     
       Type:  enhancement                       |      Status:  needs_review
   Priority:  minor                             |   Milestone:              
  Component:  categories                        |    Keywords:              
     Author:  John Palmieri, Nicolas M. ThiƩry  |    Upstream:  N/A         
   Reviewer:                                    |      Merged:              
Work_issues:                                    |  
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Comment(by jhpalmieri):

 In the sage-combinat patch, there are a few typos and some other issues:

  - in sage/categories/graded_algebras_with_basis.py, the docstring for
 "degree" says "The degree of this element in the graded polynomial
 algebra."  Delete "polynomial".

  - in sage/categories/examples/graded_algebras_with_basis.py, the
 docstring for "one_basis" contains {{{'(0,...,0`)}}}, and I think this
 should be changed to {{{``(0,...,0)``}}}.

  - in sage/categories/examples/graded_algebras_with_basis.py, the
 docstring for the main class is now outdated: it still refers to
 "basis_function" and "_basis_fcn", which don't exist any more, and also to
 "homogeneous_component", which is now part of the default implementation,
 not something specific to this example.

 I'm attaching a referee patch which fixes these.

 There are also some doctests for "basis" in
 sage/categories/graded_algebras_with_basis.py which are marked as "todo:
 not implemented".  Do we need to wait for these to be fixed, or should we
 consider this ready for review?  It may not be ideal, but we could also
 change
 {{{
 sage: A.basis(6) # todo: not implemented (output)
 Family (y^{2}, x^{3}
 }}}
 to
 {{{
 sage: A.basis(6) # todo: not implemented (output)
 Family (y^{2}, x^{3}
 sage: list(A.basis(6))
 [y^{2}, x^{3}]
 }}}
 By the way, all tests pass with this patch and with the one from #10193.
 So perhaps we could also delete the commented-out part at the beginning of
 the example, where it says
 {{{
 # TODO: double check that we can now discard this function
 }}}

-- 
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/9280#comment:6>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica, 
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