#11979: Divided power algebras
-------------------------------------+-------------------------------------
Reporter: bruce | Owner: jbandlow
Type: enhancement | Status: needs_review
Priority: minor | Milestone: sage-6.4
Component: combinatorics | Resolution:
Keywords: divided powers | Merged in:
Authors: Bruce Westbury | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
public/combinat/divided_power_alg | 103a503427c2ad28a10dad60cad3646f86e4f18b
Dependencies: | Stopgaps:
-------------------------------------+-------------------------------------
Comment (by darij):
Here is a minimal counterexample:
{{{
darij@travis-virtualbox:~/gitsage$ ./sage
┌────────────────────────────────────────────────────────────────────┐
│ Sage Version 6.4.rc0, Release Date: 2014-10-30 │
│ Type "notebook()" for the browser-based notebook interface. │
│ Type "help()" for help. │
└────────────────────────────────────────────────────────────────────┘
┏━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃ Warning: this is a prerelease version, and it may be unstable. ┃
┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛
sage: from sage.algebras.divided_power_algebra import
UnivariateDividedPowerAlgebra
sage: A = UnivariateDividedPowerAlgebra(Zmod(9)); A
The divided power algebra over Ring of integers modulo 9
sage: A = UnivariateDividedPowerAlgebra(Zmod(5)); A
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/category.py:867: UserWarning: Inconsistent
sorting results for all super categories of <class
'sage.categories.hopf_algebras.HopfAlgebras_with_category'>
self.__class__))
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/category.py:867: UserWarning: Inconsistent
sorting results for all super categories of <class
'sage.categories.hopf_algebras_with_basis.HopfAlgebrasWithBasis'>
self.__class__))
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/category.py:867: UserWarning: Inconsistent
sorting results for all super categories of <class
'sage.categories.category.JoinCategory'>
self.__class__))
Exception TypeError: 'Cannot create a consistent method resolution\norder
(MRO) for bases Bialgebras.parent_class, Algebras.parent_class,
CoalgebrasWithBasis.parent_class' in
'sage.categories.category_cy_helper.is_supercategory_of_done' ignored
---------------------------------------------------------------------------
TypeError Traceback (most recent call
last)
<ipython-input-3-008c24a8840f> in <module>()
----> 1 A = UnivariateDividedPowerAlgebra(Zmod(Integer(5))); A
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/misc/classcall_metaclass.so in
sage.misc.classcall_metaclass.ClasscallMetaclass.__call__
(build/cythonized/sage/misc/classcall_metaclass.c:1176)()
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/misc/cachefunc.so in
sage.misc.cachefunc.WeakCachedFunction.__call__
(build/cythonized/sage/misc/cachefunc.c:8031)()
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/structure/unique_representation.pyc in __classcall__(cls,
*args, **options)
1019 True
1020 """
-> 1021 instance = typecall(cls, *args, **options)
1022 assert isinstance( instance, cls )
1023 if instance.__class__.__reduce__ ==
CachedRepresentation.__reduce__:
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/misc/classcall_metaclass.so in
sage.misc.classcall_metaclass.typecall
(build/cythonized/sage/misc/classcall_metaclass.c:1603)()
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/algebras/divided_power_algebra.pyc in __init__(self, R)
123 if not R in Rings():
124 raise ValueError('R is not a ring')
--> 125 GHWBR = GradedHopfAlgebrasWithBasis(R)
126 CombinatorialFreeModule.__init__(self, R,
NonNegativeIntegers(),
127 category=GHWBR)
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/misc/classcall_metaclass.so in
sage.misc.classcall_metaclass.ClasscallMetaclass.__call__
(build/cythonized/sage/misc/classcall_metaclass.c:1176)()
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/graded_modules.pyc in __classcall__(cls,
category, *args)
110 return super(GradedModulesCategory,
cls).__classcall__(cls, category, *args)
111 else:
--> 112 return base_category_class(category, *args).Graded()
113
114 def _repr_object_names(self):
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/misc/cachefunc.so in
sage.misc.cachefunc.CachedMethodCaller.__call__
(build/cythonized/sage/misc/cachefunc.c:11291)()
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/misc/cachefunc.so in
sage.misc.cachefunc.CachedMethod._instance_call
(build/cythonized/sage/misc/cachefunc.c:14055)()
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/modules.pyc in Graded(self, base_ring)
377 assert base_ring is None or base_ring is
self.base_ring()
378 from sage.categories.graded_modules import
GradedModulesCategory
--> 379 return GradedModulesCategory.category_of(self)
380
381 @cached_method
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/misc/cachefunc.so in
sage.misc.cachefunc.CachedFunction.__call__
(build/cythonized/sage/misc/cachefunc.c:4969)()
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/covariant_functorial_construction.pyc in
category_of(cls, category, *args)
277 functor_category = getattr(category.__class__,
cls._functor_category)
278 if isinstance(functor_category, type) and
issubclass(functor_category, Category):
--> 279 return functor_category(category, *args)
280 else:
281 return cls.default_super_categories(category, *args)
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/misc/classcall_metaclass.so in
sage.misc.classcall_metaclass.ClasscallMetaclass.__call__
(build/cythonized/sage/misc/classcall_metaclass.c:1176)()
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/graded_modules.pyc in __classcall__(cls,
category, *args)
108 base_category_class = cls._base_category_class[0]
109 if isinstance(category, base_category_class):
--> 110 return super(GradedModulesCategory,
cls).__classcall__(cls, category, *args)
111 else:
112 return base_category_class(category, *args).Graded()
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/category.pyc in __classcall__(cls, *args,
**options)
440 if isinstance(cls, DynamicMetaclass):
441 cls = cls.__base__
--> 442 return super(Category, cls).__classcall__(cls, *args,
**options)
443
444 def __init__(self, s=None):
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/misc/cachefunc.so in
sage.misc.cachefunc.WeakCachedFunction.__call__
(build/cythonized/sage/misc/cachefunc.c:8031)()
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/structure/unique_representation.pyc in __classcall__(cls,
*args, **options)
1019 True
1020 """
-> 1021 instance = typecall(cls, *args, **options)
1022 assert isinstance( instance, cls )
1023 if instance.__class__.__reduce__ ==
CachedRepresentation.__reduce__:
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/misc/classcall_metaclass.so in
sage.misc.classcall_metaclass.typecall
(build/cythonized/sage/misc/classcall_metaclass.c:1603)()
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/graded_modules.pyc in __init__(self,
base_category)
35 Rational Field
36 """
---> 37 super(GradedModulesCategory, self).__init__(base_category,
base_category.base_ring())
38
39 _functor_category = "Graded"
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/covariant_functorial_construction.pyc in
__init__(self, category, *args)
302 self._base_category = category
303 self._args = args
--> 304 super(FunctorialConstructionCategory,
self).__init__(*args)
305
306 def base_category(self):
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/category_types.pyc in __init__(self, base, name)
483 isinstance(base, Category) and
base.is_subcategory(Rings())):
484 raise ValueError("base must be a ring or a subcategory
of Rings()")
--> 485 Category_over_base.__init__(self, base, name)
486
487 def base_ring(self):
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/category_types.pyc in __init__(self, base, name)
241 """
242 self.__base = base
--> 243 Category.__init__(self, name)
244
245 def _test_category_over_bases(self, **options):
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/category.pyc in __init__(self, s)
471 assert s is None
472 self.__class__ =
dynamic_class("{}_with_category".format(self.__class__.__name__),
--> 473 (self.__class__,
self.subcategory_class, ),
474 cache = False, reduction =
None,
475 doccls=self.__class__)
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/misc/lazy_attribute.so in
sage.misc.lazy_attribute._lazy_attribute.__get__
(build/cythonized/sage/misc/lazy_attribute.c:1276)()
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/category.pyc in subcategory_class(self)
1630 """
1631 return self._make_named_class('subcategory_class',
'SubcategoryMethods',
-> 1632 cache=False,
picklable=False)
1633
1634 @lazy_attribute
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/category.pyc in _make_named_class(self, name,
method_provider, cache, **options)
2791 pass
2792 result = Category._make_named_class(self, name,
method_provider,
-> 2793 cache=cache,
**options)
2794 self._make_named_class_cache[key] = result
2795 return result
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/category.pyc in _make_named_class(self, name,
method_provider, cache, picklable)
1586 reduction = None
1587 return dynamic_class(class_name,
-> 1588 tuple(getattr(cat,name) for cat in
self._super_categories_for_classes),
1589 method_provider_cls,
prepend_cls_bases = False, doccls = doccls,
1590 reduction = reduction, cache = cache)
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/misc/lazy_attribute.so in
sage.misc.lazy_attribute._lazy_attribute.__get__
(build/cythonized/sage/misc/lazy_attribute.c:1276)()
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/category.pyc in
_super_categories_for_classes(self)
1027 [Category of rngs, Category of semirings]
1028 """
-> 1029 self._all_super_categories
1030 return self._super_categories_for_classes
1031
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/misc/lazy_attribute.so in
sage.misc.lazy_attribute._lazy_attribute.__get__
(build/cythonized/sage/misc/lazy_attribute.c:1276)()
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/category.pyc in _all_super_categories(self)
860 """
861 (result, bases) =
C3_sorted_merge([cat._all_super_categories
--> 862 for cat in
self._super_categories] +
863
[self._super_categories],
864 category_sort_key)
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/misc/lazy_attribute.so in
sage.misc.lazy_attribute._lazy_attribute.__get__
(build/cythonized/sage/misc/lazy_attribute.c:1276)()
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/category.pyc in _super_categories(self)
1002 [Category of rngs, Category of semirings]
1003 """
-> 1004 return
sorted(_flatten_categories(self.super_categories(),JoinCategory), key =
category_sort_key, reverse=True)
1005
1006 @lazy_attribute
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/covariant_functorial_construction.pyc in
super_categories(self)
345 [Category of subquotients of semigroups, Category of
quotients of sets]
346 """
--> 347 return
Category.join([self.__class__.default_super_categories(self.base_category(),
*self._args)] +
348 self.extra_super_categories(),
349 as_list = True)
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/covariant_functorial_construction.pyc in
default_super_categories(cls, category, *args)
550 Category of unital subquotients of semigroups
551 """
--> 552 return Category.join([category,
super(RegressiveCovariantConstructionCategory,
cls).default_super_categories(category, *args)])
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/covariant_functorial_construction.pyc in
default_super_categories(cls, category, *args)
441 return Category.join([getattr(cat,
cls._functor_category)(*args)
442 for cat in
category._super_categories
--> 443 if hasattr(cat,
cls._functor_category)])
444
445 def is_construction_defined_by_base(self):
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/category.pyc in join(categories, as_list,
ignore_axioms, axioms)
2473
2474 # Handle axioms
-> 2475 result = join_as_tuple(cache_key, axioms, ignore_axioms)
2476 if as_list:
2477 return list(result)
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/category_cy_helper.so in
sage.categories.category_cy_helper.join_as_tuple
(build/cythonized/sage/categories/category_cy_helper.c:3145)()
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/category_cy_helper.so in
sage.categories.category_cy_helper.join_as_tuple
(build/cythonized/sage/categories/category_cy_helper.c:2425)()
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/category.pyc in is_subcategory(self, c)
1800 if c is self:
1801 return True
-> 1802 subcat_hook = c._subcategory_hook_(self)
1803 if subcat_hook is Unknown:
1804 return c in self._set_of_super_categories
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/category_types.pyc in _subcategory_hook_(self, C)
432
433 """
--> 434 if not issubclass(C.parent_class, self.parent_class):
435 return False
436 if not isinstance(C, Category_over_base):
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/misc/lazy_attribute.so in
sage.misc.lazy_attribute._lazy_attribute.__get__
(build/cythonized/sage/misc/lazy_attribute.c:1276)()
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/category.pyc in parent_class(self)
1675 :class:`sage.categories.category.JoinCategory`.
1676 """
-> 1677 return self._make_named_class('parent_class',
'ParentMethods')
1678
1679 @lazy_attribute
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/category.pyc in _make_named_class(self, name,
method_provider, cache, **options)
2791 pass
2792 result = Category._make_named_class(self, name,
method_provider,
-> 2793 cache=cache,
**options)
2794 self._make_named_class_cache[key] = result
2795 return result
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/categories/category.pyc in _make_named_class(self, name,
method_provider, cache, picklable)
1588 tuple(getattr(cat,name) for cat in
self._super_categories_for_classes),
1589 method_provider_cls,
prepend_cls_bases = False, doccls = doccls,
-> 1590 reduction = reduction, cache = cache)
1591
1592
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/structure/dynamic_class.pyc in dynamic_class(name, bases,
cls, reduction, doccls, prepend_cls_bases, cache)
314 elif cache is False:
315 # bypass the cached method
--> 316 return dynamic_class_internal.f(name, bases, cls,
reduction, doccls, prepend_cls_bases)
317 else: # cache = "ignore_reduction"
318 result = dynamic_class_internal(name, bases, cls, False,
doccls, prepend_cls_bases)
/home/darij/sage-6.3.beta6/local/lib/python2.7/site-
packages/sage/structure/dynamic_class.pyc in dynamic_class_internal(name,
bases, cls, reduction, doccls, prepend_cls_bases)
409 if isinstance(base, ClasscallMetaclass):
410 metaclass = DynamicClasscallMetaclass
--> 411 return metaclass(name, bases, methods)
412
413 class DynamicMetaclass(type):
TypeError: Cannot create a consistent method resolution
order (MRO) for bases Bialgebras.parent_class, Algebras.parent_class,
CoalgebrasWithBasis.parent_class
sage:
}}}
Compare with the following, which does not go wrong:
{{{
darij@travis-virtualbox:~/gitsage$ ./sage
┌────────────────────────────────────────────────────────────────────┐
│ Sage Version 6.4.rc0, Release Date: 2014-10-30 │
│ Type "notebook()" for the browser-based notebook interface. │
│ Type "help()" for help. │
└────────────────────────────────────────────────────────────────────┘
┏━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃ Warning: this is a prerelease version, and it may be unstable. ┃
┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛
sage: from sage.algebras.divided_power_algebra import
UnivariateDividedPowerAlgbra
sage: A = UnivariateDividedPowerAlgebra(Zmod(5)); A
The divided power algebra over Ring of integers modulo 5
sage: A = UnivariateDividedPowerAlgebra(Zmod(9)); A
The divided power algebra over Ring of integers modulo 9
sage:
}}}
(It says rc0, but it is rc1 which I have only compiled with ./sage -br
instead of make build to avoid building atlas again.)
It seems that the category refinement for the bialgebra category goes
wrong when first applied over a ring and then over a field, but goes fine
the other way roud.
--
Ticket URL: <http://trac.sagemath.org/ticket/11979#comment:22>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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