This problem can be seen clearly if we use *Matrix_generic_dense* as
element class for matrix space.
On Tuesday, March 26, 2024 at 9:42:45 PM UTC+5:30 Animesh Shree wrote:
> I was going through this code and got error. But I could not understand
> why this is the case.
>
>
>
>
> sage: T = TropicalSemiring(QQ)
> sage: T(1)
> 1
> sage: T(2)
> 2
> sage: T(-2)
> -2
> sage: -T(2)
> ---------------------------------------------------------------------------
> ArithmeticError Traceback (most recent call last)
> Cell In[26], line 1
> ----> 1 -T(Integer(2))
>
> File ~/sage/src/sage/rings/semirings/tropical_semiring.pyx:278, in
> sage.rings.semirings.tropical_semiring.TropicalSemiringElement.__neg__()
> 276 if self._val is None:
> 277 return self
> --> 278 raise ArithmeticError("cannot negate any non-infinite element")
> 279
> 280 cpdef _mul_(left, right) noexcept:
>
> ArithmeticError: cannot negate any non-infinite element
> sage:
>
>
> It looks like starting with T(-2) and reaching to -2 from T(2) by
> comparing with zero(+Inf) are different things.
> T(-2) = -2
> T(2) -T(2) = T.zero(+inf) = T(2) + (-T(2))
>
> My doubt is : if we cannot negate the elements, then how can we compute
> the determinant of a Matrix over Tropical Semiring.
> For example in 2x2 matrix : [[a,b], [c,d]] the determinant should be ad -
> bc
> But can it be expressed as ad + (-bc) -->> min ( add(a,d), -1*add(b,c) )?
>
> In fact we connot even do matrix subtraction directly.
> What can be done in these cases??
> On Thursday, March 21, 2024 at 8:48:26 PM UTC+5:30 Animesh Shree wrote:
>
>> Hello Sir,
>>
>> I have submitted my proposal.
>> Please review it and let me know necessary updates and improvements.
>>
>> I want to verify soundness of my approach and extend the proposal for
>> Multivariate Polynomials.
>>
>> Thank You
>> Animesh Shree
>> On Tuesday, March 12, 2024 at 12:26:38 AM UTC+5:30 tcscrims wrote:
>>
>>> Mathematically speaking, you can always weaken axioms. However, there
>>> are some extra advantages that additive groups have that commutative
>>> semirings don't have (mainly 0, the additive identity).
>>>
>>> That being said, there isn't anything prevent you from constructing the
>>> appropriate categories. It would be good to have a more specific use-case
>>> in mind, but that isn't necessary. However, one should be careful with the
>>> name because it would conflict with what "most" people would call an
>>> algebra (which is why we have MagmaticAlgebras).
>>>
>>> Best,
>>> Travis
>>>
>>>
>>> On Tuesday, March 12, 2024 at 2:57:29 AM UTC+9 anime...@iiits.in wrote:
>>>
>>>> Hello Sir,
>>>>
>>>> I was going through "Algebras" and I had a doubt.
>>>> Does MagmaticAlgebra and AssociativeAlgebra have to be implemented over
>>>> Ring only?
>>>> I went through internet and google uses rings to define those algebras,
>>>> but the axioms that those algebra follow (Unital, Associative) are also
>>>> preserved by commutative semirings.
>>>> Would it be good to define MagmaticAlgebra and AssociativeAlgebra over
>>>> other objects that follow those axioms too or come-up with alternative
>>>> Algebra?
>>>>
>>>
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sage: T = TropicalSemiring(QQ)
sage: m = MatrixSpace(T, 2); m
Full MatrixSpace of 2 by 2 dense matrices over Tropical semiring over Rational
Field
sage: m.element_class
<class 'sage.matrix.matrix_generic_dense.Matrix_generic_dense'>
sage:
sage:
sage:
sage:
sage: ele = m.an_element()
sage: ele
[6/5 6/5]
[6/5 6/5]
sage: ele2 = m([[1,2],[3,4]])
sage: ele2
[1 2]
[3 4]
sage: ele + ele2
[ 1 6/5]
[6/5 6/5]
sage:
sage:
sage:
sage:
sage: ele - ele2 # =============================================== Error 1 of 2
===============================================
---------------------------------------------------------------------------
ArithmeticError Traceback (most recent call last)
Cell In[27], line 1
----> 1 ele - ele2
File ~/sage/src/sage/structure/element.pyx:1345, in
sage.structure.element.Element.__sub__()
1343 cdef int cl = classify_elements(left, right)
1344 if HAVE_SAME_PARENT(cl):
-> 1345 return (<Element>left)._sub_(right)
1346 if BOTH_ARE_ELEMENT(cl):
1347 return coercion_model.bin_op(left, right, sub)
File ~/sage/src/sage/matrix/matrix_generic_dense.pyx:255, in
sage.matrix.matrix_generic_dense.Matrix_generic_dense._sub_()
253 res._entries = [None]*(self._nrows*self._ncols)
254 for k in range(self._nrows*self._ncols):
--> 255 res._entries[k] = self._entries[k] - other._entries[k]
256 return res
257
File ~/sage/src/sage/structure/element.pyx:1345, in
sage.structure.element.Element.__sub__()
1343 cdef int cl = classify_elements(left, right)
1344 if HAVE_SAME_PARENT(cl):
-> 1345 return (<Element>left)._sub_(right)
1346 if BOTH_ARE_ELEMENT(cl):
1347 return coercion_model.bin_op(left, right, sub)
File ~/sage/src/sage/structure/element.pyx:1382, in
sage.structure.element.Element._sub_()
1380 raise bin_op_exception('-', self, other)
1381 else:
-> 1382 return python_op(other)
1383
1384 def __neg__(self):
File ~/sage/src/sage/categories/additive_magmas.py:826, in
AdditiveMagmas.AdditiveUnital.ElementMethods._sub_(left, right)
807 def _sub_(left, right):
808 r"""
809 Default implementation of difference.
810
(...)
824 (0, 0)
825 """
--> 826 return left + (-right)
File ~/sage/src/sage/rings/semirings/tropical_semiring.pyx:278, in
sage.rings.semirings.tropical_semiring.TropicalSemiringElement.__neg__()
276 if self._val is None:
277 return self
--> 278 raise ArithmeticError("cannot negate any non-infinite element")
279
280 cpdef _mul_(left, right) noexcept:
ArithmeticError: cannot negate any non-infinite element
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage: ele.determinant() # =============================================== Error
2 of 2 ===============================================
---------------------------------------------------------------------------
ArithmeticError Traceback (most recent call last)
Cell In[28], line 1
----> 1 ele.determinant()
File ~/sage/src/sage/matrix/matrix2.pyx:2130, in
sage.matrix.matrix2.Matrix.determinant()
2128 d = self.get_unsafe(0, 0)
2129 elif n == 2:
-> 2130 d = self.get_unsafe(0, 0)*self.get_unsafe(1, 1) -
self.get_unsafe(1, 0)*self.get_unsafe(0, 1)
2131 elif n == 3:
2132 d = self.get_unsafe(0, 0) * (self.get_unsafe(1,
1)*self.get_unsafe(2, 2) - self.get_unsafe(1, 2)*self.get_unsafe(2, 1)) \
File ~/sage/src/sage/structure/element.pyx:1345, in
sage.structure.element.Element.__sub__()
1343 cdef int cl = classify_elements(left, right)
1344 if HAVE_SAME_PARENT(cl):
-> 1345 return (<Element>left)._sub_(right)
1346 if BOTH_ARE_ELEMENT(cl):
1347 return coercion_model.bin_op(left, right, sub)
File ~/sage/src/sage/structure/element.pyx:1382, in
sage.structure.element.Element._sub_()
1380 raise bin_op_exception('-', self, other)
1381 else:
-> 1382 return python_op(other)
1383
1384 def __neg__(self):
File ~/sage/src/sage/categories/additive_magmas.py:826, in
AdditiveMagmas.AdditiveUnital.ElementMethods._sub_(left, right)
807 def _sub_(left, right):
808 r"""
809 Default implementation of difference.
810
(...)
824 (0, 0)
825 """
--> 826 return left + (-right)
File ~/sage/src/sage/rings/semirings/tropical_semiring.pyx:278, in
sage.rings.semirings.tropical_semiring.TropicalSemiringElement.__neg__()
276 if self._val is None:
277 return self
--> 278 raise ArithmeticError("cannot negate any non-infinite element")
279
280 cpdef _mul_(left, right) noexcept:
ArithmeticError: cannot negate any non-infinite element
